Research Article A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf ·...
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Hindawi Publishing Corporation Algebra Volume 2013, Article ID 594636, 7 pages http://dx.doi.org/10.1155/2013/594636 Research Article A Study on Fuzzy Ideals of -Groups B. Davvaz 1 and O. Ratnabala Devi 2 1 Department of Mathematics, Yazd University, Yazd, Iran 2 Department of Mathematics, Manipur University, Imphal, Manipur 795003, India Correspondence should be addressed to B. Davvaz; [email protected] Received 10 February 2013; Accepted 19 June 2013 Academic Editor: Antonio M. Cegarra Copyright © 2013 B. Davvaz and O. Ratnabala Devi. 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. Using the idea of the new sort of fuzzy subnear-ring of a near-ring, fuzzy subgroups, and their generalizations defined by various researchers, we try to introduce the notion of (, ∨ )-fuzzy ideals of -groups. ese fuzzy ideals are characterized by their level ideals, and some other related properties are investigated. 1. Introduction and Basic Definitions e concept of a fuzzy set was introduced by Zadeh [1] in 1965, utilizing what Rosenfeld [2] defined as fuzzy subgroups. is was studied further in detail by different researchers in various algebraic systems. e concept of a fuzzy ideal of a ring was introduced by Liu [3]. e notion of fuzzy subnear-ring and fuzzy ideals was introduced by Abou-Zaid [4]. en in many papers, fuzzy ideals of near-rings were discussed for example, see [5–11]. In [12], the idea of fuzzy point and its belongingness to and quasi coincidence with a fuzzy set were used to define (, )-fuzzy subgroup, where , take one of the values from {, , ∧ , ∨ }, ̸ =∧. A fuzzy subgroup in the sense of Rosenfeld is in fact an (, )-fuzzy subgroup. us, the concept of (, ∨ )-fuzzy subgroup was introduced and discussed thoroughly in [7]. Bhakat and Das [13] introduced the concept of (, ∨ )- fuzzy subrings and ideals of a ring. Davvaz [14, 15], Narayanan and Manikantan [16], and Zhan and Davvaz [17] studied a new sort of fuzzy subnear-ring (ideal and prime ideal) called (, ∨ )-fuzzy subnear-ring (ideal and prime ideal) and gave characterizations in terms of the level ideals. In [18, 19], the idea of fuzzy ideals of -groups was defined, and various properties such as fundamental theorem of fuzzy ideals and fuzzy congruence were studied, respectively. In the present paper, we extend the idea of (, ∨ )-fuzzy ideals of near- rings to the case of -groups and introduce the idea of fuzzy cosets with some results. We first recall some basic concepts for the sake of com- pleteness. By a near-ring we mean a nonempty set with two binary operations “+” and “⋅” satisfying the following axioms: (i) (, +) is a group, (ii) (, ⋅) is a semigroup, (iii) ( + ) ⋅ = ⋅ + ⋅ for all , , ∈ . It is in fact a right near-ring because it satisfies the right distributive law. We will use the word “near-ring” to mean “right near-ring.” is said to be zero symmetric if 0⋅= ⋅0=0 for all ∈. We denote ⋅ by . Note that the missing leſt distributive law, ⋅ ( + ) = ⋅+⋅, has to do with linearity if is considered as a function. Example 1. Let G be a group, and let (G) be the set of all mappings from G into G. We define + and ⋅ on (G) by ( + ) () := () + () , ( ⋅ ) () := ( ()). (1) en, ((G), +, ⋅) is a near-ring. Just in the same way as -modules or vector spaces are used in ring theory, -groups are used in near-ring theory.
