Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and...

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Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice Ying Zhuang, 1 Wenqi Liu, 1 Chin-Chia Wu, 2 and Jinhai Li 1 1 Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China 2 Department of Statistics, Feng Chia University, Taichung 40724, Taiwan Correspondence should be addressed to Ying Zhuang; [email protected] Received 26 June 2014; Revised 13 July 2014; Accepted 13 July 2014; Published 23 July 2014 Academic Editor: Yunqiang Yin Copyright © 2014 Ying Zhuang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. en we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. 1. Introduction e theory of rough sets was originally proposed by Pawlak [1] in 1982 as a mathematical approach to handle imprecision, vagueness, and uncertainty in data analysis, which has been applied successfully to many areas such as knowledge rep- resentation, data mining, pattern recognition, and decision making (Sun et al. [2], Wang et al. [3]). is theory takes into consideration the indiscernibility between objects. e indiscernibility is typically characterized by an equivalence relation. Rough sets are the results of approximating crisp sets using equivalence classes. However, the requirement of an equivalence relation seems to be a very restrictive condition which may limit the applications of rough set theory since this requirement can deal only with complete information systems. erefore, some interesting and meaningful exten- sions of Pawlak’s rough set models have been proposed in the literature. For example, some interesting extensions to equivalence relations have been proposed, such as tolerance relations or similarity relations, general binary relations on the discourse, partitions and general binary relations on the neighborhood system from topological space, and general approximation spaces (examples of this approach can be found in Chen et al. [4], Wang and Hu [5], and Yin et al. [6– 8]); some general notions of rough sets such as rough fuzzy sets, fuzzy rough sets, and soſt rough sets have been proposed and discussed (examples of this approach can be found in Ali et al. [9], Feng et al. [10], Li and Yin [11], Yao et al. [12], and Zhang et al. [13]); and rough set models on two universes of discourse which can be interpreted by the notions of interval structure and generalized approximation space have been extensively studied by Li and Zhang [14], Ma and Sun [15], and Yao and Lin [16]. e relationships between lattice theory and rough sets are another topic receiving much attention in recent years. Cattaneo and Ciucci [17] focus on the study on lattices with interior and closure operators and abstract approxima- tion spaces, in which the nonequational notion of abstract approximation space for roughness theory is introduced, and its relationship with the equational definition of lattice with Tarski interior and closure operations is studied. eir categorical isomorphism is also proved, and the role of the Tarski interior and closure with an algebraic semantic of a S4-like model of modal logic is widely investigated. J¨ arvinen [18] investigates lattice-theoretical foundations of rough set theory, in which closure operators in a more general setting of ordered sets, fixpoints of Galois connections, rough set approximations and definable sets, and the lattice structures of the ordered set of all rough sets determined by different kinds of indiscernibility relations are studied in detail. e purpose of this paper is to investigate the general approxi- mation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. e relationships between the Pawlak approximation structures and these mathematic structures are established. Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 697107, 9 pages http://dx.doi.org/10.1155/2014/697107

Transcript of Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and...

Page 1: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

Research ArticlePawlak Algebra and Approximate Structure on Fuzzy Lattice

Ying Zhuang1 Wenqi Liu1 Chin-Chia Wu2 and Jinhai Li1

1 Faculty of Science Kunming University of Science and Technology Kunming 650500 China2Department of Statistics Feng Chia University Taichung 40724 Taiwan

Correspondence should be addressed to Ying Zhuang shadowzhuang163com

Received 26 June 2014 Revised 13 July 2014 Accepted 13 July 2014 Published 23 July 2014

Academic Editor Yunqiang Yin

Copyright copy 2014 Ying Zhuang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of this paper is to investigate the general approximation structure weak approximation operators and Pawlak algebra inthe framework of fuzzy lattice lattice topology and auxiliary ordering First we prove that the weak approximation operator spaceforms a complete distributive lattice Then we study the properties of transitive closure of approximation operators and apply themto rough set theory We also investigate molecule Pawlak algebra and obtain some related properties

1 Introduction

The theory of rough sets was originally proposed by Pawlak[1] in 1982 as amathematical approach to handle imprecisionvagueness and uncertainty in data analysis which has beenapplied successfully to many areas such as knowledge rep-resentation data mining pattern recognition and decisionmaking (Sun et al [2] Wang et al [3]) This theory takesinto consideration the indiscernibility between objects Theindiscernibility is typically characterized by an equivalencerelation Rough sets are the results of approximating crisp setsusing equivalence classes However the requirement of anequivalence relation seems to be a very restrictive conditionwhich may limit the applications of rough set theory sincethis requirement can deal only with complete informationsystems Therefore some interesting and meaningful exten-sions of Pawlakrsquos rough set models have been proposed inthe literature For example some interesting extensions toequivalence relations have been proposed such as tolerancerelations or similarity relations general binary relations onthe discourse partitions and general binary relations on theneighborhood system from topological space and generalapproximation spaces (examples of this approach can befound in Chen et al [4] Wang and Hu [5] and Yin et al [6ndash8]) some general notions of rough sets such as rough fuzzysets fuzzy rough sets and soft rough sets have been proposedand discussed (examples of this approach can be found in Aliet al [9] Feng et al [10] Li and Yin [11] Yao et al [12] and

Zhang et al [13]) and rough set models on two universes ofdiscourse which can be interpreted by the notions of intervalstructure and generalized approximation space have beenextensively studied by Li and Zhang [14] Ma and Sun [15]and Yao and Lin [16]

The relationships between lattice theory and rough setsare another topic receiving much attention in recent yearsCattaneo and Ciucci [17] focus on the study on latticeswith interior and closure operators and abstract approxima-tion spaces in which the nonequational notion of abstractapproximation space for roughness theory is introducedand its relationship with the equational definition of latticewith Tarski interior and closure operations is studied Theircategorical isomorphism is also proved and the role of theTarski interior and closure with an algebraic semantic of aS4-like model of modal logic is widely investigated Jarvinen[18] investigates lattice-theoretical foundations of rough settheory in which closure operators in a more general settingof ordered sets fixpoints of Galois connections rough setapproximations and definable sets and the lattice structuresof the ordered set of all rough sets determined by differentkinds of indiscernibility relations are studied in detail Thepurpose of this paper is to investigate the general approxi-mation structure weak approximation operators and Pawlakalgebra in the framework of fuzzy lattice lattice topologyand auxiliary orderingThe relationships between the Pawlakapproximation structures and these mathematic structuresare established

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 697107 9 pageshttpdxdoiorg1011552014697107

2 The Scientific World Journal

The remaining part of the paper is organized as followsSection 2 introduces the relevant definitions which will beused throughout the paper Section 3 investigates the Pawlakalgebra and weak Pawlak algebra on fuzzy lattice Section 4discusses the relationships between Pawlak algebra and aux-iliary ordering Section 5 focuses on the study of molecu-lar Pawlak algebra Section 6 investigates the properties ofPawlak algebra based on binary relation Finally Section 7concludes the paper and suggests some future research topics

2 Preliminaries

In this section we introduce some basic definitions (see [319 20]) which will be used throughout the paper

Definition 1 Apartially ordered set (119871 le) is said to be a latticeif inf119909 119910 and sup119909 119910 denoted by and and or respectivelyexist for all 119909 119910 isin 119871 A lattice 119871 is said to be complete if forevery 119860 sube 119871⋀

120572isin119860120572 and⋁

120572isin119860120572 exist

Definition 2 Let (119871 le) be a complete lattice with the max-imum element 1 and minimum element 0 and ldquo≺rdquo a binaryrelation on 119871 If the following conditions hold for all 120572 120573 120583120578 120574 isin 119871

(i) 120572 ≺ 120573 rArr 120572 le 120573(ii) 120583 le 120572 ≺ 120573 le 120578 rArr 120583 ≺ 120578(iii) 120572 ≺ 120574 120573 ≺ 120574 rArr 120572 or 120573 ≺ 120574(iv) forall120572 isin 119871 0 ≺ 120572 ≺ 1

then the relation ldquo≺rdquo is called an auxiliary ordering on 119871 If therelation ldquo≺rdquo satisfies conditions (i) (ii) and (iv) it is called aweak auxiliary ordering on119871Theweak auxiliary ordering ldquo≺rdquois called completely approximate if

forall120572 isin 119871 120572 = or 120574 | 120574 ≺ 120572 120574 isin 119871 (1)

Furthermore if the relation ldquo≺rdquo satisfies conditions (i) (ii)(iv) and

(iii)1015840 120572119905≺ 120573(119905 isin 119879) rArr ⋁

119905isin119879120572119905≺ 120573

then ldquo≺rdquo is called a strong auxiliary ordering on 119871

Definition 3 Let (119871 or and 0 1) be a completely distributivelattice If the mapping 119888 119871 rarr 119871 satisfies the followingconditions

(i) reverse law forall120572 120573 isin 119871 if 120572 le 120573 then 120573119888 le 120572119888(ii) recovery law forall120572 isin 119871 (120572119888)119888 = 120572

then (119871 or and119888 0 1) is called a fuzzy lattice

Remark 4 The notion of fuzzy lattice just given is first intro-duced in Wang [20] and is also called de Morgan completelydistributive lattice in the literature To keep consistency weadopt the term ldquofuzzy latticerdquo in this paper

Definition 5 Let (119871 or and119888 0 1) be a fuzzy lattice and 120587 sube 119871If the subset 120587 satisfies the following conditions

(i) 0 notin 120587(ii) 120572 120573 isin 120587 120572 and 120573 = 0 rArr 120572 le 120573 or 120573 le 120572(iii) 120572 120573 120574 isin 120587 120572 le 120574 120573 le 120574 rArr 120572 and 120573 = 0(iv) 120578 = or120572 | 120572 isin 120587 120572 le 120578 for all 120578 isin 119871(v) if 120587

0sube 120587 and 120587

0is linear order subset then or120587

0isin 120587

then 120587 is called amolecular set and the element of 120587 is calleda molecule The fuzzy lattice with molecule (119871(120587) or and119888 0 1)

is called amolecular lattice

Definition 6 Let (119871 or and119888 0 1) be a fuzzy lattice and 120575 sube 119871If the subset 120575 satisfies the following conditions

(1) 0 1 isin 120575(2) 120572

119905isin 120575(119905 isin 119879) rArr ⋁

119905isin119879120572119905isin 120575

(3) 120572 120573 isin 120575 rArr 120572 and 120573 isin 120575

then (119871 120575) is called a lattice topology space

3 The Pawlak Algebra and Weak PawlakAlgebra on Fuzzy Lattices

In this section we investigate the structural properties ofPawlak rough approximations on fuzzy lattices Let us beginwith introducing the following concepts

Definition 7 (see [20])) Let (119871 or and119888 0 1) be a fuzzy latticeIf the dual mappings 119886119901119903 119871 rarr 119871 and 119886119901119903 119871 rarr 119871 satisfythe following conditions

(P1) 119886119901119903(120572) = (119886119901119903(120572119888

))119888

(forall120572 isin 119871)

(P2) 119886119901119903(1) = 1

(P3) 119886119901119903(120572 and 120573) = (119886119901119903120572) and (119886119901119903120573) (forall120572 120573 isin 119871)

(P4) 120572 le 119886119901119903(120572) forall120572 isin 119871

then (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a Pawlak algebra 119886119901119903(resp 119886119901119903 pair (119886119901119903 119886119901119903) ) is called upper approximationoperator (resp lower approximation operator dual approx-imation operator) on 119871 If 119886119901119903(120572) = 119886119901119903(120572) then 120572 is calleda definable element If 119886119901119903120572 = 119886119901119903120572 then 120572 is called a roughelement

If (P3) is replaced by the following condition (P3)lowast

(P3)lowast 120572 le 120573 rArr 119886119901119903(120572) le 119886119901119903(120573)

then 119886119901119903 (resp 119886119901119903 pair (119886119901119903 119886119901119903)) is called a weak upperapproximation operator (resp weak lower approximationoperator weak dual approximation operator) on 119871 andthe system (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a weak Pawlakalgebra

Let (119871 or and119888 0 1) be a fuzzy lattice Denote by 119860119875119877(119871)

(resp 119860119875119877119882(119871)) the set of all dual approximation operators

(resp dual weak approximation operators) on 119871 respectivelyand by 120590

119886119901119903

0(119871) the set of all definable elements in the Pawlak

algebra (119871 or and119888 0 1 119886119901119903 119886119901119903)

The Scientific World Journal 3

Definition 8 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903

1

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877

119908(119871) Then the dual weak

approximation operator (1198861199011199032

1198861199011199032) is called rougher than

(1198861199011199031

1198861199011199031) denoted by (119886119901119903

1

1198861199011199031) ≺ (119886119901119903

2

1198861199011199032) if the

following inequalities hold

forall120572 isin 119871 1198861199011199032

(120572) le 1198861199011199031

(120572) le 1198861199011199031(120572) le 119886119901119903

2(120572) (2)

Proposition 9 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then 119860119875119877(119871) sube 119860119875119877

119908(119871)

Proof Let (119886119901119903 119886119901119903) isin 119860119875119877(119871) For any 120572 120573 isin 119871 with 120572 le 120573we have 120572 = 120572 and 120573 and so

119886119901119903 (120572) = 119886119901119903 (120572 and 120573) = 119886119901119903 (120572) and 119886119901119903 (120573) (3)

Therefore 119886119901119903(120572) le 119886119901119903(120573) implying that condition (P3)lowast

holds Hence (119886119901119903 119886119901119903) isin 119860119875119877119882(119871) as required

Define two dual approximation operators 119868 = (119868 119868) and119874 = (119874119874) as follows

forall120572 isin 119871 119868 (120572) = 0 120572 = 1

1 120572 = 1 119868 (120572) =

0 120572 = 0

1 120572 = 0

forall120572 isin 119871 119874 (120572) = 119874 (120572) = 120572

(4)

Then it is easy to see that 119868119874 isin 119860119875119877(119871) and119874 ≺ (119886119901119903 119886119901119903) ≺

119868 for any (119886119901119903 119886119901119903) isin 119860119875119877119908(119871)

It is possible to characterize 119860119875119877119882(119871) according to the

following result

Theorem 10 Let (119871 or and119888 0 1) be a fuzzy lattice Then(119860119875119877

119882(119871) ≺) is a complete distributive lattice with maximum

element 119868 and minimum element 119874

Proof Suppose that (119886119901119903119905

119886119901119903119905)119905isin119879

sube 119860119875119877119908(119871) where 119879 is

an index set Define⋁119905isin119879

119886119901119903119905

by

(⋁119905isin119879

119886119901119903119905

) (120572) = ⋁119905isin119879

119886119901119903119905

(120572) forall120572 isin 119871 (5)

And ⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905 and ⋀

119905isin119879119886119901119903

119905can be similarly

defined Then the following assertions hold

(1) Consider (⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905)

(⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) isin 119860119875119877

119908(119871) In fact for

any 120572 120573 isin 119871 by Definitions 6 and 7 we have

(P1) (⋁119905isin119879

119886119901119903119905

)(120572) = ⋁119905isin119879

119886119901119903119905

(120572) =⋁119905isin119879

(119886119901119903119905(120572119888))

119888 = (⋀119905isin119879

119886119901119903119905(120572119888))

119888 =((⋀

119905isin119879119886119901119903

119905)(120572119888))

119888(P2) (⋁

119905isin119879119886119901119903

119905

)(1) = ⋁119905isin119879

119886119901119903119905

(1) = 1(P3) if 120572 le 120573 then 119886119901119903

119905

(120572) le 119886119901119903119905

(120573) forall119905 isin 119879implying that⋁

119905isin119879119886119901119903

119905

(120572) le ⋁119905isin119879

119886119901119903119905(120573)

(P4) 120572 le ⋀119905isin119879

119886119901119903119905(120572) = (⋀

119905isin119879119886119901119903

119905)(120572) since

120572 le 119886119901119903119905(120572) for all 119905 isin 119879

(2) Consider (⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) isin 119860119875119877

119908(119871) The

proof is analogous to that of (1)(3) Consider inf(119886119901119903

119905

119886119901119903119905) | 119905 isin 119879 =

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) and sup(119886119901119903

119905

119886119901119903119905) | 119905 isin

119879 = (⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) Clearly (119886119901119903

119905

119886119901119903119905) ≺

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) is true for all 119905 isin 119879 In fact

let (119886119901119903 119886119901119903) isin 119860119875119877119908(119871) be such that (119886119901119903 119886119901119903) ≺

(119886119901119903119905

119886119901119903119905) for all 119905 isin 119879 Then 119886119901119903

119905

(120572) le 119886119901119903(120572) le

119886119901119903(120572) le 119886119901119903119905(120572) for all 119905 isin 119879 and 120572 isin 119871 by definition

It follows that

⋁119905isin119879

119886119901119903119905

(120572) le 119886119901119903 (120572) le 119886119901119903120572 le ⋀119905isin119879

119886119901119903119905(120572) forall120572 isin 119871 (6)

On the other hand it is obvious that (119886119901119903119905

119886119901119903119905) ≺

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) for all 119905 isin 119879 Hence inf(119886119901119903

119905

119886119901119903119905) |

119905 isin 119879 = (⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) In a similar way by duality

we have sup(119886119901119903119905

119886119901119903119905) | 119905 isin 119879 = (⋀

119905isin119879119886119901119903

119905

⋁119905isin119879

119886119901119903119905)

Summing up the above analysis (119860119875119877119882(119871) ≺) is a com-

plete lattice It is obvious that 119868 is the maximum elementand 119874 is the minimum element and that (APR

119882(119871) ≺) is

distributive This completes the proof

Proposition 11 Let (119871 or and119888 0 1) be a fuzzy lattice and (119871 120575)

a lattice topology space Define the operators 119886119901119903120575

119871 rarr 119871

and 119886119901119903120575 119871 rarr 119871 by

119886119901119903120575

(120572) = or 120574 | 120574 le 120572 120574 isin 120575

119886119901119903120575(120572) = and 120574

119888

| 120572 le 120574119888

120574 isin 120575

(7)

respectively for all 120572 isin 119871 Then the system(119871 or and119888 0 1 119886119901119903

120575

119886119901119903120575) is a Pawlak algebra

Proof It is straightforward and omitted

Proposition 12 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then (119871 120590