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Hindawi Publishing CorporationAlgebraVolume 2013 Article ID 594636 7 pageshttpdxdoiorg1011552013594636
Research ArticleA Study on Fuzzy Ideals of119873-Groups
B Davvaz1 and O Ratnabala Devi2
1 Department of Mathematics Yazd University Yazd Iran2Department of Mathematics Manipur University Imphal Manipur 795003 India
Correspondence should be addressed to B Davvaz davvazyazdacir
Received 10 February 2013 Accepted 19 June 2013
Academic Editor Antonio M Cegarra
Copyright copy 2013 B Davvaz and O Ratnabala Devi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Using the idea of the new sort of fuzzy subnear-ring of a near-ring fuzzy subgroups and their generalizations defined by variousresearchers we try to introduce the notion of (120598 120598 or 119902)-fuzzy ideals of 119873-groups These fuzzy ideals are characterized by their levelideals and some other related properties are investigated
1 Introduction and Basic Definitions
The concept of a fuzzy set was introduced by Zadeh [1] in1965 utilizing what Rosenfeld [2] defined as fuzzy subgroupsThis was studied further in detail by different researchersin various algebraic systems The concept of a fuzzy idealof a ring was introduced by Liu [3] The notion of fuzzysubnear-ring and fuzzy ideals was introduced by Abou-Zaid[4] Then in many papers fuzzy ideals of near-rings werediscussed for example see [5ndash11] In [12] the idea of fuzzypoint and its belongingness to and quasi coincidence with afuzzy set were used to define (120572 120573)-fuzzy subgroup where 120572120573 take one of the values from 120598 119902 120598 and 119902 120598 or 119902 120572 = 120598 and 119902A fuzzy subgroup in the sense of Rosenfeld is in fact an(120598 120598)-fuzzy subgroup Thus the concept of (120598 120598 or 119902)-fuzzysubgroup was introduced and discussed thoroughly in [7]Bhakat and Das [13] introduced the concept of (120598 120598 or 119902)-fuzzy subrings and ideals of a ringDavvaz [14 15]Narayananand Manikantan [16] and Zhan and Davvaz [17] studied anew sort of fuzzy subnear-ring (ideal and prime ideal) called(120598 120598 or 119902)-fuzzy subnear-ring (ideal and prime ideal) and gavecharacterizations in terms of the level ideals In [18 19] theidea of fuzzy ideals of 119873-groups was defined and variousproperties such as fundamental theorem of fuzzy ideals andfuzzy congruence were studied respectively In the presentpaper we extend the idea of (120598 120598 or 119902)-fuzzy ideals of near-rings to the case of 119873-groups and introduce the idea of fuzzycosets with some results
We first recall some basic concepts for the sake of com-pleteness
By anear-ring wemean a nonempty set119873with two binaryoperations ldquo+rdquo and ldquosdotrdquo satisfying the following axioms
(i) (119873 +) is a group(ii) (119873 sdot) is a semigroup(iii) (119909 + 119910) sdot 119911 = 119909 sdot 119911 + 119910 sdot 119911 for all 119909 119910 119911 isin 119873
It is in fact a right near-ring because it satisfies the rightdistributive law We will use the word ldquonear-ringrdquo to meanldquoright near-ringrdquo 119873 is said to be zero symmetric if 0 sdot 119909 =
119909 sdot 0 = 0 for all 119909 isin 119873 We denote 119909 sdot 119910 by 119909119910Note that the missing left distributive law 119909 sdot (119910 + 119911) =
119909 sdot 119910 + 119909 sdot 119911 has to do with linearity if 119909 is considered as afunction
Example 1 Let G be a group and let 119872(G) be the set of allmappings fromG intoG We define + and sdot on 119872(G) by
(119891 + 119892) (119909) = 119891 (119909) + 119892 (119909)
(119891 sdot 119892) (119909) = 119891 (119892 (119909))
(1)
Then (119872(G) + sdot) is a near-ring
Just in the same way as 119877-modules or vector spaces areused in ring theory 119873-groups are used in near-ring theory
2 Algebra
By an 119873-group we mean a nonempty set 119866 together witha map Φ 119873 times 119866 rarr 119866 written as Φ(119899 119892) = 119899119892 satisfying thefollowing conditions
(i) (119866 +) is a group (not necessarily abelian)
(ii) (1198991+ 1198992)119892 = 119899
1119892 + 1198992119892
(iii) (11989911198992)119892 = 119899
1(1198992119892) for all 119899
1 1198992isin 119873 119892 isin 119866
Example 2 Let 119873 be a subnear-ring of 119872(G) Then G is an119873-group via function application as operation
Example 3 The additive group (119873 +) of a near-ring (119873 + sdot)
is an 119873-group via the near-ring multiplication
An ideal 119868 of 119873-group 119866 is an additive normal subgroupof 119866 such that 119873119868 sube 119868 and 119899(119892 + ℎ) minus 119899119892 isin 119868 for all ℎ isin 119868119892 isin 119866 119899 isin 119873 A mapping between two 119873-groups 119866 and 119866
1015840
is called an 119873-homomorphism if 119891(119892 + ℎ) = 119891(119892) + 119891(ℎ) and119891(119899119892) = 119899119891(119892) for all 119892 ℎ isin 119866 119899 isin 119873
Throughout this study we use 119873 to denote a zero-symmetric near-ring and 119866 to denote an 119873-group
For any fuzzy subset 119860 of 119866 119868119898119860 = 119860(119909) | 119909 isin 119866
denotes the image of 119860 For any subset 119868 of 119866 120594119868denotes the
characteristic function of 119868
Definition 4 (see [2]) A fuzzy subset 119860 of a group 119866 is calleda fuzzy subgroup of 119866 if it satisfies the following conditions
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910)
(ii) 119860(minus119909) ge 119860(119909)
for all 119909 119910 isin 119866
Definition 5 For a fuzzy subset 119860 of 119866 119905 isin (0 1] the subset119860119905
= 119909 isin 119866 | 119860(119909) ge 119905 is called a level subset of 119866 deter-mined by 119860 and 119905
The set 119909 isin 119866 | 119860(119909) gt 0 is called the support of 119860 andis denoted by Supp119860 A fuzzy subset 119860 of 119866 of the form
119860 (119910) =
119905 ( = 0) if 119910 = 119909
0 if 119910 = 119909
(2)
is said to be a fuzzy point denoted by 119909119905 Here 119909 is called the
support point and 119905 is called its value A fuzzy point 119909119905is
said to belong to (resp quasi coincident with) a fuzzy set 119860
written as 119909119905120598119860 (resp 119909
119905119902119860) if 119860(119909) ge 119905 (resp 119860(119909) + 119905 gt 1)
If 119909119905
isin 119860 or 119909119905119902119860 then we write 119909
119905isin or119902119860 The symbols
119909119905isin119860 119909
119905119902119860 119909119905isin or119902119860 mean that 119909
119905isin 119860 119909
119905119902119860 119909119905
isin or119902119860 donot hold respectively
Definition 6 (see [7 12]) A fuzzy subset of a group 119866 is saidto be an (120598 120598 or 119902)-fuzzy subgroup of 119866 if for all 119909 119910 isin 119866 and119905 119903 isin (0 1]
(i) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
Remark 7 (see [7]) The conditions (i) and (ii) of Definition 6are respectively equivalent to
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910) 05(ii) 119860(minus119909) ge min119860(119909) 05
for all 119909 119910 isin 119866
Remark 8 For any (120598 120598 or 119902)-fuzzy subgroup119860 of 119866 such that119860(119909) ge 05 for some 119909 isin 119866 then119860(0) ge 05 and if119860(0) lt 05then 119860(119909) lt 05 for all 119909 isin 119866 So 119860 is just the usual fuzzysubgroup in the sense of Rosenfeld
Remark 9 It is noted that if 119860 is a fuzzy subgroup then it isan (120598 120598 or 119902)-fuzzy subgroup of 119866 However the converse maynot be true
Here onwards we assume that 119860 is an (120598 120598 or 119902)-fuzzysubgroup in the nontrivial sense for which case we have119860(0) ge 05
Definition 10 (see [7]) An (120598 120598or119902)-fuzzy subgroup of a group119866 is said to be (120598 120598or119902)-fuzzy normal subgroup if for any 119909 119910 isin
119866 and 119905 isin (0 1]
119909119905isin 119860 997904rArr (119910 + 119909 minus 119910)
119905isin or119902119860 (3)
Remark 11 (see [7]) The condition of (120598 120598 or 119902)-fuzzy normalsubgroup is given in the equivalent forms as