119886119901119903

0(119871)) is a zero-dimensional lattice topology

space

Proof It is evident that 1205901198861199011199030

(119871) = 120572 isin 119871 | 120572 = 119886119901119903(120572) cap

120572 isin 119871 | 120572 = 119886119901119903(120572) And we conclude that the followingassertions hold

(a) 120572 isin 119871 | 120572 = 119886119901119903(120572) is union-closed Suppose that120572

119905| 119905 isin 119879 sube 120572 isin 119871 | 120572 = 119886119901119903(120572) where 119879 is an

index set Now for any 119904 isin 119879 we have 120572119904le ⋁

119905isin119879120572119905

and so 120572119904= 119886119901119903(120572

119904) le 119886119901119903(⋁

119905isin119879120572119905) implying that

⋁119905isin119879

120572119905le 119886119901119903(⋁

119905isin119879120572119905) Since ⋁

119905isin119879120572119905ge 119886119901119903(⋁

119905isin119879120572119905)

we have⋁119905isin119879

120572119905= 119886119901119903(⋁

119905isin119879120572119905) by conditions (P1) and

(P4) as required(b) 120572 isin 119871 | 120572 = 119886119901119903(120572) is intersection-closed Suppose

that 120572119905| 119905 isin 119879 sube 120572 isin 119871 | 120572 = 119886119901119903(120572) where 119879

4 The Scientific World Journal

is an index set For any 119904 isin 119879 we have 120572119904ge ⋀

119905isin119879120572119905

and so 120572119904= 119886119901119903(120572

119904) ge 119886119901119903(⋀

119905isin119879120572119905) implying that

⋀119905isin119879

120572119905ge 119886119901119903(⋀

119905isin119879120572119905) By condition (P4) we have

⋀119905isin119879

120572119905= 119886119901119903(⋀

119905isin119879120572119905) as required

(c) 120572 isin 119871 | 120572 = 119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572)

is complement-closed For any 120572 isin 120572 isin 119871 | 120572 =

119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572) by condition (P1)we have 119886119901119903(120572119888) = ((119886119901119903(120572119888))

119888

)119888 = (119886119901119903(120572))

119888

= 120572implying that 120572119888 isin 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 =

119886119901119903(120572) Hence 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 = 119886119901119903(120572) iscomplement-closed

Summing up the above analysis (119871 1205901198861199011199030

(119871)) is a zero-dimensional lattice topology space

Now let us turn our attention to the study of the transitiveclosure of approximation operators

Let (119871 or and119888 0 1) be a fuzzy lattice Define the operatorldquo∘rdquo on 119860119875119877(119871) as follows

(1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) = (119886119901119903

1

∘ 1198861199011199032

1198861199011199031∘ 119886119901119903

2)

(8)

That is

forall120572 isin 119871 (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) (120572)

= (1198861199011199031

(1198861199011199032

(120572)) 1198861199011199031(119886119901119903

2(120572)))

(9)

where (1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871) For any positive

integer 119896 define (119886119901119903 119886119901119903)(119896+1)

= (119886119901119903 119886119901119903)(119896)

∘ (119886119901119903 119886119901119903)Denote cl(119886119901119903 119886119901119903) = ⋁

infin

119896=1(119886119901119903 119886119901119903)

(119896) which is called thetransitive closure of (119886119901119903 119886119901119903) Then we obtain the followingresult

Lemma 13 Let (119871 or and119888 0 1) be a fuzzy lattice Then

(1) (1198861199011199031

1198861199011199031) ≺ (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) for all

(1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(2) (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(3) cl(119886119901119903 119886119901119903) is a dual approximation operator

Proof (1) It is straightforward and omitted(2) It suffices to prove that (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032)

satisfies conditions (P1)ndash(P4) in Definition 7 In fact for any120572 120573 isin 119871 we have

(P1) 1198861199011199031

(1198861199011199032

(120572)) = 1198861199011199031((119886119901119903

2

(120572))119888

)119888

=

(1198861199011199031((119886119901119903

2(120572119888))

119888

)119888

)119888

= (1198861199011199031(119886119901119903

2(120572119888)))

119888(P2) (119886119901119903

1

∘ 1198861199011199032

)(1) = 1198861199011199031

(1198861199011199032

(1)) = 1198861199011199031

(1) = 1

(P3) (1198861199011199031

∘ 1198861199011199032

)(120572 and 120573) = 1198861199011199031

(1198861199011199032

(120572 and 120573))

= 1198861199011199031

(1198861199011199032

(120572) and 1198861199011199032

(120573)) = (1198861199011199031

(1198861199011199032

(120572))) and

(1198861199011199031

(1198861199011199032

(120573)))= (1198861199011199031

∘1198861199011199032

)(120572))and(1198861199011199031

∘1198861199011199032

)(120573))

(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903

1(120572) le

1198861199011199031(119886119901119903

2(120572)) = (119886119901119903

1∘ 119886119901119903

2)(120572)

(3) From the proof of Theorem 10 we have

cl (119886119901119903 119886119901119903) = (

infin

⋀119896=1

119886119901119903(119896)

infin

⋁119896=1

119886119901119903(119896)

) (10)

And it is easy to prove by mathematical induction that

119886119901119903(119896)

(120572 and 120573)

= 119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573) for all positive integer 119896

(11)

Therefore

(

infin

⋀119896=1

119886119901119903(119896)

)(120572 and 120573) =

infin

⋀119896=1

119886119901119903(119896)

(120572 and 120573)

=

infin

⋀119896=1

(119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573))

= (

infin

⋀119896=1

119886119901119903(119896)

(120572)) and (

infin

⋀119896=1

119886119901119903(119896)

(120573))

(12)

It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator

Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871

Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has

(

infin

⋀119896=1

119886119901119903(119896)

) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)

(

infin

⋁119896=1

119886119901119903(119896)

) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)

Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879

119886119901119903 for any 120572 isin 119871 we have

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 120572

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903 (120572)

The Scientific World Journal 5

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(2)

(120572)

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(119896)

(120572)

(15)

implying that or120574 | 120574 le 120572 119886119901119903(120574) = 120574 le (⋀infin

119896=1119886119901119903(119896))(120572)

Conversely for any 120572 isin 119871

119886119901119903((

infin

⋀119896=1

119886119901119903(119896)

) (120572)) = (

infin

⋀119896=1

119886119901119903(119896)

) (120572) le 120572 (16)

It follows that (⋀infin

119896=1119886119901119903(119896))(120572) isin 119879

119886119901119903 and so (⋀

infin

119896=1119886119901119903(119896))(120572)

le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds

As a consequence ofTheorem 15 we obtain the followingresult

Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903

lowast= ⋁

infin

119896=1119886119901119903

(119896) is a closure operator on 119871

4 The Relationships between Pawlak Algebraand Auxiliary Ordering

Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately

The following results present the relationships betweenPawlak algebra and strong auxiliary ordering

Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903

≺

119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows

forall120572 isin 119871 119886119901119903≺

(120572) = or 120574 | 120574 ≺ 120572 120574 isin 119871

119886119901119903≺(120572) = (119886119901119903

≺

(120572119888

))119888

(17)

then (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

Proof It is obvious that condition (P1) holds In what followswe prove that conditions (P2)ndash(P4) are satisfied

(1) The strong auxiliary ordering ≺ implies that 1 ≺ 1 istrue and so we have

119886119901119903≺

(1) = or 120574 | 120574 ≺ 1 120574 isin 119871 = 1 (18)

implying that condition (P2) holds

(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺

120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120572 120574 isin 119871

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120573 120574 isin 119871 (19)

Therefore

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120572 120574 isin 119871

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120573 120574 isin 119871 (20)

It follows that 119886119901119903≺

(120572 and 120573) le 119886119901119903≺

(120572) and 119886119901119903≺

(120572 and 120573) le

119886119901119903≺

(120573) and hence

119886119901119903≺

(120572 and 120573) le (119886119901119903≺

(120572)) and (119886119901119903≺

(120573)) (21)

On the other hand by condition (iii) in Definition 3 wehave

or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572

or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573(22)

It follows from

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572 le 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573 le 120573

(23)

that

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120573(24)

Hence

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572 and 120573

(25)

implying that

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) isin 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 (26)

Thus 119886119901119903≺

(120572) and 119886119901119903≺

(120573) le 119886119901119903≺

(120572 and 120573) and so 119886119901119903≺

(120572) and

119886119901119903≺

(120573) = 119886119901119903≺

(120572 and 120573) Hence condition (P3) holds

(3) By the duality of approximation operators it sufficesto prove that 119886119901119903

≺

(120572) le 120572 And this is clearly true bythe definition of 119886119901119903

≺

(120572)

Summing up the above statements (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

6 The Scientific World Journal

Theorem18 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak algebraDefine a binary relation ldquo≺rdquo as follows

forall120572 120573 isin 119871 120572 ≺ 120573 lArrrArr 119886119901119903 (120572) le 120573 (27)

Then ldquo≺rdquo is an auxiliary ordering on 119871

Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573

(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582

(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus

119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)

implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary

ordering on 119871

5 Molecular Pawlak Algebraand Rough Topology

In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set

Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular

Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863

a molec-ular net and 120572 isin 120587 If there exists 119889

0isin 119863 such that 119886119901119903(120572) le

119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit

of 119878 denoted by 119878119886119901119903

997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878

In the sequel we provide two examples of molecularPawlak algebra in topology space

Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =

119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define

forall119860 isin 119871 119886119901119903119860 = ⋃119909isin119860

119861 (119909 1) 119886119901119903119860 = (119886119901119903119860119888

)119888

(29)

where 119861(119909 1) is the closed ball Then (Φ(119880)(120587) cup cap119888

119886119901119903

119886119901119903) is a molecular Pawlak algebra Let 1199090be the 119886119901119903-limit

of the molecular sequence 119909119899119899isin119873

(ie 119909119899

119886119901119903

997888997888rarr 1199090) which

means that there exists 1198990

isin 119873 such that 120588(119909119899 119909

0) le 1 for

119899 ge 119899119900 Obviously 119909

119899rarr 119909

0implies 119909

119899

119886119901119903

997888997888rarr 1199090(119899 rarr infin)

Example 21 Let119880 be a nonempty set119877 an equivalent relationon 119880 119871 = F(119880) fuzzy power set on 119880 120587 = 119909

120582 | 119909 isin 119880 120582 isin

(0 1] and119880119877 = 119883119894119894isin119868 For any119860 isin F(119880) define two fuzzy

sets 119865119860 and 119865119860on 119880119877 as

forall119894 isin 119868 119865119860

(119883119894) = sup 119860 (119909) | 119909 isin 119883

119894

119865119860(119883

119894) = inf 119860 (119909) | 119909 isin 119883

119894

(30)

Then we have for all 119860 119861 isin 119865(119880)

(1) 119865119860cup119861 = 119865119860 cup 119865119861

(2) 119865119860cap119861

= 119865119860cap 119865

119861

(3) 119865119860cap119861 sube 119865119860 cap 119865119861

(4) 119865119860cup119861

supe 119865119860cup 119865

119861

(5) 119865119860sube 119865119860

Moreover for any 119860 isin 119865(119880) define 119886119901119903119860 and 119886119901119903119860 as

(119886119901119903119860) (119909) = 119865119860 (119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(31)

(119886119901119903119860) (119909) = 119865119860(119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(32)

for any 119909 isin 119880 Then (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) is a molecular

Pawlak algebraIn the system (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) the rough neigh-

borhood 119886119901119903(119909120582) of molecule 119909

120582is represented as

119886119901119903 (119909120582) (119910) =

120582 119877 (119909 119910) = 1

0 119877 (119909 119910) = 0(33)

119878119886119901119903

997888997888rarr 119909120582if and only if there exists 119889

0isin 119863 such that 119909119877119910119889 and

120582119889le 120582 for 119889 ge 119889

0 where 119878(119889) = 119910119889

120582119889

and119883119894is poly-point set

If 119880119877 = 119909 | 119909 isin 119880 namely 119877(119909 119910) = 1 if and only if119909 = 119910 then we have

119886119901119903 (119909120582) = 119886119901119903 (119909

120582) = 119909

120582 (34)

for fuzzy point 119909120582and it is know that 119878

119886119901119903

997888997888rarr 119909120582if and only if

there exists 1198890isin 119863 such that 119878(119889) = 119909

120582for 119889 ge 119889

0

Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap

119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)

The following is now straightforward

Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers

Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra

The Scientific World Journal 7

Proof Set ℎ(0) = 0 and define

ℎ (120574) = ⋁120572le120574120572isin120587

ℎ (120572) ℎ (120574) = (ℎ (120574119888

))119888

(35)

Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra

6 The Application of ApproximationOperators to Rough Sets

Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880

119903 (119909) = 119910 | (119909 119910) isin 119877 (36)

and defined general dual approximation operators 119886119901119903119877

and119886119901119903

119877as for all119883 sube 119880

119886119901119903119877

(119883) = 119909 | 119903 (119909) sube 119883

119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =

(37)

In this section we further investigate the properties ofapproximation operators induced by a binary relation

The following results recall some basic properties ofapproximation operators induced by a binary relation

Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880

(R1) 119886119901119903119877

(119883) = [119886119901119903119877(119883119888)]

119888 119886119901119903119877(119883) = [119886119901119903

119877

(119883119888)]119888

(R2) 119886119901119903119877

() = 119886119901119903119877() = 119886119901119903

119877

(119880) = 119886119901119903119877(119880) = 119880

(R3) 119886119901119903119877

(119883 cap 119884) = 119886119901119903119877

(119883) cap 119886119901119903119877

(119884) 119886119901119903119877(119883 cap 119884) sube

119886119901119903119877(119883) cap 119886119901119903

119877(119884)

(R4) 119886119901119903119877

(119883 cup 119884) supe 119886119901119903119877

(119883) cup 119886119901119903119877

(119884) 119886119901119903119877(119883 cup 119884) =

119886119901119903119877(119883) cup 119886119901119903

119877(119884)

(R5) 119886119901119903119877

(119883) sube 119883 sube 119886119901119903119877(119883)

(R6) 119909 isin 119886119901119903119877119910 hArr 119910 isin 119886119901119903

119877119909

Proposition 25 indicates that the system (Φ(119880) cup

cap119888 119880 119886119901119903119877

119886119901119903119877) is a Pawlak algebra where Φ(119880)

denotes the set of all subsets of 119880

Proposition 26 (see [22]) Let 119877 sube 119880times119880 be a binary relationon 119880 Then

(1) 119877 is reflexive if and only if 119886119901119903119877

(119883) sube 119883 sube

119886119901119903119877(119883) (forall119883 sube 119880)

(2) 119877 is symmetric if and only if 119909 isin 119886119901119903119877119910 is equivalent

to 119910 isin 119886119901119903119877119909

(3) the reflexive relation 119877 is transitive if and only if 119886119901119903119877

is a closure operator

Proposition 27 Let (119901 119901) isin 119860119875119877(Φ(119880)) where119901 is a closureoperator Then

119909 isin 119901 119910 119894119891119891 119901 119909 sube 119901 119910 (38)

Proof It is straightforward and omitted

Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following

(1) the binary relation 119877119901defined as

119877119901= (119909 119910) | 119909 isin 119901 119910 (39)

is an equivalent relation on 119880(2) (119886119901119903

119877119901

119886119901119903119877119901

) ≺ (119901 119901) Moreover if 119880 is a finiteuniverse or the cover 119901119909

119909isin119880of 119880 is finite then

(119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Proof (1) It follows from Proposition 29 and condition (P6)that

119877119901= (119909 119910) | 119901 119909 = 119901 119910 (40)

Then it is easy to see that 119877119901is an equivalent relation where

[119909]119877= 119901119909 for all 119909 isin 119880

(2) For any119883 sube 119880 we have

119886119901119903119877119901

(119883) = 119909 | [119909]119877119901

cap 119883 =

= 119909 | 119910 | 119909 isin 119901 119910 cap 119883 =

= 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

sube 119901 (119883)

(41)

On the other hand 119886119901119903119877119901

(119883119888) sube 119901(119883119888) implies that(119901(119883119888))

119888

sube (119886119901119903119877119901

(119883119888))119888 hence 119901(119883) sube 119886119901119903

119877119901

(119883) Thus we

have (119886119901119903119877119901

119886119901119903119877119901

) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =

1199101 119910

2 119910

119898 sube 119880 Then we have

119901 (119883) = ⋃119910119896isin119883

119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901

= 119909 | [119909]119877119901

cap 119883 =

= 119886119901119903119877119901

(119883)

(42)

Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901

(119883) It

follows that (119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Now suppose that the cover 119901119909119909isin119880

of 119880 is finitethat is there exist 119909

119896isin 119880 119896 = 1 2 119898 such that

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

2 The Scientific World Journal

The remaining part of the paper is organized as followsSection 2 introduces the relevant definitions which will beused throughout the paper Section 3 investigates the Pawlakalgebra and weak Pawlak algebra on fuzzy lattice Section 4discusses the relationships between Pawlak algebra and aux-iliary ordering Section 5 focuses on the study of molecu-lar Pawlak algebra Section 6 investigates the properties ofPawlak algebra based on binary relation Finally Section 7concludes the paper and suggests some future research topics

2 Preliminaries

In this section we introduce some basic definitions (see [319 20]) which will be used throughout the paper

Definition 1 Apartially ordered set (119871 le) is said to be a latticeif inf119909 119910 and sup119909 119910 denoted by and and or respectivelyexist for all 119909 119910 isin 119871 A lattice 119871 is said to be complete if forevery 119860 sube 119871⋀

120572isin119860120572 and⋁

120572isin119860120572 exist

Definition 2 Let (119871 le) be a complete lattice with the max-imum element 1 and minimum element 0 and ldquo≺rdquo a binaryrelation on 119871 If the following conditions hold for all 120572 120573 120583120578 120574 isin 119871

(i) 120572 ≺ 120573 rArr 120572 le 120573(ii) 120583 le 120572 ≺ 120573 le 120578 rArr 120583 ≺ 120578(iii) 120572 ≺ 120574 120573 ≺ 120574 rArr 120572 or 120573 ≺ 120574(iv) forall120572 isin 119871 0 ≺ 120572 ≺ 1

then the relation ldquo≺rdquo is called an auxiliary ordering on 119871 If therelation ldquo≺rdquo satisfies conditions (i) (ii) and (iv) it is called aweak auxiliary ordering on119871Theweak auxiliary ordering ldquo≺rdquois called completely approximate if

forall120572 isin 119871 120572 = or 120574 | 120574 ≺ 120572 120574 isin 119871 (1)