(i) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05(ii) 119860(119909 + 119910) ge min119860(119910 + 119909) 05(iii) 119860([119909 119910]) ge min119860(119909) 05 for all 119909 119910 isin 119866
Here [119909 119910] denotes the commutator of 119909 119910 in 119866
In the light of this fact the condition of Definition 10 canbe replaced by any one of the above conditions in Remark 8
Definition 12 (see [18]) Let 119860 be a fuzzy subset of an 119873-group 119866 It is called a fuzzy 119873-subgroup of 119866 if it satisfies thefollowing conditions
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910)(ii) 119860(119899119909) ge 119860(119909)
for all 119909 119910 isin 119866 119899 isin 119873
Remark 13 If 119866 is a unitary 119873-group the above conditionsare equivalent to conditions119860(119909minus119910) ge min119860(119909) 119860(119910) and119860(119899119909) ge 119860(119909) for all 119909 119910 isin 119866 119899 isin 119873
Definition 14 (see [18 19]) A nonempty fuzzy subset 119860 of an119873-group 119866 is called a fuzzy ideal if it satisfies the followingconditions
(i) 119860(119909 minus 119910) ge min119860(119909) 119860(119910)(ii) 119860(119899119909) ge 119860(119909)(iii) 119860(119910 + 119909 minus 119910) ge 119860(119909)(iv) 119860(119899(119909 + 119910) minus 119899119909) ge 119860(119910)
for all 119909 119910 isin 119866 119899 isin 119873
Algebra 3
Definition 15 (see [14]) A fuzzy set 119860 of a near-ring 119873 iscalled an (120598 120598or119902)-fuzzy subnear-ring of119873 if for all 119905 119903 isin (0 1]and 119909 119910 isin 119873
(i) (a) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(b) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
(ii) 119909119905 119910119903isin 119860 rArr (119909119910)min119905119903 isin or119902119860
119860 is called an (120598 120598or119902)-fuzzy ideal of119873 if it is (120598 120598or119902)-fuzzy subnear-ring of 119873 and
(iii) 119909119905isin 119860 rArr (119910 + 119909 minus 119910) isin or119902119860
(iv) 119910119903isin 119860 119909 isin 119873 rArr (119910119909)
119903isin or119902119860
(v) 119886119905isin 119860 rArr (119910(119909 + 119886) minus 119910119909)
119905isin or119902119860
for all 119909 119910 119886 isin 119873
2 Generalized Fuzzy Ideals
In this section we give the definition of (120598 120598 or 119902)-fuzzysubgroup and ideal of an 119873-group 119866 based on Definitions 14and 15
Definition 16 A fuzzy subset 119860 of an 119873-group 119866 is said to bean (120598 120598 or 119902)-fuzzy subgroup of 119866 if 119909 119910 isin 119866 119899 isin 119873 119905 119903 isin
(0 1]
(i) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
(iii) 119909119905isin 119860 119899 isin 119873 rArr (119899119909)
119905isin or119902119860
Lemma 17 Let119860 be a fuzzy subset of119866 and 119905 119903 isin (0 1]Then
(i) 119909119905 119910119903
isin 119860 rArr (119909 + 119910)min119905119903120598 or 119902119860 hArr 119860(119909 + 119910) ge
min119860(119909) 119860(119910) 05(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860 hArr 119860(minus119909) ge min119860(119909) 05
for all 119909 isin 119866(iii) 119909
119905isin 119860 rArr (119899119909)
119905120598 or 119902119860 hArr 119860(119899119909) ge min119860(119909) 05
for all 119909 isin 119866 119899 isin 119873
Proof (i) Let 119909 119910 isin 119866 Consider the case (a) min119860(119909)
119860(119910) lt 05Assume that 119860(119909 + 119910) lt min119860(119909) 119860(119910) 05 =
min119860(119909) 119860(119910) Choose 119905 such that 119860(119909 + 119910) lt 119905 lt
min119860(119909) 119860(119910) which implies that 119909119905
isin 119860 119910119905
isin 119860 but(119909 + 119910)
119905isin or119902119860 [as 119860(119909 + 119910) + 119905 lt 1 and 119860(119909 + 119910) lt 119905]
Consider the case (b) min119860(119909) 119860(119910) ge 05 Assume that119860(119909 + 119910) lt min119860(119909) 119860(119910) 05 = 05 Choose 119905 such that119860(119909 + 119910) lt 119905 lt 05 so that 119909
119905 119910119905isin 119860 but (119909 + 119910)
119905isin or119902119860
Conversely let 119909119905 119910119903