Furthermore if the relation ldquo≺rdquo satisfies conditions (i) (ii)(iv) and

(iii)1015840 120572119905≺ 120573(119905 isin 119879) rArr ⋁

119905isin119879120572119905≺ 120573

then ldquo≺rdquo is called a strong auxiliary ordering on 119871

Definition 3 Let (119871 or and 0 1) be a completely distributivelattice If the mapping 119888 119871 rarr 119871 satisfies the followingconditions

(i) reverse law forall120572 120573 isin 119871 if 120572 le 120573 then 120573119888 le 120572119888(ii) recovery law forall120572 isin 119871 (120572119888)119888 = 120572

then (119871 or and119888 0 1) is called a fuzzy lattice

Remark 4 The notion of fuzzy lattice just given is first intro-duced in Wang [20] and is also called de Morgan completelydistributive lattice in the literature To keep consistency weadopt the term ldquofuzzy latticerdquo in this paper

Definition 5 Let (119871 or and119888 0 1) be a fuzzy lattice and 120587 sube 119871If the subset 120587 satisfies the following conditions

(i) 0 notin 120587(ii) 120572 120573 isin 120587 120572 and 120573 = 0 rArr 120572 le 120573 or 120573 le 120572(iii) 120572 120573 120574 isin 120587 120572 le 120574 120573 le 120574 rArr 120572 and 120573 = 0(iv) 120578 = or120572 | 120572 isin 120587 120572 le 120578 for all 120578 isin 119871(v) if 120587

0sube 120587 and 120587

0is linear order subset then or120587

0isin 120587

then 120587 is called amolecular set and the element of 120587 is calleda molecule The fuzzy lattice with molecule (119871(120587) or and119888 0 1)

is called amolecular lattice

Definition 6 Let (119871 or and119888 0 1) be a fuzzy lattice and 120575 sube 119871If the subset 120575 satisfies the following conditions

(1) 0 1 isin 120575(2) 120572

119905isin 120575(119905 isin 119879) rArr ⋁

119905isin119879120572119905isin 120575

(3) 120572 120573 isin 120575 rArr 120572 and 120573 isin 120575

then (119871 120575) is called a lattice topology space

3 The Pawlak Algebra and Weak PawlakAlgebra on Fuzzy Lattices

In this section we investigate the structural properties ofPawlak rough approximations on fuzzy lattices Let us beginwith introducing the following concepts

Definition 7 (see [20])) Let (119871 or and119888 0 1) be a fuzzy latticeIf the dual mappings 119886119901119903 119871 rarr 119871 and 119886119901119903 119871 rarr 119871 satisfythe following conditions

(P1) 119886119901119903(120572) = (119886119901119903(120572119888

))119888

(forall120572 isin 119871)

(P2) 119886119901119903(1) = 1

(P3) 119886119901119903(120572 and 120573) = (119886119901119903120572) and (119886119901119903120573) (forall120572 120573 isin 119871)

(P4) 120572 le 119886119901119903(120572) forall120572 isin 119871

then (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a Pawlak algebra 119886119901119903(resp 119886119901119903 pair (119886119901119903 119886119901119903) ) is called upper approximationoperator (resp lower approximation operator dual approx-imation operator) on 119871 If 119886119901119903(120572) = 119886119901119903(120572) then 120572 is calleda definable element If 119886119901119903120572 = 119886119901119903120572 then 120572 is called a roughelement

If (P3) is replaced by the following condition (P3)lowast

(P3)lowast 120572 le 120573 rArr 119886119901119903(120572) le 119886119901119903(120573)

then 119886119901119903 (resp 119886119901119903 pair (119886119901119903 119886119901119903)) is called a weak upperapproximation operator (resp weak lower approximationoperator weak dual approximation operator) on 119871 andthe system (119871 or and119888 0 1 119886119901119903 119886119901119903) is called a weak Pawlakalgebra

Let (119871 or and119888 0 1) be a fuzzy lattice Denote by 119860119875119877(119871)

(resp 119860119875119877119882(119871)) the set of all dual approximation operators

(resp dual weak approximation operators) on 119871 respectivelyand by 120590

119886119901119903

0(119871) the set of all definable elements in the Pawlak

algebra (119871 or and119888 0 1 119886119901119903 119886119901119903)

The Scientific World Journal 3

Definition 8 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903

1

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877

119908(119871) Then the dual weak

approximation operator (1198861199011199032

1198861199011199032) is called rougher than

(1198861199011199031

1198861199011199031) denoted by (119886119901119903

1

1198861199011199031) ≺ (119886119901119903

2

1198861199011199032) if the

following inequalities hold

forall120572 isin 119871 1198861199011199032

(120572) le 1198861199011199031

(120572) le 1198861199011199031(120572) le 119886119901119903

2(120572) (2)

Proposition 9 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then 119860119875119877(119871) sube 119860119875119877

119908(119871)

Proof Let (119886119901119903 119886119901119903) isin 119860119875119877(119871) For any 120572 120573 isin 119871 with 120572 le 120573we have 120572 = 120572 and 120573 and so

119886119901119903 (120572) = 119886119901119903 (120572 and 120573) = 119886119901119903 (120572) and 119886119901119903 (120573) (3)

Therefore 119886119901119903(120572) le 119886119901119903(120573) implying that condition (P3)lowast

holds Hence (119886119901119903 119886119901119903) isin 119860119875119877119882(119871) as required

Define two dual approximation operators 119868 = (119868 119868) and119874 = (119874119874) as follows

forall120572 isin 119871 119868 (120572) = 0 120572 = 1

1 120572 = 1 119868 (120572) =

0 120572 = 0

1 120572 = 0

forall120572 isin 119871 119874 (120572) = 119874 (120572) = 120572

(4)

Then it is easy to see that 119868119874 isin 119860119875119877(119871) and119874 ≺ (119886119901119903 119886119901119903) ≺

119868 for any (119886119901119903 119886119901119903) isin 119860119875119877119908(119871)

It is possible to characterize 119860119875119877119882(119871) according to the

following result

Theorem 10 Let (119871 or and119888 0 1) be a fuzzy lattice Then(119860119875119877

119882(119871) ≺) is a complete distributive lattice with maximum

element 119868 and minimum element 119874

Proof Suppose that (119886119901119903119905

119886119901119903119905)119905isin119879

sube 119860119875119877119908(119871) where 119879 is

an index set Define⋁119905isin119879

119886119901119903119905

by

(⋁119905isin119879

119886119901119903119905

) (120572) = ⋁119905isin119879

119886119901119903119905

(120572) forall120572 isin 119871 (5)

And ⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905 and ⋀

119905isin119879119886119901119903

119905can be similarly

defined Then the following assertions hold

(1) Consider (⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905)

(⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) isin 119860119875119877

119908(119871) In fact for

any 120572 120573 isin 119871 by Definitions 6 and 7 we have

(P1) (⋁119905isin119879

119886119901119903119905

)(120572) = ⋁119905isin119879

119886119901119903119905

(120572) =⋁119905isin119879

(119886119901119903119905(120572119888))

119888 = (⋀119905isin119879

119886119901119903119905(120572119888))

119888 =((⋀

119905isin119879119886119901119903

119905)(120572119888))

119888(P2) (⋁

119905isin119879119886119901119903

119905

)(1) = ⋁119905isin119879

119886119901119903119905

(1) = 1(P3) if 120572 le 120573 then 119886119901119903

119905

(120572) le 119886119901119903119905

(120573) forall119905 isin 119879implying that⋁

119905isin119879119886119901119903

119905

(120572) le ⋁119905isin119879

119886119901119903119905(120573)

(P4) 120572 le ⋀119905isin119879

119886119901119903119905(120572) = (⋀

119905isin119879119886119901119903

119905)(120572) since

120572 le 119886119901119903119905(120572) for all 119905 isin 119879

(2) Consider (⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) isin 119860119875119877

119908(119871) The

proof is analogous to that of (1)(3) Consider inf(119886119901119903

119905

119886119901119903119905) | 119905 isin 119879 =

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) and sup(119886119901119903

119905

119886119901119903119905) | 119905 isin

119879 = (⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) Clearly (119886119901119903

119905

119886119901119903119905) ≺

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) is true for all 119905 isin 119879 In fact

let (119886119901119903 119886119901119903) isin 119860119875119877119908(119871) be such that (119886119901119903 119886119901119903) ≺

(119886119901119903119905

119886119901119903119905) for all 119905 isin 119879 Then 119886119901119903

119905

(120572) le 119886119901119903(120572) le

119886119901119903(120572) le 119886119901119903119905(120572) for all 119905 isin 119879 and 120572 isin 119871 by definition

It follows that

⋁119905isin119879

119886119901119903119905

(120572) le 119886119901119903 (120572) le 119886119901119903120572 le ⋀119905isin119879

119886119901119903119905(120572) forall120572 isin 119871 (6)

On the other hand it is obvious that (119886119901119903119905

119886119901119903119905) ≺

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) for all 119905 isin 119879 Hence inf(119886119901119903

119905

119886119901119903119905) |

119905 isin 119879 = (⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) In a similar way by duality

we have sup(119886119901119903119905

119886119901119903119905) | 119905 isin 119879 = (⋀

119905isin119879119886119901119903

119905

⋁119905isin119879

119886119901119903119905)

Summing up the above analysis (119860119875119877119882(119871) ≺) is a com-

plete lattice It is obvious that 119868 is the maximum elementand 119874 is the minimum element and that (APR

119882(119871) ≺) is

distributive This completes the proof

Proposition 11 Let (119871 or and119888 0 1) be a fuzzy lattice and (119871 120575)

a lattice topology space Define the operators 119886119901119903120575

119871 rarr 119871

and 119886119901119903120575 119871 rarr 119871 by

119886119901119903120575

(120572) = or 120574 | 120574 le 120572 120574 isin 120575

119886119901119903120575(120572) = and 120574

119888

| 120572 le 120574119888

120574 isin 120575

(7)

respectively for all 120572 isin 119871 Then the system(119871 or and119888 0 1 119886119901119903

120575

119886119901119903120575) is a Pawlak algebra

Proof It is straightforward and omitted

Proposition 12 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then (119871 120590

119886119901119903

0(119871)) is a zero-dimensional lattice topology

space

Proof It is evident that 1205901198861199011199030

(119871) = 120572 isin 119871 | 120572 = 119886119901119903(120572) cap

120572 isin 119871 | 120572 = 119886119901119903(120572) And we conclude that the followingassertions hold

(a) 120572 isin 119871 | 120572 = 119886119901119903(120572) is union-closed Suppose that120572

119905| 119905 isin 119879 sube 120572 isin 119871 | 120572 = 119886119901119903(120572) where 119879 is an

index set Now for any 119904 isin 119879 we have 120572119904le ⋁

119905isin119879120572119905

and so 120572119904= 119886119901119903(120572

119904) le 119886119901119903(⋁

119905isin119879120572119905) implying that

⋁119905isin119879

120572119905le 119886119901119903(⋁

119905isin119879120572119905) Since ⋁

119905isin119879120572119905ge 119886119901119903(⋁

119905isin119879120572119905)

we have⋁119905isin119879

120572119905= 119886119901119903(⋁

119905isin119879120572119905) by conditions (P1) and

(P4) as required(b) 120572 isin 119871 | 120572 = 119886119901119903(120572) is intersection-closed Suppose

that 120572119905| 119905 isin 119879 sube 120572 isin 119871 | 120572 = 119886119901119903(120572) where 119879

4 The Scientific World Journal

is an index set For any 119904 isin 119879 we have 120572119904ge ⋀

119905isin119879120572119905

and so 120572119904= 119886119901119903(120572

119904) ge 119886119901119903(⋀

119905isin119879120572119905) implying that

⋀119905isin119879

120572119905ge 119886119901119903(⋀

119905isin119879120572119905) By condition (P4) we have

⋀119905isin119879

120572119905= 119886119901119903(⋀

119905isin119879120572119905) as required

(c) 120572 isin 119871 | 120572 = 119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572)

is complement-closed For any 120572 isin 120572 isin 119871 | 120572 =

119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572) by condition (P1)we have 119886119901119903(120572119888) = ((119886119901119903(120572119888))

119888

)119888 = (119886119901119903(120572))

119888

= 120572implying that 120572119888 isin 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 =

119886119901119903(120572) Hence 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 = 119886119901119903(120572) iscomplement-closed

Summing up the above analysis (119871 1205901198861199011199030

(119871)) is a zero-dimensional lattice topology space

Now let us turn our attention to the study of the transitiveclosure of approximation operators

Let (119871 or and119888 0 1) be a fuzzy lattice Define the operatorldquo∘rdquo on 119860119875119877(119871) as follows

(1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) = (119886119901119903

1

∘ 1198861199011199032

1198861199011199031∘ 119886119901119903

2)

(8)

That is

forall120572 isin 119871 (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) (120572)

= (1198861199011199031

(1198861199011199032

(120572)) 1198861199011199031(119886119901119903

2(120572)))

(9)

where (1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871) For any positive

integer 119896 define (119886119901119903 119886119901119903)(119896+1)

= (119886119901119903 119886119901119903)(119896)

∘ (119886119901119903 119886119901119903)Denote cl(119886119901119903 119886119901119903) = ⋁

infin

119896=1(119886119901119903 119886119901119903)

(119896) which is called thetransitive closure of (119886119901119903 119886119901119903) Then we obtain the followingresult

Lemma 13 Let (119871 or and119888 0 1) be a fuzzy lattice Then

(1) (1198861199011199031

1198861199011199031) ≺ (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) for all

(1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(2) (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(3) cl(119886119901119903 119886119901119903) is a dual approximation operator

Proof (1) It is straightforward and omitted(2) It suffices to prove that (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032)

satisfies conditions (P1)ndash(P4) in Definition 7 In fact for any120572 120573 isin 119871 we have

(P1) 1198861199011199031

(1198861199011199032

(120572)) = 1198861199011199031((119886119901119903

2

(120572))119888

)119888

=

(1198861199011199031((119886119901119903

2(120572119888))

119888

)119888

)119888

= (1198861199011199031(119886119901119903

2(120572119888)))

119888(P2) (119886119901119903

1

∘ 1198861199011199032

)(1) = 1198861199011199031

(1198861199011199032

(1)) = 1198861199011199031

(1) = 1

(P3) (1198861199011199031

∘ 1198861199011199032

)(120572 and 120573) = 1198861199011199031

(1198861199011199032

(120572 and 120573))

= 1198861199011199031

(1198861199011199032

(120572) and 1198861199011199032

(120573)) = (1198861199011199031

(1198861199011199032

(120572))) and

(1198861199011199031

(1198861199011199032

(120573)))= (1198861199011199031

∘1198861199011199032

)(120572))and(1198861199011199031

∘1198861199011199032

)(120573))

(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903

1(120572) le

1198861199011199031(119886119901119903

2(120572)) = (119886119901119903

1∘ 119886119901119903

2)(120572)

(3) From the proof of Theorem 10 we have

cl (119886119901119903 119886119901119903) = (

infin

⋀119896=1

119886119901119903(119896)

infin

⋁119896=1

119886119901119903(119896)

) (10)

And it is easy to prove by mathematical induction that

119886119901119903(119896)

(120572 and 120573)

= 119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573) for all positive integer 119896

(11)

Therefore

(

infin

⋀119896=1

119886119901119903(119896)

)(120572 and 120573) =

infin

⋀119896=1

119886119901119903(119896)

(120572 and 120573)

=

infin

⋀119896=1

(119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573))

= (

infin

⋀119896=1

119886119901119903(119896)

(120572)) and (

infin

⋀119896=1

119886119901119903(119896)

(120573))

(12)

It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator

Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871

Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has

(

infin

⋀119896=1

119886119901119903(119896)

) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)

(

infin

⋁119896=1

119886119901119903(119896)

) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)

Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879

119886119901119903 for any 120572 isin 119871 we have

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 120572

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903 (120572)

The Scientific World Journal 5

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(2)

(120572)

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(119896)

(120572)

(15)

implying that or120574 | 120574 le 120572 119886119901119903(120574) = 120574 le (⋀infin

119896=1119886119901119903(119896))(120572)

Conversely for any 120572 isin 119871

119886119901119903((

infin

⋀119896=1

119886119901119903(119896)

) (120572)) = (

infin

⋀119896=1

119886119901119903(119896)

) (120572) le 120572 (16)

It follows that (⋀infin

119896=1119886119901119903(119896))(120572) isin 119879

119886119901119903 and so (⋀

infin

119896=1119886119901119903(119896))(120572)

le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds

As a consequence ofTheorem 15 we obtain the followingresult

Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903

lowast= ⋁

infin

119896=1119886119901119903

(119896) is a closure operator on 119871

4 The Relationships between Pawlak Algebraand Auxiliary Ordering

Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately

The following results present the relationships betweenPawlak algebra and strong auxiliary ordering

Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903

≺

119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows

forall120572 isin 119871 119886119901119903≺

(120572) = or 120574 | 120574 ≺ 120572 120574 isin 119871

119886119901119903≺(120572) = (119886119901119903

≺

(120572119888

))119888

(17)

then (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

Proof It is obvious that condition (P1) holds In what followswe prove that conditions (P2)ndash(P4) are satisfied

(1) The strong auxiliary ordering ≺ implies that 1 ≺ 1 istrue and so we have

119886119901119903≺

(1) = or 120574 | 120574 ≺ 1 120574 isin 119871 = 1 (18)

implying that condition (P2) holds

(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺

120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120572 120574 isin 119871

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120573 120574 isin 119871 (19)

Therefore

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120572 120574 isin 119871

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120573 120574 isin 119871 (20)

It follows that 119886119901119903≺

(120572 and 120573) le 119886119901119903≺

(120572) and 119886119901119903≺

(120572 and 120573) le

119886119901119903≺

(120573) and hence

119886119901119903≺

(120572 and 120573) le (119886119901119903≺

(120572)) and (119886119901119903≺

(120573)) (21)

On the other hand by condition (iii) in Definition 3 wehave

or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572

or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573(22)

It follows from

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572 le 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573 le 120573

(23)

that

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120573(24)

Hence

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572 and 120573

(25)

implying that

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) isin 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 (26)