isin 119860 rArr 119860(119909) ge 119905 119860(119910) ge 119903 Then119860(119909 + 119910) ge min119860(119909) 119860(119910) 05 ge min119905 119903 05 Thus 119860(119909 +
119910) ge min119905 119903 if either 119905 or 119903 le 05 and 119860(119909 + 119910) ge 05 if both119905 and 119903 gt 05 which means (119909 + 119910)min119905119903120598 or 119902119860
(ii) Let 119909 isin 119866 min119860(119909) 05 le 05 Suppose 119860(minus119909) lt
min119860(119909) 05 le 05 Choose 119903 such that 119860(minus119909) lt 119903 lt
min119860(119909) 05 le 05 Then 119909119903
isin 119860 but (minus119909)119903isin or119902119860 which
contradicts the hypothesis So 119860(minus119909) ge min119860(119909) 05 forall 119909 isin 119866
Conversely let 119909119905
isin 119860 Then 119860(119909) ge 119905 But we have119860(minus119909) ge min119860(119909) 05 ge min119905 05 rArr 119860(minus119909) ge 119905 or119860(minus119909) ge 05 according as 119905 le 05 or 119905 gt 05 rArr (minus119909)
119905isin or119902119860
(iii) Let119909 isin 119866 andmin119860(119909) 05 le 05 Suppose119860(119899119909) lt
min119860(119909) 05 le 05 Choose 119903 such that 119860(119899119909) lt 119903 lt
min119860(119909) 05 le 05 Then 119860(119909) gt 119903 that is 119909119903
isin 119860 but(119899119909)119903isin or119902119860 as 119860(119899119909) lt 119903 and 119860(119899119909) + 119903 le 1
Conversely let 119909119905isin 119860 119899 isin 119873 then119860(119909) ge 119905 But119860(119899119909) ge
min119860(119909) 05 ge min119905 05 rArr 119860(119899119909) ge 119905 or 119860(119899119909) ge 05
according as 119905 le 05 or 119903 gt 05 rArr 119860(119899119909) ge 119905 or 119860(119899119909) + 119905 gt
1 rArr (119899119909)119905isin or119902119860
Theorem 18 Let119860 be a fuzzy subset of119866 Then119860 is an (120598 120598or
119902)-fuzzy subgroup of 119866 if and only if the following conditionsare satisfied
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910) 05(ii) 119860(minus119909) ge min119860(119909) 05(iii) 119860(119899119909) ge min119860(119909) 05
for all 119909 119910 isin 119866 119899 isin 119873
Proof It follows from the previous lemma
Definition 19 A fuzzy subset 119860 of an 119873-group 119866 is said to be(120598 120598 or 119902)-fuzzy ideal of 119866 if it is an (120598 120598 or 119902)-fuzzy subgroupand satisfies the following conditions
(i) 119909119905isin 119860 rArr (119910 + 119909 minus 119910)
119905isin V119902119860
(ii) 119886119905isin 119860 rArr (119899(119909 + 119886) minus 119899119909)
119905isin V119902119860
for any 119899 isin 119873 119909 119886 isin 119866
Lemma20 Let119860 be a fuzzy subset of119866 and 119905 119903 isin (0 1]Then
(i) 119909119905
isin 119860 rArr (119910 + 119909 minus 119910)119905
isin V119902119860 hArr 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05(ii) 119886119905isin 119860 rArr (119899(119909+119886)minus119899119909)
119905isin V119902119860 hArr 119860(119899(119909+119886)minus119899119909) ge
min119860(119909) 05
Proof (i) Assume that 119860(119910 + 119909 minus 119910) lt min119860(119909) 05Choose 119905 such that 119860(119910 + 119909 minus 119910) lt 119905 lt min119860(119909) 05 Butmin119860(119909) 05 le 119860(119909) or 05 according as 119860(119909) lt 05 or119860(119909) ge 05 So 119860(119909) gt 119905 or 119860(119909) ge 05 rArr 119909
119905isin 119860 or 119909
05isin 119860
But (119910 + 119909 minus 119910)119905isin V119902119860 or (119910 + 119909 minus 119910)
05isin V119902119860 respectively
which contradicts the hypothesisConversely assume that 119909
119905isin 119860 then 119860(119909) ge 119905 For any
119910 isin 119866 we have119860(119910+119909minus119910) ge min119860(119909) 05 ge min119905 05 rArr
119860(119910 + 119909 minus 119910) ge 119905 or 05 according as 119905 lt 05 or 119905 ge 05 rArr
(119910+119909minus119910)119905isin 119860 or119860(119910+119909minus119910)+119905 gt 1 SorArr (119910+119909minus119910)
119905isin V119902119860
(ii) Assume that119860(119899(119909+119886)minus119899119909) lt min119860(119886) 05 = 119860(119886)