Thus 119886119901119903≺

(120572) and 119886119901119903≺

(120573) le 119886119901119903≺

(120572 and 120573) and so 119886119901119903≺

(120572) and

119886119901119903≺

(120573) = 119886119901119903≺

(120572 and 120573) Hence condition (P3) holds

(3) By the duality of approximation operators it sufficesto prove that 119886119901119903

≺

(120572) le 120572 And this is clearly true bythe definition of 119886119901119903

≺

(120572)

Summing up the above statements (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

6 The Scientific World Journal

Theorem18 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak algebraDefine a binary relation ldquo≺rdquo as follows

forall120572 120573 isin 119871 120572 ≺ 120573 lArrrArr 119886119901119903 (120572) le 120573 (27)

Then ldquo≺rdquo is an auxiliary ordering on 119871

Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573

(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582

(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus

119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)

implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary

ordering on 119871

5 Molecular Pawlak Algebraand Rough Topology

In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set

Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular

Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863

a molec-ular net and 120572 isin 120587 If there exists 119889

0isin 119863 such that 119886119901119903(120572) le

119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit

of 119878 denoted by 119878119886119901119903

997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878

In the sequel we provide two examples of molecularPawlak algebra in topology space

Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =

119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define

forall119860 isin 119871 119886119901119903119860 = ⋃119909isin119860

119861 (119909 1) 119886119901119903119860 = (119886119901119903119860119888

)119888

(29)

where 119861(119909 1) is the closed ball Then (Φ(119880)(120587) cup cap119888

119886119901119903

119886119901119903) is a molecular Pawlak algebra Let 1199090be the 119886119901119903-limit

of the molecular sequence 119909119899119899isin119873

(ie 119909119899

119886119901119903

997888997888rarr 1199090) which

means that there exists 1198990

isin 119873 such that 120588(119909119899 119909

0) le 1 for

119899 ge 119899119900 Obviously 119909

119899rarr 119909

0implies 119909

119899

119886119901119903

997888997888rarr 1199090(119899 rarr infin)

Example 21 Let119880 be a nonempty set119877 an equivalent relationon 119880 119871 = F(119880) fuzzy power set on 119880 120587 = 119909

120582 | 119909 isin 119880 120582 isin

(0 1] and119880119877 = 119883119894119894isin119868 For any119860 isin F(119880) define two fuzzy

sets 119865119860 and 119865119860on 119880119877 as

forall119894 isin 119868 119865119860

(119883119894) = sup 119860 (119909) | 119909 isin 119883

119894

119865119860(119883

119894) = inf 119860 (119909) | 119909 isin 119883

119894

(30)

Then we have for all 119860 119861 isin 119865(119880)

(1) 119865119860cup119861 = 119865119860 cup 119865119861

(2) 119865119860cap119861

= 119865119860cap 119865

119861

(3) 119865119860cap119861 sube 119865119860 cap 119865119861

(4) 119865119860cup119861

supe 119865119860cup 119865

119861

(5) 119865119860sube 119865119860

Moreover for any 119860 isin 119865(119880) define 119886119901119903119860 and 119886119901119903119860 as

(119886119901119903119860) (119909) = 119865119860 (119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(31)

(119886119901119903119860) (119909) = 119865119860(119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(32)

for any 119909 isin 119880 Then (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) is a molecular

Pawlak algebraIn the system (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) the rough neigh-

borhood 119886119901119903(119909120582) of molecule 119909

120582is represented as

119886119901119903 (119909120582) (119910) =

120582 119877 (119909 119910) = 1

0 119877 (119909 119910) = 0(33)

119878119886119901119903

997888997888rarr 119909120582if and only if there exists 119889

0isin 119863 such that 119909119877119910119889 and

120582119889le 120582 for 119889 ge 119889

0 where 119878(119889) = 119910119889

120582119889

and119883119894is poly-point set

If 119880119877 = 119909 | 119909 isin 119880 namely 119877(119909 119910) = 1 if and only if119909 = 119910 then we have

119886119901119903 (119909120582) = 119886119901119903 (119909

120582) = 119909

120582 (34)

for fuzzy point 119909120582and it is know that 119878

119886119901119903

997888997888rarr 119909120582if and only if

there exists 1198890isin 119863 such that 119878(119889) = 119909

120582for 119889 ge 119889

0

Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap

119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)

The following is now straightforward

Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers

Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra

The Scientific World Journal 7

Proof Set ℎ(0) = 0 and define

ℎ (120574) = ⋁120572le120574120572isin120587

ℎ (120572) ℎ (120574) = (ℎ (120574119888

))119888

(35)

Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra

6 The Application of ApproximationOperators to Rough Sets

Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880

119903 (119909) = 119910 | (119909 119910) isin 119877 (36)

and defined general dual approximation operators 119886119901119903119877

and119886119901119903

119877as for all119883 sube 119880

119886119901119903119877

(119883) = 119909 | 119903 (119909) sube 119883

119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =

(37)

In this section we further investigate the properties ofapproximation operators induced by a binary relation

The following results recall some basic properties ofapproximation operators induced by a binary relation

Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880

(R1) 119886119901119903119877

(119883) = [119886119901119903119877(119883119888)]

119888 119886119901119903119877(119883) = [119886119901119903

119877

(119883119888)]119888

(R2) 119886119901119903119877

() = 119886119901119903119877() = 119886119901119903

119877

(119880) = 119886119901119903119877(119880) = 119880

(R3) 119886119901119903119877

(119883 cap 119884) = 119886119901119903119877

(119883) cap 119886119901119903119877

(119884) 119886119901119903119877(119883 cap 119884) sube

119886119901119903119877(119883) cap 119886119901119903

119877(119884)

(R4) 119886119901119903119877

(119883 cup 119884) supe 119886119901119903119877

(119883) cup 119886119901119903119877

(119884) 119886119901119903119877(119883 cup 119884) =

119886119901119903119877(119883) cup 119886119901119903

119877(119884)

(R5) 119886119901119903119877

(119883) sube 119883 sube 119886119901119903119877(119883)

(R6) 119909 isin 119886119901119903119877119910 hArr 119910 isin 119886119901119903

119877119909

Proposition 25 indicates that the system (Φ(119880) cup

cap119888 119880 119886119901119903119877

119886119901119903119877) is a Pawlak algebra where Φ(119880)

denotes the set of all subsets of 119880

Proposition 26 (see [22]) Let 119877 sube 119880times119880 be a binary relationon 119880 Then

(1) 119877 is reflexive if and only if 119886119901119903119877

(119883) sube 119883 sube

119886119901119903119877(119883) (forall119883 sube 119880)

(2) 119877 is symmetric if and only if 119909 isin 119886119901119903119877119910 is equivalent

to 119910 isin 119886119901119903119877119909

(3) the reflexive relation 119877 is transitive if and only if 119886119901119903119877

is a closure operator

Proposition 27 Let (119901 119901) isin 119860119875119877(Φ(119880)) where119901 is a closureoperator Then

119909 isin 119901 119910 119894119891119891 119901 119909 sube 119901 119910 (38)

Proof It is straightforward and omitted

Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following

(1) the binary relation 119877119901defined as

119877119901= (119909 119910) | 119909 isin 119901 119910 (39)

is an equivalent relation on 119880(2) (119886119901119903

119877119901

119886119901119903119877119901

) ≺ (119901 119901) Moreover if 119880 is a finiteuniverse or the cover 119901119909

119909isin119880of 119880 is finite then

(119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Proof (1) It follows from Proposition 29 and condition (P6)that

119877119901= (119909 119910) | 119901 119909 = 119901 119910 (40)

Then it is easy to see that 119877119901is an equivalent relation where

[119909]119877= 119901119909 for all 119909 isin 119880

(2) For any119883 sube 119880 we have

119886119901119903119877119901

(119883) = 119909 | [119909]119877119901

cap 119883 =

= 119909 | 119910 | 119909 isin 119901 119910 cap 119883 =

= 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

sube 119901 (119883)

(41)

On the other hand 119886119901119903119877119901

(119883119888) sube 119901(119883119888) implies that(119901(119883119888))

119888

sube (119886119901119903119877119901

(119883119888))119888 hence 119901(119883) sube 119886119901119903

119877119901

(119883) Thus we

have (119886119901119903119877119901

119886119901119903119877119901

) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =

1199101 119910

2 119910

119898 sube 119880 Then we have

119901 (119883) = ⋃119910119896isin119883

119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901

= 119909 | [119909]119877119901

cap 119883 =

= 119886119901119903119877119901

(119883)

(42)

Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901

(119883) It

follows that (119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Now suppose that the cover 119901119909119909isin119880

of 119880 is finitethat is there exist 119909

119896isin 119880 119896 = 1 2 119898 such that

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

The Scientific World Journal 3

Definition 8 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903

1

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877

119908(119871) Then the dual weak

approximation operator (1198861199011199032

1198861199011199032) is called rougher than

(1198861199011199031

1198861199011199031) denoted by (119886119901119903

1

1198861199011199031) ≺ (119886119901119903

2

1198861199011199032) if the

following inequalities hold

forall120572 isin 119871 1198861199011199032

(120572) le 1198861199011199031

(120572) le 1198861199011199031(120572) le 119886119901119903

2(120572) (2)

Proposition 9 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then 119860119875119877(119871) sube 119860119875119877

119908(119871)

Proof Let (119886119901119903 119886119901119903) isin 119860119875119877(119871) For any 120572 120573 isin 119871 with 120572 le 120573we have 120572 = 120572 and 120573 and so

119886119901119903 (120572) = 119886119901119903 (120572 and 120573) = 119886119901119903 (120572) and 119886119901119903 (120573) (3)

Therefore 119886119901119903(120572) le 119886119901119903(120573) implying that condition (P3)lowast

holds Hence (119886119901119903 119886119901119903) isin 119860119875119877119882(119871) as required

Define two dual approximation operators 119868 = (119868 119868) and119874 = (119874119874) as follows

forall120572 isin 119871 119868 (120572) = 0 120572 = 1

1 120572 = 1 119868 (120572) =

0 120572 = 0

1 120572 = 0

forall120572 isin 119871 119874 (120572) = 119874 (120572) = 120572

(4)

Then it is easy to see that 119868119874 isin 119860119875119877(119871) and119874 ≺ (119886119901119903 119886119901119903) ≺

119868 for any (119886119901119903 119886119901119903) isin 119860119875119877119908(119871)

It is possible to characterize 119860119875119877119882(119871) according to the

following result

Theorem 10 Let (119871 or and119888 0 1) be a fuzzy lattice Then(119860119875119877

119882(119871) ≺) is a complete distributive lattice with maximum

element 119868 and minimum element 119874

Proof Suppose that (119886119901119903119905

119886119901119903119905)119905isin119879

sube 119860119875119877119908(119871) where 119879 is

an index set Define⋁119905isin119879

119886119901119903119905

by

(⋁119905isin119879

119886119901119903119905

) (120572) = ⋁119905isin119879

119886119901119903119905

(120572) forall120572 isin 119871 (5)

And ⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905 and ⋀

119905isin119879119886119901119903

119905can be similarly

defined Then the following assertions hold

(1) Consider (⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905)

(⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) isin 119860119875119877

119908(119871) In fact for

any 120572 120573 isin 119871 by Definitions 6 and 7 we have

(P1) (⋁119905isin119879

119886119901119903119905

)(120572) = ⋁119905isin119879

119886119901119903119905

(120572) =⋁119905isin119879

(119886119901119903119905(120572119888))

119888 = (⋀119905isin119879

119886119901119903119905(120572119888))

119888 =((⋀

119905isin119879119886119901119903

119905)(120572119888))

119888(P2) (⋁

119905isin119879119886119901119903

119905

)(1) = ⋁119905isin119879

119886119901119903119905

(1) = 1(P3) if 120572 le 120573 then 119886119901119903

119905

(120572) le 119886119901119903119905

(120573) forall119905 isin 119879implying that⋁

119905isin119879119886119901119903

119905

(120572) le ⋁119905isin119879

119886119901119903119905(120573)

(P4) 120572 le ⋀119905isin119879

119886119901119903119905(120572) = (⋀

119905isin119879119886119901119903

119905)(120572) since

120572 le 119886119901119903119905(120572) for all 119905 isin 119879

(2) Consider (⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) isin 119860119875119877

119908(119871) The

proof is analogous to that of (1)(3) Consider inf(119886119901119903

119905

119886119901119903119905) | 119905 isin 119879 =

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) and sup(119886119901119903

119905

119886119901119903119905) | 119905 isin

119879 = (⋀119905isin119879

119886119901119903119905

⋁119905isin119879

119886119901119903119905) Clearly (119886119901119903

119905

119886119901119903119905) ≺

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) is true for all 119905 isin 119879 In fact

let (119886119901119903 119886119901119903) isin 119860119875119877119908(119871) be such that (119886119901119903 119886119901119903) ≺

(119886119901119903119905

119886119901119903119905) for all 119905 isin 119879 Then 119886119901119903

119905

(120572) le 119886119901119903(120572) le

119886119901119903(120572) le 119886119901119903119905(120572) for all 119905 isin 119879 and 120572 isin 119871 by definition

It follows that

⋁119905isin119879

119886119901119903119905

(120572) le 119886119901119903 (120572) le 119886119901119903120572 le ⋀119905isin119879

119886119901119903119905(120572) forall120572 isin 119871 (6)

On the other hand it is obvious that (119886119901119903119905

119886119901119903119905) ≺

(⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) for all 119905 isin 119879 Hence inf(119886119901119903

119905

119886119901119903119905) |

119905 isin 119879 = (⋁119905isin119879

119886119901119903119905

⋀119905isin119879

119886119901119903119905) In a similar way by duality

we have sup(119886119901119903119905

119886119901119903119905) | 119905 isin 119879 = (⋀

119905isin119879119886119901119903

119905

⋁119905isin119879

119886119901119903119905)

Summing up the above analysis (119860119875119877119882(119871) ≺) is a com-

plete lattice It is obvious that 119868 is the maximum elementand 119874 is the minimum element and that (APR

119882(119871) ≺) is

distributive This completes the proof

Proposition 11 Let (119871 or and119888 0 1) be a fuzzy lattice and (119871 120575)

a lattice topology space Define the operators 119886119901119903120575

119871 rarr 119871

and 119886119901119903120575 119871 rarr 119871 by

119886119901119903120575

(120572) = or 120574 | 120574 le 120572 120574 isin 120575

119886119901119903120575(120572) = and 120574

119888

| 120572 le 120574119888

120574 isin 120575

(7)

respectively for all 120572 isin 119871 Then the system(119871 or and119888 0 1 119886119901119903

120575

119886119901119903120575) is a Pawlak algebra

Proof It is straightforward and omitted

Proposition 12 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak alge-bra Then (119871 120590

119886119901119903

0(119871)) is a zero-dimensional lattice topology

space

Proof It is evident that 1205901198861199011199030

(119871) = 120572 isin 119871 | 120572 = 119886119901119903(120572) cap

120572 isin 119871 | 120572 = 119886119901119903(120572) And we conclude that the followingassertions hold

(a) 120572 isin 119871 | 120572 = 119886119901119903(120572) is union-closed Suppose that120572

119905| 119905 isin 119879 sube 120572 isin 119871 | 120572 = 119886119901119903(120572) where 119879 is an

index set Now for any 119904 isin 119879 we have 120572119904le ⋁

119905isin119879120572119905

and so 120572119904= 119886119901119903(120572

119904) le 119886119901119903(⋁

119905isin119879120572119905) implying that

⋁119905isin119879

120572119905le 119886119901119903(⋁

119905isin119879120572119905) Since ⋁

119905isin119879120572119905ge 119886119901119903(⋁

119905isin119879120572119905)

we have⋁119905isin119879

120572119905= 119886119901119903(⋁

119905isin119879120572119905) by conditions (P1) and

(P4) as required(b) 120572 isin 119871 | 120572 = 119886119901119903(120572) is intersection-closed Suppose

that 120572119905| 119905 isin 119879 sube 120572 isin 119871 | 120572 = 119886119901119903(120572) where 119879

4 The Scientific World Journal

is an index set For any 119904 isin 119879 we have 120572119904ge ⋀

119905isin119879120572119905

and so 120572119904= 119886119901119903(120572

119904) ge 119886119901119903(⋀

119905isin119879120572119905) implying that

⋀119905isin119879

120572119905ge 119886119901119903(⋀

119905isin119879120572119905) By condition (P4) we have

⋀119905isin119879

120572119905= 119886119901119903(⋀

119905isin119879120572119905) as required

(c) 120572 isin 119871 | 120572 = 119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572)

is complement-closed For any 120572 isin 120572 isin 119871 | 120572 =

119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572) by condition (P1)we have 119886119901119903(120572119888) = ((119886119901119903(120572119888))

119888

)119888 = (119886119901119903(120572))

119888

= 120572implying that 120572119888 isin 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 =

119886119901119903(120572) Hence 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 = 119886119901119903(120572) iscomplement-closed

Summing up the above analysis (119871 1205901198861199011199030

(119871)) is a zero-dimensional lattice topology space

Now let us turn our attention to the study of the transitiveclosure of approximation operators

Let (119871 or and119888 0 1) be a fuzzy lattice Define the operatorldquo∘rdquo on 119860119875119877(119871) as follows

(1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) = (119886119901119903

1

∘ 1198861199011199032

1198861199011199031∘ 119886119901119903

2)

(8)

That is

forall120572 isin 119871 (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) (120572)

= (1198861199011199031

(1198861199011199032

(120572)) 1198861199011199031(119886119901119903

2(120572)))

(9)

where (1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871) For any positive

integer 119896 define (119886119901119903 119886119901119903)(119896+1)

= (119886119901119903 119886119901119903)(119896)

∘ (119886119901119903 119886119901119903)Denote cl(119886119901119903 119886119901119903) = ⋁

infin

119896=1(119886119901119903 119886119901119903)

(119896) which is called thetransitive closure of (119886119901119903 119886119901119903) Then we obtain the followingresult

Lemma 13 Let (119871 or and119888 0 1) be a fuzzy lattice Then

(1) (1198861199011199031

1198861199011199031) ≺ (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) for all

(1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(2) (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(3) cl(119886119901119903 119886119901119903) is a dual approximation operator

Proof (1) It is straightforward and omitted(2) It suffices to prove that (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032)

satisfies conditions (P1)ndash(P4) in Definition 7 In fact for any120572 120573 isin 119871 we have

(P1) 1198861199011199031

(1198861199011199032

(120572)) = 1198861199011199031((119886119901119903

2

(120572))119888

)119888

=

(1198861199011199031((119886119901119903

2(120572119888))

119888

)119888

)119888

= (1198861199011199031(119886119901119903

2(120572119888)))

119888(P2) (119886119901119903

1

∘ 1198861199011199032

)(1) = 1198861199011199031

(1198861199011199032

(1)) = 1198861199011199031

(1) = 1

(P3) (1198861199011199031

∘ 1198861199011199032

)(120572 and 120573) = 1198861199011199031

(1198861199011199032

(120572 and 120573))

= 1198861199011199031

(1198861199011199032

(120572) and 1198861199011199032

(120573)) = (1198861199011199031

(1198861199011199032

(120572))) and

(1198861199011199031

(1198861199011199032

(120573)))= (1198861199011199031

∘1198861199011199032

)(120572))and(1198861199011199031

∘1198861199011199032

)(120573))

(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903

1(120572) le

1198861199011199031(119886119901119903

2(120572)) = (119886119901119903

1∘ 119886119901119903

2)(120572)

(3) From the proof of Theorem 10 we have

cl (119886119901119903 119886119901119903) = (

infin

⋀119896=1

119886119901119903(119896)

infin

⋁119896=1

119886119901119903(119896)

) (10)

And it is easy to prove by mathematical induction that

119886119901119903(119896)

(120572 and 120573)

= 119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573) for all positive integer 119896

(11)

Therefore

(

infin

⋀119896=1

119886119901119903(119896)

)(120572 and 120573) =

infin

⋀119896=1

119886119901119903(119896)

(120572 and 120573)

=

infin

⋀119896=1

(119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573))

= (

infin

⋀119896=1

119886119901119903(119896)

(120572)) and (

infin

⋀119896=1

119886119901119903(119896)

(120573))

(12)

It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator

Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871

Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has

(

infin

⋀119896=1

119886119901119903(119896)

) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)

(

infin

⋁119896=1

119886119901119903(119896)

) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)

Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879

119886119901119903 for any 120572 isin 119871 we have

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 120572

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903 (120572)

The Scientific World Journal 5

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(2)

(120572)

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(119896)

(120572)

(15)

implying that or120574 | 120574 le 120572 119886119901119903(120574) = 120574 le (⋀infin

119896=1119886119901119903(119896))(120572)

Conversely for any 120572 isin 119871

119886119901119903((

infin

⋀119896=1

119886119901119903(119896)

) (120572)) = (

infin

⋀119896=1

119886119901119903(119896)

) (120572) le 120572 (16)

It follows that (⋀infin

119896=1119886119901119903(119896))(120572) isin 119879

119886119901119903 and so (⋀

infin

119896=1119886119901119903(119896))(120572)

le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds

As a consequence ofTheorem 15 we obtain the followingresult

Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903

lowast= ⋁

infin

119896=1119886119901119903

(119896) is a closure operator on 119871

4 The Relationships between Pawlak Algebraand Auxiliary Ordering

Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately

The following results present the relationships betweenPawlak algebra and strong auxiliary ordering

Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903

≺

119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows

forall120572 isin 119871 119886119901119903≺

(120572) = or 120574 | 120574 ≺ 120572 120574 isin 119871

119886119901119903≺(120572) = (119886119901119903

≺

(120572119888

))119888

(17)

then (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

Proof It is obvious that condition (P1) holds In what followswe prove that conditions (P2)ndash(P4) are satisfied

(1) The strong auxiliary ordering ≺ implies that 1 ≺ 1 istrue and so we have

119886119901119903≺

(1) = or 120574 | 120574 ≺ 1 120574 isin 119871 = 1 (18)

implying that condition (P2) holds

(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺

120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120572 120574 isin 119871

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120573 120574 isin 119871 (19)

Therefore

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120572 120574 isin 119871

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120573 120574 isin 119871 (20)

It follows that 119886119901119903≺

(120572 and 120573) le 119886119901119903≺

(120572) and 119886119901119903≺

(120572 and 120573) le

119886119901119903≺

(120573) and hence

119886119901119903≺

(120572 and 120573) le (119886119901119903≺

(120572)) and (119886119901119903≺

(120573)) (21)

On the other hand by condition (iii) in Definition 3 wehave

or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572

or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573(22)

It follows from

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572 le 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573 le 120573

(23)

that

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120573(24)

Hence

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572 and 120573

(25)

implying that

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) isin 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 (26)

Thus 119886119901119903≺

(120572) and 119886119901119903≺

(120573) le 119886119901119903≺

(120572 and 120573) and so 119886119901119903≺

(120572) and

119886119901119903≺

(120573) = 119886119901119903≺

(120572 and 120573) Hence condition (P3) holds

(3) By the duality of approximation operators it sufficesto prove that 119886119901119903

≺

(120572) le 120572 And this is clearly true bythe definition of 119886119901119903

≺

(120572)

Summing up the above statements (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

6 The Scientific World Journal

Theorem18 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak algebraDefine a binary relation ldquo≺rdquo as follows

forall120572 120573 isin 119871 120572 ≺ 120573 lArrrArr 119886119901119903 (120572) le 120573 (27)

Then ldquo≺rdquo is an auxiliary ordering on 119871

Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573

(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582

(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus

119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)

implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary

ordering on 119871

5 Molecular Pawlak Algebraand Rough Topology

In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set

Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular

Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863

a molec-ular net and 120572 isin 120587 If there exists 119889

0isin 119863 such that 119886119901119903(120572) le

119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit

of 119878 denoted by 119878119886119901119903

997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878

In the sequel we provide two examples of molecularPawlak algebra in topology space

Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =

119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define

forall119860 isin 119871 119886119901119903119860 = ⋃119909isin119860

119861 (119909 1) 119886119901119903119860 = (119886119901119903119860119888

)119888

(29)

where 119861(119909 1) is the closed ball Then (Φ(119880)(120587) cup cap119888

119886119901119903

119886119901119903) is a molecular Pawlak algebra Let 1199090be the 119886119901119903-limit

of the molecular sequence 119909119899119899isin119873

(ie 119909119899

119886119901119903

997888997888rarr 1199090) which

means that there exists 1198990

isin 119873 such that 120588(119909119899 119909

0) le 1 for

119899 ge 119899119900 Obviously 119909

119899rarr 119909

0implies 119909

119899

119886119901119903

997888997888rarr 1199090(119899 rarr infin)

Example 21 Let119880 be a nonempty set119877 an equivalent relationon 119880 119871 = F(119880) fuzzy power set on 119880 120587 = 119909

120582 | 119909 isin 119880 120582 isin

(0 1] and119880119877 = 119883119894119894isin119868 For any119860 isin F(119880) define two fuzzy

sets 119865119860 and 119865119860on 119880119877 as

forall119894 isin 119868 119865119860

(119883119894) = sup 119860 (119909) | 119909 isin 119883

119894

119865119860(119883

119894) = inf 119860 (119909) | 119909 isin 119883

119894

(30)

Then we have for all 119860 119861 isin 119865(119880)

(1) 119865119860cup119861 = 119865119860 cup 119865119861

(2) 119865119860cap119861

= 119865119860cap 119865

119861

(3) 119865119860cap119861 sube 119865119860 cap 119865119861

(4) 119865119860cup119861

supe 119865119860cup 119865

119861

(5) 119865119860sube 119865119860

Moreover for any 119860 isin 119865(119880) define 119886119901119903119860 and 119886119901119903119860 as

(119886119901119903119860) (119909) = 119865119860 (119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(31)

(119886119901119903119860) (119909) = 119865119860(119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(32)

for any 119909 isin 119880 Then (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) is a molecular

Pawlak algebraIn the system (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) the rough neigh-

borhood 119886119901119903(119909120582) of molecule 119909

120582is represented as

119886119901119903 (119909120582) (119910) =

120582 119877 (119909 119910) = 1

0 119877 (119909 119910) = 0(33)

119878119886119901119903

997888997888rarr 119909120582if and only if there exists 119889

0isin 119863 such that 119909119877119910119889 and

120582119889le 120582 for 119889 ge 119889

0 where 119878(119889) = 119910119889

120582119889

and119883119894is poly-point set

If 119880119877 = 119909 | 119909 isin 119880 namely 119877(119909 119910) = 1 if and only if119909 = 119910 then we have

119886119901119903 (119909120582) = 119886119901119903 (119909

120582) = 119909

120582 (34)

for fuzzy point 119909120582and it is know that 119878

119886119901119903

997888997888rarr 119909120582if and only if

there exists 1198890isin 119863 such that 119878(119889) = 119909

120582for 119889 ge 119889

0

Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap

119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)

The following is now straightforward

Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers

Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra

The Scientific World Journal 7

Proof Set ℎ(0) = 0 and define

ℎ (120574) = ⋁120572le120574120572isin120587

ℎ (120572) ℎ (120574) = (ℎ (120574119888

))119888

(35)

Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra

6 The Application of ApproximationOperators to Rough Sets

Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880

119903 (119909) = 119910 | (119909 119910) isin 119877 (36)

and defined general dual approximation operators 119886119901119903119877

and119886119901119903

119877as for all119883 sube 119880

119886119901119903119877

(119883) = 119909 | 119903 (119909) sube 119883

119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =

(37)

In this section we further investigate the properties ofapproximation operators induced by a binary relation

The following results recall some basic properties ofapproximation operators induced by a binary relation

Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880

(R1) 119886119901119903119877

(119883) = [119886119901119903119877(119883119888)]

119888 119886119901119903119877(119883) = [119886119901119903

119877

(119883119888)]119888

(R2) 119886119901119903119877

() = 119886119901119903119877() = 119886119901119903

119877

(119880) = 119886119901119903119877(119880) = 119880

(R3) 119886119901119903119877

(119883 cap 119884) = 119886119901119903119877

(119883) cap 119886119901119903119877

(119884) 119886119901119903119877(119883 cap 119884) sube

119886119901119903119877(119883) cap 119886119901119903

119877(119884)

(R4) 119886119901119903119877

(119883 cup 119884) supe 119886119901119903119877

(119883) cup 119886119901119903119877

(119884) 119886119901119903119877(119883 cup 119884) =

119886119901119903119877(119883) cup 119886119901119903

119877(119884)

(R5) 119886119901119903119877

(119883) sube 119883 sube 119886119901119903119877(119883)

(R6) 119909 isin 119886119901119903119877119910 hArr 119910 isin 119886119901119903

119877119909

Proposition 25 indicates that the system (Φ(119880) cup

cap119888 119880 119886119901119903119877

119886119901119903119877) is a Pawlak algebra where Φ(119880)

denotes the set of all subsets of 119880

Proposition 26 (see [22]) Let 119877 sube 119880times119880 be a binary relationon 119880 Then

(1) 119877 is reflexive if and only if 119886119901119903119877

(119883) sube 119883 sube

119886119901119903119877(119883) (forall119883 sube 119880)

(2) 119877 is symmetric if and only if 119909 isin 119886119901119903119877119910 is equivalent

to 119910 isin 119886119901119903119877119909

(3) the reflexive relation 119877 is transitive if and only if 119886119901119903119877

is a closure operator

Proposition 27 Let (119901 119901) isin 119860119875119877(Φ(119880)) where119901 is a closureoperator Then

119909 isin 119901 119910 119894119891119891 119901 119909 sube 119901 119910 (38)

Proof It is straightforward and omitted

Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following

(1) the binary relation 119877119901defined as

119877119901= (119909 119910) | 119909 isin 119901 119910 (39)

is an equivalent relation on 119880(2) (119886119901119903

119877119901

119886119901119903119877119901

) ≺ (119901 119901) Moreover if 119880 is a finiteuniverse or the cover 119901119909

119909isin119880of 119880 is finite then

(119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Proof (1) It follows from Proposition 29 and condition (P6)that

119877119901= (119909 119910) | 119901 119909 = 119901 119910 (40)

Then it is easy to see that 119877119901is an equivalent relation where

[119909]119877= 119901119909 for all 119909 isin 119880

(2) For any119883 sube 119880 we have

119886119901119903119877119901

(119883) = 119909 | [119909]119877119901

cap 119883 =

= 119909 | 119910 | 119909 isin 119901 119910 cap 119883 =

= 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

sube 119901 (119883)

(41)

On the other hand 119886119901119903119877119901

(119883119888) sube 119901(119883119888) implies that(119901(119883119888))

119888

sube (119886119901119903119877119901

(119883119888))119888 hence 119901(119883) sube 119886119901119903

119877119901

(119883) Thus we

have (119886119901119903119877119901

119886119901119903119877119901

) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =

1199101 119910

2 119910

119898 sube 119880 Then we have

119901 (119883) = ⋃119910119896isin119883

119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901

= 119909 | [119909]119877119901

cap 119883 =

= 119886119901119903119877119901

(119883)

(42)

Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901

(119883) It

follows that (119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Now suppose that the cover 119901119909119909isin119880

of 119880 is finitethat is there exist 119909

119896isin 119880 119896 = 1 2 119898 such that

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

4 The Scientific World Journal

is an index set For any 119904 isin 119879 we have 120572119904ge ⋀

119905isin119879120572119905

and so 120572119904= 119886119901119903(120572

119904) ge 119886119901119903(⋀

119905isin119879120572119905) implying that

⋀119905isin119879

120572119905ge 119886119901119903(⋀

119905isin119879120572119905) By condition (P4) we have

⋀119905isin119879

120572119905= 119886119901119903(⋀

119905isin119879120572119905) as required

(c) 120572 isin 119871 | 120572 = 119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572)

is complement-closed For any 120572 isin 120572 isin 119871 | 120572 =

119886119901119903(120572) cap 120572 isin 119871 | 120572 = 119886119901119903(120572) by condition (P1)we have 119886119901119903(120572119888) = ((119886119901119903(120572119888))

119888

)119888 = (119886119901119903(120572))

119888

= 120572implying that 120572119888 isin 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 =

119886119901119903(120572) Hence 120572 | 120572 = 119886119901119903(120572) cap 120572 | 120572 = 119886119901119903(120572) iscomplement-closed

Summing up the above analysis (119871 1205901198861199011199030

(119871)) is a zero-dimensional lattice topology space

Now let us turn our attention to the study of the transitiveclosure of approximation operators

Let (119871 or and119888 0 1) be a fuzzy lattice Define the operatorldquo∘rdquo on 119860119875119877(119871) as follows

(1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) = (119886119901119903

1

∘ 1198861199011199032

1198861199011199031∘ 119886119901119903

2)

(8)

That is

forall120572 isin 119871 (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) (120572)

= (1198861199011199031

(1198861199011199032

(120572)) 1198861199011199031(119886119901119903

2(120572)))

(9)

where (1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871) For any positive

integer 119896 define (119886119901119903 119886119901119903)(119896+1)

= (119886119901119903 119886119901119903)(119896)

∘ (119886119901119903 119886119901119903)Denote cl(119886119901119903 119886119901119903) = ⋁

infin

119896=1(119886119901119903 119886119901119903)

(119896) which is called thetransitive closure of (119886119901119903 119886119901119903) Then we obtain the followingresult

Lemma 13 Let (119871 or and119888 0 1) be a fuzzy lattice Then

(1) (1198861199011199031

1198861199011199031) ≺ (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) for all

(1198861199011199031

1198861199011199031) (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(2) (1198861199011199031

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032) isin 119860119875119877(119871)

(3) cl(119886119901119903 119886119901119903) is a dual approximation operator

Proof (1) It is straightforward and omitted(2) It suffices to prove that (119886119901119903

1

1198861199011199031) ∘ (119886119901119903

2

1198861199011199032)

satisfies conditions (P1)ndash(P4) in Definition 7 In fact for any120572 120573 isin 119871 we have

(P1) 1198861199011199031

(1198861199011199032

(120572)) = 1198861199011199031((119886119901119903

2

(120572))119888

)119888

=

(1198861199011199031((119886119901119903

2(120572119888))

119888

)119888

)119888

= (1198861199011199031(119886119901119903

2(120572119888)))

119888(P2) (119886119901119903

1

∘ 1198861199011199032

)(1) = 1198861199011199031

(1198861199011199032

(1)) = 1198861199011199031

(1) = 1

(P3) (1198861199011199031

∘ 1198861199011199032

)(120572 and 120573) = 1198861199011199031

(1198861199011199032

(120572 and 120573))

= 1198861199011199031

(1198861199011199032

(120572) and 1198861199011199032

(120573)) = (1198861199011199031

(1198861199011199032

(120572))) and

(1198861199011199031

(1198861199011199032

(120573)))= (1198861199011199031

∘1198861199011199032

)(120572))and(1198861199011199031

∘1198861199011199032

)(120573))

(P4) forall120572 isin 119871 120572 le 1198861199011199032(120572) rArr 120572 le 119886119901119903

1(120572) le

1198861199011199031(119886119901119903

2(120572)) = (119886119901119903

1∘ 119886119901119903

2)(120572)

(3) From the proof of Theorem 10 we have

cl (119886119901119903 119886119901119903) = (

infin

⋀119896=1

119886119901119903(119896)

infin

⋁119896=1

119886119901119903(119896)

) (10)

And it is easy to prove by mathematical induction that

119886119901119903(119896)

(120572 and 120573)

= 119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573) for all positive integer 119896

(11)

Therefore

(

infin

⋀119896=1

119886119901119903(119896)

)(120572 and 120573) =

infin

⋀119896=1

119886119901119903(119896)

(120572 and 120573)

=

infin

⋀119896=1

(119886119901119903(119896)

(120572) and 119886119901119903(119896)

(120573))

= (

infin

⋀119896=1

119886119901119903(119896)

(120572)) and (

infin

⋀119896=1

119886119901119903(119896)

(120573))

(12)

It follows that cl(119886119901119903 119886119901119903) satisfies condition (P3) Similarlywe can prove that cl(119886119901119903 119886119901119903) also satisfies conditions (P1)(P2) and (P4) Hence cl(119886119901119903 119886119901119903) is a dual approximationoperator

Theorem 14 (see [21]) If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlakalgebra then the subset 119879apr = 120574 | 119886119901119903(120574) = 120574 is a latticetopology on 119871

Theorem 15 Let (119871 or and119888 0 1) be a fuzzy lattice and(119886119901119903 119886119901119903) isin 119860119875119877(119871) Then for any 120572 isin 119871 one has

(

infin

⋀119896=1

119886119901119903(119896)

) (120572) = or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 (13)