or 05 for some 119899 isin 119873 119909 119886 isin 119866 According as 119860(119886) lt 05 or119860(119886) ge 05 Choose 119905 isin (0 1] such that 119860(119899(119909 + 119886) minus 119899119909) lt
119905 lt min119860(119886) 05 le 05 In either case 119860(119899(119909 + 119886) minus 119899119909) lt 119905
and 119860(119899(119909 + 119886) minus 119899119909) + 119905 lt 1 So (119899(119909 + 119886) minus 119899119909)119905isin V119902119860 a
contradictionConversely assume that119860(119899(119909+119886)minus119899119909) ge min119860(119886) 05
for all 119886 119909 isin 119866 119899 isin 119873 Let 119886119905
isin 119860 Then 119860(119886) ge 119905 So119860(119899(119909 + 119886) minus 119899119909) ge min119905 05 le 05 = 119905 or 05 according as119905 le 05 or 119905 gt 05 So 119860(119899(119909 + 119886) minus 119899119909)
119905isin V119902119860
4 Algebra
Theorem 21 Let 119860 be an (120598 120598 or 119902) fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 if and only if
(i) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 for all 119909 119910 isin 119866
(ii) 119860(119899(119909 + 119886) minus 119899119909) ge min119860(119886) 05 for all 119899 isin 119873119909 119886 isin 119866
Proof It is immediate from Lemma 20
By definition a fuzzy ideal of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 But the converse is not true in general as shown by thefollowing example
Example 22 Consider 119866 = S3
= 119894 1205881 1205882 1205911 1205912 1205913 (written
additively) to be a Z-group Define a fuzzy subset 119860 of 119866
as 119860(119894) = 1 119860(1205881) = 119860(120588
2) = 119860(120591
2) = 119860(120591
3) = 06
119860(1205911) = 08 which is not fuzzy ideal as 119860[2(120591
1+ 1205912) minus 2120591
2] =
119860(1205881) = 119860(120588
2) = 06 lt 119860(120591
1) it contradicts the condition
(iv) of Definition 14 As 119860(119909 minus 119910) 119860(119899119909) 119860(119910 + 119909 minus 119910) and119860(119899(119909+119886)minus119899119909) = 06 or 08 ge min05 06 or 08 = 05 thusthe notion of (120598 120598or119902)-fuzzy ideal is a successful generalizationof fuzzy ideals of 119866 as introduced in [18]
Theorem 23 Let 119860119894 119894 isin 119869 be any family of (120598 120598 or 119902)-fuzzy
ideals of 119866 Then 119860 = ⋂119860119894is an (120598 120598 or 119902)-fuzzy ideal of 119866
Proof It is straightforward
Theorem 24 A nonempty subset 119868 of 119866 is an ideal of 119866 if andonly if 120594
119868is an (isin isin or119902)-fuzzy ideal of 119866
Proof If 119868 is an ideal of119866 it is clear from [18 Proposition 211]that 120594
119868is fuzzy ideal of 119866 Since every fuzzy ideal is (120598 120598 or 119902)-
fuzzy ideal 120594119868is (120598 120598 or 119902)-fuzzy ideal of 119866
Conversely let 120594119868be an (120598 120598 or 119902)-fuzzy ideal of 119866 Let
119909 119910 isin 119868 120594119868(119909 minus 119910) ge min120594
119868(119909) 120594119868(119910) 05 = 05 So
120594119868(119909 minus 119910) = 1 rArr 119909 minus 119910 isin 119868 Let 119899 isin 119873 119909 isin 119868 120594
119868(119899119909) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899119909) = 1 rArr 119899119909 isin 119868119910 isin 119866 119909 isin 119868
120594119868(119910 + 119909 minus 119910) ge min120594
119868(119909) 05 = 05 rArr 120594
119868(119910 + 119909 minus 119910) =
1 rArr 119910 + 119909 minus 119910 isin 119868119899 isin 119873 119909 isin 119868 119910 isin 119866 120594119868(119899(119910 + 119909) minus 119899119910) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899(119910 + 119909) minus 119899119910) = 1 rArr 119899(119910 + 119909) minus
119899119910 isin 119868 Then 119868 is an ideal of 119866
Theorem 25 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy(subgroup) ideal of 119866 if and only if the level subset 119860
119905is a
(subgroup) ideal for 0 lt 119905 le 05
Proof We prove the result for (120598 120598or119902)-fuzzy ideal 119878 Let119860 bean (120598 120598 or 119902)-fuzzy ideal of 119866 Let 119905 le 05 119909 119910 119894 isin 119860