(

infin

⋁119896=1

119886119901119903(119896)

) (120572) = and 120591 | 120572 le 120591 119886119901119903 (120591) = 120591 (14)

Proof It suffices to prove (13) Equation (14) can be proved bythe duality of approximation operators By the infinite-union-closing in 119879

119886119901119903 for any 120572 isin 119871 we have

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 120572

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903 (120572)

The Scientific World Journal 5

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(2)

(120572)

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(119896)

(120572)

(15)

implying that or120574 | 120574 le 120572 119886119901119903(120574) = 120574 le (⋀infin

119896=1119886119901119903(119896))(120572)

Conversely for any 120572 isin 119871

119886119901119903((

infin

⋀119896=1

119886119901119903(119896)

) (120572)) = (

infin

⋀119896=1

119886119901119903(119896)

) (120572) le 120572 (16)

It follows that (⋀infin

119896=1119886119901119903(119896))(120572) isin 119879

119886119901119903 and so (⋀

infin

119896=1119886119901119903(119896))(120572)

le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds

As a consequence ofTheorem 15 we obtain the followingresult

Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903

lowast= ⋁

infin

119896=1119886119901119903

(119896) is a closure operator on 119871

4 The Relationships between Pawlak Algebraand Auxiliary Ordering

Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately

The following results present the relationships betweenPawlak algebra and strong auxiliary ordering

Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903

≺

119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows

forall120572 isin 119871 119886119901119903≺

(120572) = or 120574 | 120574 ≺ 120572 120574 isin 119871

119886119901119903≺(120572) = (119886119901119903

≺

(120572119888

))119888

(17)

then (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

Proof It is obvious that condition (P1) holds In what followswe prove that conditions (P2)ndash(P4) are satisfied

(1) The strong auxiliary ordering ≺ implies that 1 ≺ 1 istrue and so we have

119886119901119903≺

(1) = or 120574 | 120574 ≺ 1 120574 isin 119871 = 1 (18)

implying that condition (P2) holds

(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺

120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120572 120574 isin 119871

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120573 120574 isin 119871 (19)

Therefore

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120572 120574 isin 119871

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120573 120574 isin 119871 (20)

It follows that 119886119901119903≺

(120572 and 120573) le 119886119901119903≺

(120572) and 119886119901119903≺

(120572 and 120573) le

119886119901119903≺

(120573) and hence

119886119901119903≺

(120572 and 120573) le (119886119901119903≺

(120572)) and (119886119901119903≺

(120573)) (21)

On the other hand by condition (iii) in Definition 3 wehave

or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572

or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573(22)

It follows from

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572 le 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573 le 120573

(23)

that

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120573(24)

Hence

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572 and 120573

(25)

implying that

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) isin 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 (26)

Thus 119886119901119903≺

(120572) and 119886119901119903≺

(120573) le 119886119901119903≺

(120572 and 120573) and so 119886119901119903≺

(120572) and

119886119901119903≺

(120573) = 119886119901119903≺

(120572 and 120573) Hence condition (P3) holds

(3) By the duality of approximation operators it sufficesto prove that 119886119901119903

≺

(120572) le 120572 And this is clearly true bythe definition of 119886119901119903

≺

(120572)

Summing up the above statements (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

6 The Scientific World Journal

Theorem18 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak algebraDefine a binary relation ldquo≺rdquo as follows

forall120572 120573 isin 119871 120572 ≺ 120573 lArrrArr 119886119901119903 (120572) le 120573 (27)

Then ldquo≺rdquo is an auxiliary ordering on 119871

Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573

(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582

(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus

119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)

implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary

ordering on 119871

5 Molecular Pawlak Algebraand Rough Topology

In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set

Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular

Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863

a molec-ular net and 120572 isin 120587 If there exists 119889

0isin 119863 such that 119886119901119903(120572) le

119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit

of 119878 denoted by 119878119886119901119903

997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878

In the sequel we provide two examples of molecularPawlak algebra in topology space

Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =

119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define

forall119860 isin 119871 119886119901119903119860 = ⋃119909isin119860

119861 (119909 1) 119886119901119903119860 = (119886119901119903119860119888

)119888

(29)

where 119861(119909 1) is the closed ball Then (Φ(119880)(120587) cup cap119888

119886119901119903

119886119901119903) is a molecular Pawlak algebra Let 1199090be the 119886119901119903-limit

of the molecular sequence 119909119899119899isin119873

(ie 119909119899

119886119901119903

997888997888rarr 1199090) which

means that there exists 1198990

isin 119873 such that 120588(119909119899 119909

0) le 1 for

119899 ge 119899119900 Obviously 119909

119899rarr 119909

0implies 119909

119899

119886119901119903

997888997888rarr 1199090(119899 rarr infin)

Example 21 Let119880 be a nonempty set119877 an equivalent relationon 119880 119871 = F(119880) fuzzy power set on 119880 120587 = 119909

120582 | 119909 isin 119880 120582 isin

(0 1] and119880119877 = 119883119894119894isin119868 For any119860 isin F(119880) define two fuzzy

sets 119865119860 and 119865119860on 119880119877 as

forall119894 isin 119868 119865119860

(119883119894) = sup 119860 (119909) | 119909 isin 119883

119894

119865119860(119883

119894) = inf 119860 (119909) | 119909 isin 119883

119894

(30)

Then we have for all 119860 119861 isin 119865(119880)

(1) 119865119860cup119861 = 119865119860 cup 119865119861

(2) 119865119860cap119861

= 119865119860cap 119865

119861

(3) 119865119860cap119861 sube 119865119860 cap 119865119861

(4) 119865119860cup119861

supe 119865119860cup 119865

119861

(5) 119865119860sube 119865119860

Moreover for any 119860 isin 119865(119880) define 119886119901119903119860 and 119886119901119903119860 as

(119886119901119903119860) (119909) = 119865119860 (119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(31)

(119886119901119903119860) (119909) = 119865119860(119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(32)

for any 119909 isin 119880 Then (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) is a molecular

Pawlak algebraIn the system (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) the rough neigh-

borhood 119886119901119903(119909120582) of molecule 119909

120582is represented as

119886119901119903 (119909120582) (119910) =

120582 119877 (119909 119910) = 1

0 119877 (119909 119910) = 0(33)

119878119886119901119903

997888997888rarr 119909120582if and only if there exists 119889

0isin 119863 such that 119909119877119910119889 and

120582119889le 120582 for 119889 ge 119889

0 where 119878(119889) = 119910119889

120582119889

and119883119894is poly-point set

If 119880119877 = 119909 | 119909 isin 119880 namely 119877(119909 119910) = 1 if and only if119909 = 119910 then we have

119886119901119903 (119909120582) = 119886119901119903 (119909

120582) = 119909

120582 (34)

for fuzzy point 119909120582and it is know that 119878

119886119901119903

997888997888rarr 119909120582if and only if

there exists 1198890isin 119863 such that 119878(119889) = 119909

120582for 119889 ge 119889

0

Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap

119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)

The following is now straightforward

Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers

Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra

The Scientific World Journal 7

Proof Set ℎ(0) = 0 and define

ℎ (120574) = ⋁120572le120574120572isin120587

ℎ (120572) ℎ (120574) = (ℎ (120574119888

))119888

(35)

Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra

6 The Application of ApproximationOperators to Rough Sets

Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880

119903 (119909) = 119910 | (119909 119910) isin 119877 (36)

and defined general dual approximation operators 119886119901119903119877

and119886119901119903

119877as for all119883 sube 119880

119886119901119903119877

(119883) = 119909 | 119903 (119909) sube 119883

119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =

(37)

In this section we further investigate the properties ofapproximation operators induced by a binary relation

The following results recall some basic properties ofapproximation operators induced by a binary relation

Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880

(R1) 119886119901119903119877

(119883) = [119886119901119903119877(119883119888)]

119888 119886119901119903119877(119883) = [119886119901119903

119877

(119883119888)]119888

(R2) 119886119901119903119877

() = 119886119901119903119877() = 119886119901119903

119877

(119880) = 119886119901119903119877(119880) = 119880

(R3) 119886119901119903119877

(119883 cap 119884) = 119886119901119903119877

(119883) cap 119886119901119903119877

(119884) 119886119901119903119877(119883 cap 119884) sube

119886119901119903119877(119883) cap 119886119901119903

119877(119884)

(R4) 119886119901119903119877

(119883 cup 119884) supe 119886119901119903119877

(119883) cup 119886119901119903119877

(119884) 119886119901119903119877(119883 cup 119884) =

119886119901119903119877(119883) cup 119886119901119903

119877(119884)

(R5) 119886119901119903119877

(119883) sube 119883 sube 119886119901119903119877(119883)

(R6) 119909 isin 119886119901119903119877119910 hArr 119910 isin 119886119901119903

119877119909

Proposition 25 indicates that the system (Φ(119880) cup

cap119888 119880 119886119901119903119877

119886119901119903119877) is a Pawlak algebra where Φ(119880)

denotes the set of all subsets of 119880

Proposition 26 (see [22]) Let 119877 sube 119880times119880 be a binary relationon 119880 Then

(1) 119877 is reflexive if and only if 119886119901119903119877

(119883) sube 119883 sube

119886119901119903119877(119883) (forall119883 sube 119880)

(2) 119877 is symmetric if and only if 119909 isin 119886119901119903119877119910 is equivalent

to 119910 isin 119886119901119903119877119909

(3) the reflexive relation 119877 is transitive if and only if 119886119901119903119877

is a closure operator

Proposition 27 Let (119901 119901) isin 119860119875119877(Φ(119880)) where119901 is a closureoperator Then

119909 isin 119901 119910 119894119891119891 119901 119909 sube 119901 119910 (38)

Proof It is straightforward and omitted

Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following

(1) the binary relation 119877119901defined as

119877119901= (119909 119910) | 119909 isin 119901 119910 (39)

is an equivalent relation on 119880(2) (119886119901119903

119877119901

119886119901119903119877119901

) ≺ (119901 119901) Moreover if 119880 is a finiteuniverse or the cover 119901119909

119909isin119880of 119880 is finite then

(119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Proof (1) It follows from Proposition 29 and condition (P6)that

119877119901= (119909 119910) | 119901 119909 = 119901 119910 (40)

Then it is easy to see that 119877119901is an equivalent relation where

[119909]119877= 119901119909 for all 119909 isin 119880

(2) For any119883 sube 119880 we have

119886119901119903119877119901

(119883) = 119909 | [119909]119877119901

cap 119883 =

= 119909 | 119910 | 119909 isin 119901 119910 cap 119883 =

= 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

sube 119901 (119883)

(41)

On the other hand 119886119901119903119877119901

(119883119888) sube 119901(119883119888) implies that(119901(119883119888))

119888

sube (119886119901119903119877119901

(119883119888))119888 hence 119901(119883) sube 119886119901119903

119877119901

(119883) Thus we

have (119886119901119903119877119901

119886119901119903119877119901

) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =

1199101 119910

2 119910

119898 sube 119880 Then we have

119901 (119883) = ⋃119910119896isin119883

119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901

= 119909 | [119909]119877119901

cap 119883 =

= 119886119901119903119877119901

(119883)

(42)

Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901

(119883) It

follows that (119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Now suppose that the cover 119901119909119909isin119880

of 119880 is finitethat is there exist 119909

119896isin 119880 119896 = 1 2 119898 such that

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

The Scientific World Journal 5

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(2)

(120572)

119886119901119903 [or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574]

= or 120574 | 120574 le 120572 119886119901119903 (120574) = 120574 le 119886119901119903(119896)

(120572)

(15)

implying that or120574 | 120574 le 120572 119886119901119903(120574) = 120574 le (⋀infin

119896=1119886119901119903(119896))(120572)

Conversely for any 120572 isin 119871

119886119901119903((

infin

⋀119896=1

119886119901119903(119896)

) (120572)) = (

infin

⋀119896=1

119886119901119903(119896)

) (120572) le 120572 (16)

It follows that (⋀infin

119896=1119886119901119903(119896))(120572) isin 119879

119886119901119903 and so (⋀

infin

119896=1119886119901119903(119896))(120572)

le or120574 | 120574 le 120572 119886119901119903(120574) = 120574 Hence (13) holds

As a consequence ofTheorem 15 we obtain the followingresult

Corollary 16 If (119871 or and119888 0 1 119886119901119903 119886119901119903) is a Pawlak algebrathen 119886119901119903

lowast= ⋁

infin

119896=1119886119901119903

(119896) is a closure operator on 119871

4 The Relationships between Pawlak Algebraand Auxiliary Ordering

Gierz [19] introduces the concept of auxiliary ordering for thestudy of continuous lattice In fact the auxiliary ordering isan order relation which is rougher than the initial orderingand can be regarded as the approximation of initial orderingInspired by this idea we can use approximation operator todescribe order approximately

The following results present the relationships betweenPawlak algebra and strong auxiliary ordering

Theorem 17 Let (119871 or and119888 0 1) be a fuzzy lattice and ldquo≺rdquo astrong auxiliary ordering on 119871 Define two operators 119886119901119903

≺

119871 rarr 119871 and 119886119901119903≺ 119871 rarr 119871 as follows

forall120572 isin 119871 119886119901119903≺

(120572) = or 120574 | 120574 ≺ 120572 120574 isin 119871

119886119901119903≺(120572) = (119886119901119903

≺

(120572119888

))119888

(17)

then (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

Proof It is obvious that condition (P1) holds In what followswe prove that conditions (P2)ndash(P4) are satisfied

(1) The strong auxiliary ordering ≺ implies that 1 ≺ 1 istrue and so we have

119886119901119903≺

(1) = or 120574 | 120574 ≺ 1 120574 isin 119871 = 1 (18)

implying that condition (P2) holds

(2) Let 120572 120573 120574 isin 119871 be such that 120574 ≺ 120572 and 120573 Then 120574 le 120574 ≺

120572 and 120573 le 120572 and 120574 le 120574 ≺ 120572 and 120573 le 120573 By condition (ii) inDefinition 1 we have 120574 ≺ 120572 120574 ≺ 120573 and so

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120572 120574 isin 119871

120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 sube 120574 | 120574 ≺ 120573 120574 isin 119871 (19)

Therefore

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120572 120574 isin 119871

or 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 le or 120574 | 120574 ≺ 120573 120574 isin 119871 (20)

It follows that 119886119901119903≺

(120572 and 120573) le 119886119901119903≺

(120572) and 119886119901119903≺

(120572 and 120573) le

119886119901119903≺

(120573) and hence

119886119901119903≺

(120572 and 120573) le (119886119901119903≺

(120572)) and (119886119901119903≺

(120573)) (21)

On the other hand by condition (iii) in Definition 3 wehave

or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572

or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573(22)

It follows from

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120572 120574 isin 119871 ≺ 120572 le 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) le or 120574 | 120574 ≺ 120573 120574 isin 119871 ≺ 120573 le 120573

(23)

that

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120573(24)

Hence

(or 120574 | 120574 ≺ 120572 120574 isin 119871) and (or 120574 | 120574 ≺ 120573 120574 isin 119871) ≺ 120572 and 120573

(25)

implying that

(or 120574 | 120574 ≺ 120572 120574 isin 119871)

and (or 120574 | 120574 ≺ 120573 120574 isin 119871) isin 120574 | 120574 ≺ 120572 and 120573 120574 isin 119871 (26)

Thus 119886119901119903≺

(120572) and 119886119901119903≺

(120573) le 119886119901119903≺

(120572 and 120573) and so 119886119901119903≺

(120572) and

119886119901119903≺

(120573) = 119886119901119903≺

(120572 and 120573) Hence condition (P3) holds

(3) By the duality of approximation operators it sufficesto prove that 119886119901119903

≺

(120572) le 120572 And this is clearly true bythe definition of 119886119901119903

≺

(120572)

Summing up the above statements (119871 or and119888 0 1 119886119901119903≺

119886119901119903≺) is a Pawlak algebra

6 The Scientific World Journal

Theorem18 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak algebraDefine a binary relation ldquo≺rdquo as follows

forall120572 120573 isin 119871 120572 ≺ 120573 lArrrArr 119886119901119903 (120572) le 120573 (27)

Then ldquo≺rdquo is an auxiliary ordering on 119871

Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573

(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582

(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus

119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)

implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary

ordering on 119871

5 Molecular Pawlak Algebraand Rough Topology

In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set

Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular

Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863

a molec-ular net and 120572 isin 120587 If there exists 119889

0isin 119863 such that 119886119901119903(120572) le

119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit

of 119878 denoted by 119878119886119901119903

997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878

In the sequel we provide two examples of molecularPawlak algebra in topology space

Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =

119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define

forall119860 isin 119871 119886119901119903119860 = ⋃119909isin119860

119861 (119909 1) 119886119901119903119860 = (119886119901119903119860119888

)119888

(29)

where 119861(119909 1) is the closed ball Then (Φ(119880)(120587) cup cap119888

119886119901119903

119886119901119903) is a molecular Pawlak algebra Let 1199090be the 119886119901119903-limit

of the molecular sequence 119909119899119899isin119873

(ie 119909119899

119886119901119903

997888997888rarr 1199090) which

means that there exists 1198990

isin 119873 such that 120588(119909119899 119909

0) le 1 for

119899 ge 119899119900 Obviously 119909

119899rarr 119909

0implies 119909

119899

119886119901119903

997888997888rarr 1199090(119899 rarr infin)

Example 21 Let119880 be a nonempty set119877 an equivalent relationon 119880 119871 = F(119880) fuzzy power set on 119880 120587 = 119909

120582 | 119909 isin 119880 120582 isin

(0 1] and119880119877 = 119883119894119894isin119868 For any119860 isin F(119880) define two fuzzy

sets 119865119860 and 119865119860on 119880119877 as

forall119894 isin 119868 119865119860

(119883119894) = sup 119860 (119909) | 119909 isin 119883

119894

119865119860(119883

119894) = inf 119860 (119909) | 119909 isin 119883

119894

(30)

Then we have for all 119860 119861 isin 119865(119880)

(1) 119865119860cup119861 = 119865119860 cup 119865119861

(2) 119865119860cap119861

= 119865119860cap 119865

119861

(3) 119865119860cap119861 sube 119865119860 cap 119865119861

(4) 119865119860cup119861

supe 119865119860cup 119865

119861

(5) 119865119860sube 119865119860

Moreover for any 119860 isin 119865(119880) define 119886119901119903119860 and 119886119901119903119860 as

(119886119901119903119860) (119909) = 119865119860 (119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(31)

(119886119901119903119860) (119909) = 119865119860(119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(32)

for any 119909 isin 119880 Then (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) is a molecular

Pawlak algebraIn the system (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) the rough neigh-

borhood 119886119901119903(119909120582) of molecule 119909

120582is represented as

119886119901119903 (119909120582) (119910) =

120582 119877 (119909 119910) = 1

0 119877 (119909 119910) = 0(33)

119878119886119901119903

997888997888rarr 119909120582if and only if there exists 119889

0isin 119863 such that 119909119877119910119889 and

120582119889le 120582 for 119889 ge 119889

0 where 119878(119889) = 119910119889

120582119889

and119883119894is poly-point set

If 119880119877 = 119909 | 119909 isin 119880 namely 119877(119909 119910) = 1 if and only if119909 = 119910 then we have

119886119901119903 (119909120582) = 119886119901119903 (119909

120582) = 119909

120582 (34)

for fuzzy point 119909120582and it is know that 119878

119886119901119903

997888997888rarr 119909120582if and only if

there exists 1198890isin 119863 such that 119878(119889) = 119909

120582for 119889 ge 119889

0

Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap

119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)

The following is now straightforward

Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers

Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra

The Scientific World Journal 7

Proof Set ℎ(0) = 0 and define

ℎ (120574) = ⋁120572le120574120572isin120587

ℎ (120572) ℎ (120574) = (ℎ (120574119888

))119888

(35)

Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra

6 The Application of ApproximationOperators to Rough Sets

Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880

119903 (119909) = 119910 | (119909 119910) isin 119877 (36)

and defined general dual approximation operators 119886119901119903119877

and119886119901119903

119877as for all119883 sube 119880

119886119901119903119877

(119883) = 119909 | 119903 (119909) sube 119883

119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =

(37)

In this section we further investigate the properties ofapproximation operators induced by a binary relation

The following results recall some basic properties ofapproximation operators induced by a binary relation

Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880

(R1) 119886119901119903119877

(119883) = [119886119901119903119877(119883119888)]

119888 119886119901119903119877(119883) = [119886119901119903

119877

(119883119888)]119888

(R2) 119886119901119903119877

() = 119886119901119903119877() = 119886119901119903

119877

(119880) = 119886119901119903119877(119880) = 119880

(R3) 119886119901119903119877

(119883 cap 119884) = 119886119901119903119877

(119883) cap 119886119901119903119877

(119884) 119886119901119903119877(119883 cap 119884) sube

119886119901119903119877(119883) cap 119886119901119903

119877(119884)

(R4) 119886119901119903119877

(119883 cup 119884) supe 119886119901119903119877

(119883) cup 119886119901119903119877

(119884) 119886119901119903119877(119883 cup 119884) =

119886119901119903119877(119883) cup 119886119901119903

119877(119884)

(R5) 119886119901119903119877

(119883) sube 119883 sube 119886119901119903119877(119883)

(R6) 119909 isin 119886119901119903119877119910 hArr 119910 isin 119886119901119903

119877119909

Proposition 25 indicates that the system (Φ(119880) cup

cap119888 119880 119886119901119903119877

119886119901119903119877) is a Pawlak algebra where Φ(119880)

denotes the set of all subsets of 119880

Proposition 26 (see [22]) Let 119877 sube 119880times119880 be a binary relationon 119880 Then

(1) 119877 is reflexive if and only if 119886119901119903119877

(119883) sube 119883 sube

119886119901119903119877(119883) (forall119883 sube 119880)

(2) 119877 is symmetric if and only if 119909 isin 119886119901119903119877119910 is equivalent

to 119910 isin 119886119901119903119877119909

(3) the reflexive relation 119877 is transitive if and only if 119886119901119903119877

is a closure operator

Proposition 27 Let (119901 119901) isin 119860119875119877(Φ(119880)) where119901 is a closureoperator Then

119909 isin 119901 119910 119894119891119891 119901 119909 sube 119901 119910 (38)

Proof It is straightforward and omitted

Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following

(1) the binary relation 119877119901defined as

119877119901= (119909 119910) | 119909 isin 119901 119910 (39)

is an equivalent relation on 119880(2) (119886119901119903

119877119901

119886119901119903119877119901

) ≺ (119901 119901) Moreover if 119880 is a finiteuniverse or the cover 119901119909

119909isin119880of 119880 is finite then

(119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Proof (1) It follows from Proposition 29 and condition (P6)that

119877119901= (119909 119910) | 119901 119909 = 119901 119910 (40)

Then it is easy to see that 119877119901is an equivalent relation where

[119909]119877= 119901119909 for all 119909 isin 119880

(2) For any119883 sube 119880 we have

119886119901119903119877119901

(119883) = 119909 | [119909]119877119901

cap 119883 =

= 119909 | 119910 | 119909 isin 119901 119910 cap 119883 =

= 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

sube 119901 (119883)

(41)

On the other hand 119886119901119903119877119901

(119883119888) sube 119901(119883119888) implies that(119901(119883119888))

119888

sube (119886119901119903119877119901

(119883119888))119888 hence 119901(119883) sube 119886119901119903

119877119901

(119883) Thus we

have (119886119901119903119877119901

119886119901119903119877119901

) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =

1199101 119910

2 119910

119898 sube 119880 Then we have

119901 (119883) = ⋃119910119896isin119883

119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901

= 119909 | [119909]119877119901

cap 119883 =

= 119886119901119903119877119901

(119883)

(42)

Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901

(119883) It

follows that (119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Now suppose that the cover 119901119909119909isin119880

of 119880 is finitethat is there exist 119909

119896isin 119880 119896 = 1 2 119898 such that

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

6 The Scientific World Journal

Theorem18 Let (119871 or and119888 0 1 119886119901119903 119886119901119903) be a Pawlak algebraDefine a binary relation ldquo≺rdquo as follows

forall120572 120573 isin 119871 120572 ≺ 120573 lArrrArr 119886119901119903 (120572) le 120573 (27)

Then ldquo≺rdquo is an auxiliary ordering on 119871

Proof (1) Let 120572 120573 isin 119871be such that 120572 ≺ 120573 Then 120572 le 120573 since120572 le 119886119901119903(120572) and 119886119901119903(120572) le 120573

(2) Let 120572 120573 120578 120582 isin 119871 be such that 120578 le 120572 ≺ 120573 le 120582 Then119886119901119903(120578) le 119886119901119903(120572) and 119886119901119903(120572) le 120573 le 120582 It follows that 119886119901119903(120578) le120582 that is 120578 ≺ 120582

(3) Let 120572 120573 120582 isin 119871 be such that 120572 ≺ 120574 and 120573 ≺ 120574 Then119886119901119903(120572) le 120574 and 119886119901119903(120573) le 120574 Thus

119886119901119903 (120572 or 120573) = 119886119901119903 (120572) or 119886119901119903 (120573) le 120574 (28)

implying that 120572 or 120573 ≺ 120574(4) It follows from 0 = 119886119901119903(0) le 120572 that 0 ≺ 120572 for all 120572 isin 119871Summing up the above statements ldquo≺rdquo is an auxiliary

ordering on 119871

5 Molecular Pawlak Algebraand Rough Topology

In this section we focus on the approximate structure onfuzzy lattices This structure can be regarded as abstractsystem of rough set

Definition 19 Let (119871(120587) or and119888 119886119901119903 119886119901119903) be a molecular

Pawlak algebra (119863 ge) a directed set and 119878(119889)119889isin119863

a molec-ular net and 120572 isin 120587 If there exists 119889

0isin 119863 such that 119886119901119903(120572) le

119878(119889) le 119886119901119903(120572) for any 119889 ge 1198890 then 120572 is called a rough limit

of 119878 denoted by 119878119886119901119903

997888997888rarr 120572 The set of all rough limits of 119878 isdenoted by lim 119878

In the sequel we provide two examples of molecularPawlak algebra in topology space

Example 20 Let (119880 120588) be a metric space 119871 = Φ(119880) and 120587 =

119909 | 119909 isin 119880 where Φ(119880) denotes the set of all subsets of 119880Define

forall119860 isin 119871 119886119901119903119860 = ⋃119909isin119860

119861 (119909 1) 119886119901119903119860 = (119886119901119903119860119888

)119888

(29)

where 119861(119909 1) is the closed ball Then (Φ(119880)(120587) cup cap119888

119886119901119903

119886119901119903) is a molecular Pawlak algebra Let 1199090be the 119886119901119903-limit

of the molecular sequence 119909119899119899isin119873

(ie 119909119899

119886119901119903

997888997888rarr 1199090) which

means that there exists 1198990

isin 119873 such that 120588(119909119899 119909

0) le 1 for

119899 ge 119899119900 Obviously 119909

119899rarr 119909

0implies 119909

119899

119886119901119903

997888997888rarr 1199090(119899 rarr infin)

Example 21 Let119880 be a nonempty set119877 an equivalent relationon 119880 119871 = F(119880) fuzzy power set on 119880 120587 = 119909

120582 | 119909 isin 119880 120582 isin

(0 1] and119880119877 = 119883119894119894isin119868 For any119860 isin F(119880) define two fuzzy

sets 119865119860 and 119865119860on 119880119877 as

forall119894 isin 119868 119865119860

(119883119894) = sup 119860 (119909) | 119909 isin 119883

119894

119865119860(119883

119894) = inf 119860 (119909) | 119909 isin 119883

119894

(30)

Then we have for all 119860 119861 isin 119865(119880)

(1) 119865119860cup119861 = 119865119860 cup 119865119861

(2) 119865119860cap119861

= 119865119860cap 119865

119861

(3) 119865119860cap119861 sube 119865119860 cap 119865119861

(4) 119865119860cup119861

supe 119865119860cup 119865

119861

(5) 119865119860sube 119865119860

Moreover for any 119860 isin 119865(119880) define 119886119901119903119860 and 119886119901119903119860 as

(119886119901119903119860) (119909) = 119865119860 (119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(31)

(119886119901119903119860) (119909) = 119865119860(119883

119894) 119909 isin 119883

119894

0 119909 notin 119883119894

(32)

for any 119909 isin 119880 Then (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) is a molecular

Pawlak algebraIn the system (F(119880)(120587) cup cap119888 119886119901119903 119886119901119903) the rough neigh-

borhood 119886119901119903(119909120582) of molecule 119909

120582is represented as

119886119901119903 (119909120582) (119910) =

120582 119877 (119909 119910) = 1

0 119877 (119909 119910) = 0(33)

119878119886119901119903

997888997888rarr 119909120582if and only if there exists 119889

0isin 119863 such that 119909119877119910119889 and

120582119889le 120582 for 119889 ge 119889

0 where 119878(119889) = 119910119889

120582119889

and119883119894is poly-point set

If 119880119877 = 119909 | 119909 isin 119880 namely 119877(119909 119910) = 1 if and only if119909 = 119910 then we have

119886119901119903 (119909120582) = 119886119901119903 (119909

120582) = 119909

120582 (34)

for fuzzy point 119909120582and it is know that 119878

119886119901119903

997888997888rarr 119909120582if and only if

there exists 1198890isin 119863 such that 119878(119889) = 119909

120582for 119889 ge 119889

0

Definition 22 A molecular net is called 119877-fuzzy-rough-convergent if and only if it is 119886119901119903-convergent in the molecularPawlak algebra (F(119880)(120587) cup cap

119888 119886119901119903 119886119901119903) defined by formu-las (31) and (32)

The following is now straightforward

Proposition 23 The 119877-fuzzy-rough-convergent classes satisfythe Moore-Smith conditions can induce a topology called 119877-rough fuzzy topology which is a nullity topology with squaremembers

Theorem 24 Let (119871(120587) or and119888 0 1) be a molecular latticeThen any mapping ℎ 120587 rarr 119871 satisfying 120572 le ℎ(120572) can atleast induce a molecular Pawlak algebra

The Scientific World Journal 7

Proof Set ℎ(0) = 0 and define

ℎ (120574) = ⋁120572le120574120572isin120587

ℎ (120572) ℎ (120574) = (ℎ (120574119888

))119888

(35)

Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra

6 The Application of ApproximationOperators to Rough Sets

Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880

119903 (119909) = 119910 | (119909 119910) isin 119877 (36)

and defined general dual approximation operators 119886119901119903119877

and119886119901119903

119877as for all119883 sube 119880

119886119901119903119877

(119883) = 119909 | 119903 (119909) sube 119883

119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =

(37)

In this section we further investigate the properties ofapproximation operators induced by a binary relation

The following results recall some basic properties ofapproximation operators induced by a binary relation

Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880

(R1) 119886119901119903119877

(119883) = [119886119901119903119877(119883119888)]

119888 119886119901119903119877(119883) = [119886119901119903

119877

(119883119888)]119888

(R2) 119886119901119903119877

() = 119886119901119903119877() = 119886119901119903

119877

(119880) = 119886119901119903119877(119880) = 119880

(R3) 119886119901119903119877

(119883 cap 119884) = 119886119901119903119877

(119883) cap 119886119901119903119877

(119884) 119886119901119903119877(119883 cap 119884) sube

119886119901119903119877(119883) cap 119886119901119903

119877(119884)

(R4) 119886119901119903119877

(119883 cup 119884) supe 119886119901119903119877

(119883) cup 119886119901119903119877

(119884) 119886119901119903119877(119883 cup 119884) =

119886119901119903119877(119883) cup 119886119901119903

119877(119884)

(R5) 119886119901119903119877

(119883) sube 119883 sube 119886119901119903119877(119883)

(R6) 119909 isin 119886119901119903119877119910 hArr 119910 isin 119886119901119903

119877119909

Proposition 25 indicates that the system (Φ(119880) cup

cap119888 119880 119886119901119903119877

119886119901119903119877) is a Pawlak algebra where Φ(119880)

denotes the set of all subsets of 119880

Proposition 26 (see [22]) Let 119877 sube 119880times119880 be a binary relationon 119880 Then

(1) 119877 is reflexive if and only if 119886119901119903119877

(119883) sube 119883 sube

119886119901119903119877(119883) (forall119883 sube 119880)

(2) 119877 is symmetric if and only if 119909 isin 119886119901119903119877119910 is equivalent

to 119910 isin 119886119901119903119877119909

(3) the reflexive relation 119877 is transitive if and only if 119886119901119903119877

is a closure operator

Proposition 27 Let (119901 119901) isin 119860119875119877(Φ(119880)) where119901 is a closureoperator Then

119909 isin 119901 119910 119894119891119891 119901 119909 sube 119901 119910 (38)

Proof It is straightforward and omitted

Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following

(1) the binary relation 119877119901defined as

119877119901= (119909 119910) | 119909 isin 119901 119910 (39)

is an equivalent relation on 119880(2) (119886119901119903

119877119901

119886119901119903119877119901

) ≺ (119901 119901) Moreover if 119880 is a finiteuniverse or the cover 119901119909

119909isin119880of 119880 is finite then

(119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Proof (1) It follows from Proposition 29 and condition (P6)that

119877119901= (119909 119910) | 119901 119909 = 119901 119910 (40)

Then it is easy to see that 119877119901is an equivalent relation where

[119909]119877= 119901119909 for all 119909 isin 119880

(2) For any119883 sube 119880 we have

119886119901119903119877119901

(119883) = 119909 | [119909]119877119901

cap 119883 =

= 119909 | 119910 | 119909 isin 119901 119910 cap 119883 =

= 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

sube 119901 (119883)

(41)

On the other hand 119886119901119903119877119901

(119883119888) sube 119901(119883119888) implies that(119901(119883119888))

119888

sube (119886119901119903119877119901

(119883119888))119888 hence 119901(119883) sube 119886119901119903

119877119901

(119883) Thus we

have (119886119901119903119877119901

119886119901119903119877119901

) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =

1199101 119910

2 119910

119898 sube 119880 Then we have

119901 (119883) = ⋃119910119896isin119883

119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901

= 119909 | [119909]119877119901

cap 119883 =

= 119886119901119903119877119901

(119883)

(42)

Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901

(119883) It

follows that (119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Now suppose that the cover 119901119909119909isin119880

of 119880 is finitethat is there exist 119909

119896isin 119880 119896 = 1 2 119898 such that

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

The Scientific World Journal 7

Proof Set ℎ(0) = 0 and define

ℎ (120574) = ⋁120572le120574120572isin120587

ℎ (120572) ℎ (120574) = (ℎ (120574119888

))119888

(35)

Then it is evident that (119871(120587) or and119888 ℎ ℎ) is amolecular Pawlakalgebra

6 The Application of ApproximationOperators to Rough Sets

Yao and Lin [16] introduce the concept of rough sets withgeneral binary relation They defined 119877-neighborhood of 119909from a universe 119880 by the binary relation 119877 on 119880

119903 (119909) = 119910 | (119909 119910) isin 119877 (36)

and defined general dual approximation operators 119886119901119903119877

and119886119901119903

119877as for all119883 sube 119880

119886119901119903119877

(119883) = 119909 | 119903 (119909) sube 119883

119886119901119903119877(119883) = 119909 | 119903 (119909) cap 119883 =

(37)

In this section we further investigate the properties ofapproximation operators induced by a binary relation

The following results recall some basic properties ofapproximation operators induced by a binary relation

Proposition 25 (see [16]) Let 119877 sube 119880times119880 be a similar relationon 119880 Then the following assertions hold for all 119909 119910 isin 119880 and119883 119884 sube 119880

(R1) 119886119901119903119877

(119883) = [119886119901119903119877(119883119888)]

119888 119886119901119903119877(119883) = [119886119901119903

119877

(119883119888)]119888

(R2) 119886119901119903119877

() = 119886119901119903119877() = 119886119901119903

119877

(119880) = 119886119901119903119877(119880) = 119880

(R3) 119886119901119903119877

(119883 cap 119884) = 119886119901119903119877

(119883) cap 119886119901119903119877

(119884) 119886119901119903119877(119883 cap 119884) sube

119886119901119903119877(119883) cap 119886119901119903

119877(119884)

(R4) 119886119901119903119877

(119883 cup 119884) supe 119886119901119903119877

(119883) cup 119886119901119903119877

(119884) 119886119901119903119877(119883 cup 119884) =

119886119901119903119877(119883) cup 119886119901119903

119877(119884)

(R5) 119886119901119903119877

(119883) sube 119883 sube 119886119901119903119877(119883)

(R6) 119909 isin 119886119901119903119877119910 hArr 119910 isin 119886119901119903

119877119909

Proposition 25 indicates that the system (Φ(119880) cup

cap119888 119880 119886119901119903119877

119886119901119903119877) is a Pawlak algebra where Φ(119880)

denotes the set of all subsets of 119880

Proposition 26 (see [22]) Let 119877 sube 119880times119880 be a binary relationon 119880 Then

(1) 119877 is reflexive if and only if 119886119901119903119877

(119883) sube 119883 sube

119886119901119903119877(119883) (forall119883 sube 119880)

(2) 119877 is symmetric if and only if 119909 isin 119886119901119903119877119910 is equivalent

to 119910 isin 119886119901119903119877119909

(3) the reflexive relation 119877 is transitive if and only if 119886119901119903119877

is a closure operator

Proposition 27 Let (119901 119901) isin 119860119875119877(Φ(119880)) where119901 is a closureoperator Then

119909 isin 119901 119910 119894119891119891 119901 119909 sube 119901 119910 (38)

Proof It is straightforward and omitted

Theorem 28 Let (Φ(119880) cup cap119888 119880 119901 119901) be a Pawlak alge-bra where 119901 is a closure operator that satisfies (P6) for all119909 119910 isin 119880 119909 isin 119901119910 hArr 119910 isin 119901119909 Then we have the following

(1) the binary relation 119877119901defined as

119877119901= (119909 119910) | 119909 isin 119901 119910 (39)

is an equivalent relation on 119880(2) (119886119901119903

119877119901

119886119901119903119877119901

) ≺ (119901 119901) Moreover if 119880 is a finiteuniverse or the cover 119901119909

119909isin119880of 119880 is finite then

(119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Proof (1) It follows from Proposition 29 and condition (P6)that

119877119901= (119909 119910) | 119901 119909 = 119901 119910 (40)

Then it is easy to see that 119877119901is an equivalent relation where

[119909]119877= 119901119909 for all 119909 isin 119880

(2) For any119883 sube 119880 we have

119886119901119903119877119901

(119883) = 119909 | [119909]119877119901

cap 119883 =

= 119909 | 119910 | 119909 isin 119901 119910 cap 119883 =

= 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

sube 119901 (119883)

(41)

On the other hand 119886119901119903119877119901

(119883119888) sube 119901(119883119888) implies that(119901(119883119888))

119888

sube (119886119901119903119877119901

(119883119888))119888 hence 119901(119883) sube 119886119901119903

119877119901

(119883) Thus we

have (119886119901119903119877119901

119886119901119903119877119901

) ≺ (119901 119901)Suppose that universe 119880 is finite and let 119883 =

1199101 119910

2 119910

119898 sube 119880 Then we have

119901 (119883) = ⋃119910119896isin119883

119901 119910119896 = 119909 | exist119910 isin 119883 such that 119909 isin 119901 119910

= 119909 | exist119910 isin 119883 such that 119910 isin [119909]119877119901

= 119909 | [119909]119877119901

cap 119883 =

= 119886119901119903119877119901

(119883)

(42)

Analogous to the above proof we have 119901(119883) = 119886119901119903119877119901

(119883) It

follows that (119886119901119903119877119901

119886119901119903119877119901

) = (119901 119901)

Now suppose that the cover 119901119909119909isin119880

of 119880 is finitethat is there exist 119909

119896isin 119880 119896 = 1 2 119898 such that

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

8 The Scientific World Journal

119901119909119896119896isin12119898

is a cover of119880 Analogous to the above proofto prove that (119886119901119903

119877119901

119886119901119903119877119901

) = (119901 119901) it suffices to prove that

for all 119883 sube 119880 119901(119883) sube 119886119901119903119877(119883) In fact it follows from

119901119909119896119896isin12119898

being a cover of119880 that119883 sube 119880 sube ⋃119898

119896=1119901119909

119896

Hence

119901 (119883) sube 119901(

119898

⋃119896=1

119901 119909119896)

=

119898

⋃119896=1

119901 119909119896

= 119909 | there exists 119909119896isin 119883 such that 119909 isin 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119901 119909 = 119901 119909

119896

= 119909 | there exists 119909119896isin 119883 such that 119909

119896isin [119909]

119877119901

= 119909 | [119909]119877119901

cap 119883 = = 119886119901119903119877119901

(119883)

(43)

as required

In the sequel denote by R the set of all similar relationson119880 Then it is evident thatR is infinite-intersection-closedAnd the following result holds

Proposition 29 LetR be the set of all similar relations on 119880Then we have

(1) [119909]⋂119905isin119879

119877119905= ⋂

119905isin119879[119909]

119877119905for 119877

119905119905isin119879

sube R where 119879 is anindex set

(2) forall1198771 119877

2isin R 119877

1sube 119877

2hArr (119886119901119903

1198771

1198861199011199031198771

) ≺

(1198861199011199031198772

1198861199011199031198772

)

Proof (1) Consider [119909]⋂119905isin119879

119877119905= 119910 | (119909 119910) isin ⋂

119905isin119879119877119905 = 119910 |

forall119905 isin 119879 (119909 119910) isin 119877119905 =⋂

119905isin119879119910 | (119909 119910) isin 119877

119905 =⋂

119905isin119879[119909]

119877119905

(2) Let 1198771 119877

2isin R be such that 119877

1sube 119877

2 Then for

any 119909 isin 119880 we have [119909]1198771

sube [119909]1198772 Now let 119883 sube 119880

If 119909 isin 1198861199011199031198772

(119883) then [119909]1198772

sube 119883 implying that [119909]1198771

=

[119909]1198771cap1198772

= [119909]1198771

cap [119909]1198772

sube [119909]1198772

sube 119883 that is 119909 isin 1198861199011199031198771

(119883)Therefore 119886119901119903

1198772

(119883) sube 1198861199011199031198771

(119883) and 1198861199011199031198771

(119883) sube 1198861199011199031198772

(119883)

by the duality It follows that (1198861199011199031198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)Conversely suppose that (119886119901119903

1198771

1198861199011199031198771

) ≺ (1198861199011199031198772

1198861199011199031198772

)

and (119909 119910) isin 1198771 It follows that 119909 isin 119886119901119903

1198771

119910 Since119886119901119903

1198771

(119883) sube 1198861199011199031198772

(119883) for any 119883 sube 119880 by the assumption weknow that 119886119901119903

1198771

119910 sube 1198861199011199031198772

119910 and hence 119909 isin 1198861199011199031198772

119910implying that (119909 119910) isin 119877

2 Therefore 119877

1sube 119877

2

Theorem 30 Let 119880 be a finite set and 119877 a similar relationon 119880 Then there exists an equivalent relation 119877 such thatcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) and that 119877 is the transitive

closure of 119877

Proof It follows from the proof of Theorem 10 thatcl(119886119901119903

119877

119886119901119903119877) = (⋀

infin

119896=1119886119901119903

119877

(119896) ⋁infin

119896=1119886119901119903

119877

(119896)

) and that

⋁infin

119896=1119886119901119903

119877

(119896) is a closure operator satisfying (R6) Now wedefine a binary relation 119877 on 119880 as follows

119877 = (119909 119910) | 119909 isin (

infin

⋃119896=1

119886119901119903119877

(119896)

)119910 (44)

Then it is evident that 119877 is a similar relation on 119880 andcl(119886119901119903

119877

119886119901119903119877) = (119886119901119903

119877

119886119901119903119877) by Theorem 10 In the sequel

we prove that 119877 is the transitive closure of 119877 which alsoimplies that 119877 is an equivalent relation on 119880

Suppose that 119877lowast is an equivalent relation such that 119877 sube

119877lowast By Proposition 29 we have (119886119901119903119877

119886119901119903119877) ≺ (119886119901119903

119877lowast 119886119901119903

119877lowast)

and 119886119901119903119877lowast is a closure operator and hence (119886119901119903

119877

119886119901119903119877) ≺

(119886119901119903119877lowast 119886119901119903

119877lowast) Thus by Proposition 29 119877 sube 119877lowast

7 Conclusions

In this paper we have investigated the general approximationstructure weak approximation operators and Pawlak algebrain the framework of fuzzy lattice lattice topology andauxiliary ordering The relationships between the Pawlakapproximation structures and these mathematic structuresare established and some related properties are presentedThese works would provide a new direction for the studyof rough set theory and information systems As for futureresearch it will be interesting to continue the study ofmolecular Pawlak algebra and general partial approximationspaces

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 61305057 and 71301022) and theNatural Science Research Foundation of KunmingUniversityof Science and Technology (no 14118760)

References

[1] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[2] B Sun W Ma and H Zhao ldquoA fuzzy rough set approach toemergency material demand prediction over two universesrdquoApplied Mathematical Modelling vol 37 no 10-11 pp 7062ndash7070 2013

[3] C Wang D Chen B Sun and Q Hu ldquoCommunicationbetween information systems with covering based rough setsrdquoInformation Sciences vol 216 pp 17ndash33 2012

[4] D Chen W Li X Zhang and S Kwong ldquoEvidence-theory-based numerical algorithms of attribute reduction withneighborhood-covering rough setsrdquo International Journal ofApproximate Reasoning vol 55 no 3 pp 908ndash923 2014

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

The Scientific World Journal 9

[5] C YWang andBQHu ldquoFuzzy rough sets based on generalizedresiduated latticesrdquo Information Sciences vol 248 pp 31ndash492013

[6] Y Yin and X Huang ldquoFuzzy roughness in hyperrings basedon a complete residuated latticerdquo International Journal of FuzzySystems vol 13 no 3 pp 185ndash194 2011

[7] YQ Yin JM Zhan andPCorsini ldquo119871-fuzzy roughness of 119899-arypolygroupsrdquo Acta Mathematica Sinica vol 27 no 1 pp 97ndash1182011

[8] Y Yin J Zhan and P Corsini ldquoFuzzy roughness of n-aryhypergroups based on a complete residuated latticerdquo NeuralComputing and Applications vol 20 no 1 pp 41ndash57 2011

[9] M I Ali B Davvaz and M Shabir ldquoSome properties ofgeneralized rough setsrdquo Information Sciences vol 224 pp 170ndash179 2013

[10] F Feng X Liu V Leoreanu-Fotea and Y B Jun ldquoSoft sets andsoft rough setsrdquo Information Sciences vol 181 no 6 pp 1125ndash1137 2011

[11] F Li and Y Yin ldquoThe 120579-lower and 119879-upper fuzzy roughapproximation operators on a semigrouprdquo Information Sciencesvol 195 pp 241ndash255 2012

[12] Y Yao J Mi and Z Li ldquoA novel variable precision (120579 120590)-fuzzy rough set model based on fuzzy granulesrdquo Fuzzy Sets andSystems vol 236 pp 58ndash72 2014

[13] H Zhang Y Leung and L Zhou ldquoVariable-precision-dominance-based rough set approach to interval-valued infor-mation systemsrdquo Information Sciences vol 244 pp 75ndash91 2013

[14] T J Li and W X Zhang ldquoRough fuzzy approximations on twouniverses of discourserdquo Information Sciences vol 178 no 3 pp892ndash906 2008

[15] W Ma and B Sun ldquoProbabilistic rough set over two universesand rough entropyrdquo International Journal of Approximate Rea-soning vol 53 no 4 pp 608ndash619 2012

[16] Y Y Yao and T Y Lin ldquoGeneralization of rough sets usingmodal logicrdquo Intelligent Automation and Soft Computing vol 2no 2 pp 103ndash120 1996

[17] G Cattaneo and D Ciucci ldquoLattices with interior and closureoperators and abstract approximation spacesrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) vol 5656 pp67ndash116 2009

[18] J Jarvinen ldquoLattice theory for rough Setsrdquo Transactions onRough Sets vol 6 pp 400ndash498 2007

[19] G Gierz A Compendium of Continuous Lattice Springer NewYork NY USA 1980

[20] G J Wang ldquoOn the structure of fuzzy latticesrdquo Acta Mathemat-ica Sinica vol 29 no 4 pp 539ndash543 1986

[21] L A Zadeh ldquoToward a generalized theory of uncertainty(GTU)mdashan outlinerdquo Information Sciences vol 172 no 1-2 pp1ndash40 2005

[22] Y Y Yao ldquoConstructive and algebraic methods of the theoryof rough setsrdquo Information Sciences vol 109 no 1ndash4 pp 21ndash471998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Pawlak Algebra and Approximate Structure ...Research Article Pawlak Algebra and Approximate Structure on Fuzzy Lattice YingZhuang, 1 WenqiLiu, 1 Chin-ChiaWu, 2 andJinhaiLi

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Adam Pawlak Engineering work in the information society adam.pawlak@polsl.pl.

Adam Pawlak Engineering work in the information society [email protected].

"Zagadkowa koperta listonosza Artura", Gérard Moncomble, Paweł Pawlak

"Zagadkowa koperta listonosza Artura", Gérard Moncomble, Paweł Pawlak

Tomasz Pawlak, 2009if.pw.edu.pl/~pawlak/wyklady/nukleus/nuk-03.pdf · accelerator glossary (tandem) generator linac (linear acc) radio-frequency cavity cyclotron synchrocyclotron

Tomasz Pawlak, 2009if.pw.edu.pl/~pawlak/wyklady/nukleus/nuk-03.pdf · accelerator glossary (tandem) generator linac (linear acc) radio-frequency cavity cyclotron synchrocyclotron

Zdzisław Pawlak - Muzeum Historii Polskibazhum.muzhp.pl/media//files/Studia_Wloclawskie/Studia... · 2017. 9. 5. · Zdzisław Pawlak Ks. Antoni Warmuz - profesor filozofii Seminarium

Zdzisław Pawlak - Muzeum Historii Polskibazhum.muzhp.pl/media//files/Studia_Wloclawskie/Studia... · 2017. 9. 5. · Zdzisław Pawlak Ks. Antoni Warmuz - profesor filozofii Seminarium

ACTA UNI VERSITATIS LODZIENSIS Agnieszka Pawlak POLSKA I ...

ACTA UNI VERSITATIS LODZIENSIS Agnieszka Pawlak POLSKA I ...

Wiesław Pawlak „Grecyzm Juwenala” Glosa do prologu Kroniki ... · 21 „Grecyzm Juwenala Glosa do rologu Kroniki Mistrza Wincentego Wiesław Pawlak katolicki uniwersytet lubelski

Wiesław Pawlak „Grecyzm Juwenala” Glosa do prologu Kroniki ... · 21 „Grecyzm Juwenala Glosa do rologu Kroniki Mistrza Wincentego Wiesław Pawlak katolicki uniwersytet lubelski

Zbigniew Pawlak, Andrzej Smoleń · 2020. 7. 17. · 85 ROCZNIK NAUKOWY WYDZIAŁU ZARZĄDZANIA W CIECHANOWIE 1-4 (VI) 2012 Zbigniew Pawlak, Andrzej Smoleń ZASADY BUDOWY WSPÓŁCZESNYCH

Zbigniew Pawlak, Andrzej Smoleń · 2020. 7. 17. · 85 ROCZNIK NAUKOWY WYDZIAŁU ZARZĄDZANIA W CIECHANOWIE 1-4 (VI) 2012 Zbigniew Pawlak, Andrzej Smoleń ZASADY BUDOWY WSPÓŁCZESNYCH

2011 Pawlak Jureczko Czapla Modeling the Structural

2011 Pawlak Jureczko Czapla Modeling the Structural

IlAttachment & Trauma Network ene Pawlak and Julie Simmons ... · Ilene Pawlak and Julie Simmons Attachment & Trauma Network Creating Trauma Sensitive Schools Conference February,

IlAttachment & Trauma Network ene Pawlak and Julie Simmons ... · Ilene Pawlak and Julie Simmons Attachment & Trauma Network Creating Trauma Sensitive Schools Conference February,

Zbigniew Pawlak, Jerzy Wlazło, "Pęknięte miasto. Biesłan"

Zbigniew Pawlak, Jerzy Wlazło, "Pęknięte miasto. Biesłan"

Pawlak Kazimierz - Glass Fibre / Wlokno szklane

Pawlak Kazimierz - Glass Fibre / Wlokno szklane

ANA CAROLINA MENEZES PAWLAK PEDROSO

ANA CAROLINA MENEZES PAWLAK PEDROSO

Dorota A. Pawlak Emil Tymicki - kpk.gov.pl · Materiały o ulepszonej fotoluminescencji RE dla wzmacniaczy optycznych M. Gajc, … D. A. Pawlak, Adv. Funct. Mat. (2013), 23, 3443.

Dorota A. Pawlak Emil Tymicki - kpk.gov.pl · Materiały o ulepszonej fotoluminescencji RE dla wzmacniaczy optycznych M. Gajc, … D. A. Pawlak, Adv. Funct. Mat. (2013), 23, 3443.

Algebra I (PreAP, Pt. 1, Pt. 2) Curriculum Guide · ALGEBRA I (PreAP, PT 1, and PT 2) CURRICULUM GUIDE (Revised 2016) PRINCE WILLIAM COUNTY SCHOOLS 3 PreAP Algebra I Objectives Approximate

Algebra I (PreAP, Pt. 1, Pt. 2) Curriculum Guide · ALGEBRA I (PreAP, PT 1, and PT 2) CURRICULUM GUIDE (Revised 2016) PRINCE WILLIAM COUNTY SCHOOLS 3 PreAP Algebra I Objectives Approximate

Approximate Computing with Approximate Circuits: …jhan8/publications/esweek17... · 2017-10-11 · Approximate Computing with Approximate Circuits: Methodologies and Applications

Approximate Computing with Approximate Circuits: …jhan8/publications/esweek17... · 2017-10-11 · Approximate Computing with Approximate Circuits: Methodologies and Applications

Ill-posed Computational Algebra with Approximate Data

Ill-posed Computational Algebra with Approximate Data

Mirosław Pawlak: On the effectiveness of options in grammar teaching

Mirosław Pawlak: On the effectiveness of options in grammar teaching

Fotografia - Grzegorz Pawlak

Fotografia - Grzegorz Pawlak

In: Gianni Jacucci, Adam Pawlak, Kurt Sandkuhl (ed ...

In: Gianni Jacucci, Adam Pawlak, Kurt Sandkuhl (ed ...

Laila pawlak, Why corporates should be entrepreneurial super weights

Laila pawlak, Why corporates should be entrepreneurial super weights