119905 119899 isin 119873
(i) 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 ge min119905 05 = 119905 rArr
119909 minus 119910 isin 119860119905
(ii) 119860(119899119909) ge min119860(119886) 05 ge min119905 05 = 119905 rArr 119899119909 isin
119860119905
(iii) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 ge min119905 05 = 119905 rArr
119910 + 119909 minus 119910 isin 119860119905 119910 isin 119866
(iv) 119860(119899(119910 + 119909) minus 119899119910) ge min119860(119909) 05 ge min119905 05 =
119905 rArr 119899(119910 + 119909) minus 119899119910 isin 119860119905 119910 isin 119866 119899 isin 119873
Hence 119860119905is an ideal of 119866 Again let 119860
119905be an ideal of 119866 for
all 119905 le 05 If possible let there exist 119909 119910 isin 119866 such that 119860(119909 minus
119910) lt 119905 lt min119860(119909) 119860(119910) 05 Let 119905 be such that 119860(119909 minus 119910) lt
119905 lt min119860(119909) 119860(119910) 05 rArr 119909 119910 isin 119860119905and 119909 minus 119910 notin 119860
119905 a
contradiction So 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 for all119909 119910 isin 119866 For 119899 isin 119873 119909 isin 119866 let 119860(119899119909) lt min119860(119909) 05 Ifpossible let 119905 be such that 119860(119899119909) lt 119905 lt min119860(119909) 05 Thisimplies 119909 isin 119860
119905 but 119899119909 notin 119860
119905 a contradiction Similarly we
can prove that 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 119860[119899(119910 + 119909) minus
119899119910] ge min119860(119909) 05 119909 119910 isin 119866 119899 isin 119873
Remark 26 For 119905 isin (05 1) 119860 may be an (120598 120598 or 119902)-fuzzyideal of 119866 but 119860
119905may not be an ideal of 119866 Let 119905 = 08 in
Example 22 Then 119860119905= 119894 120591
1 119860119905is not an ideal of S
3as it is
not a normal subgroup of S3
We are looking for a corresponding result when 119860119905is an
ideal of 119866 for all 119905 isin (05 1]
Theorem 27 Let 119860 be a fuzzy subset of an 119873-group 119866 Then119860119905
= 120601 is an ideal of119866 for all 119905 isin (05 1] if and only if119860 satisfiesthe following conditions
(i) max119860(119909 minus 119910) 05 ge min119860(119909) 119860(119910)
(ii) max119860(119899119909) 05 ge 119860(119909)
(iii) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)
(iv) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
for all 119909 119910 isin 119866 119899 isin 119873
Proof Suppose that 119860119905
= 120601 is an ideal of 119866 for all 119905 isin
(05 1] In order to prove (i) suppose that for some 119909 119910 isin
119866 max119860(119909 minus 119910) 05 lt min119860(119909) 119860(119910) Let 119905 =
min119860(119909) 119860(119910) So 119909 119910 isin 119860119905and 119905 isin (05 1] Since 119860
119905is
an ideal 119909 minus 119910 isin 119860119905 So 119860(119909 minus 119910) ge 119905 gt max119860(119909 minus 119910) 05
a contradiction In order to prove (ii) suppose that 119909 isin 119866119899 isin 119873 and max119860(119899119909) 05 lt 119860(119909) = 119905 (say) Then119909 isin 119860
119905rArr 119899119909 isin 119860
119905rArr 119860(119899119909) ge 119905 gt max119860(119899119909) 05 a
contradiction Similarly we can prove (iii) and (iv)Conversely suppose that conditions (i) to (iv) hold We
show that 119860119905is an ideal of 119866 for all 119905 isin (05 1] Let 119909 119910 isin 119860
119905
Then 05 lt 119905 le min119860(119909) 119860(119910) le max119860(119909 minus 119910) 05 =
119860(119909 minus 119910) So 119909 minus 119910 isin 119860119905 Let 119899 isin 119873 119909 isin 119860
119905 Then 05 lt
119905 le 119860(119909) le max119860(119909) 05 = 119860(119909) so 119899119909 isin 119860119905 For 119909 isin 119860
119905
119910 isin 119866 05 lt 119905 le 119860(119909) le max119860(119910 + 119909 minus 119910) 05 = 119860(119910 + 119909 minus
119910) rArr 119910 + 119909 minus 119910 isin 119860119905 Also if 119899 isin 119873 119909 isin 119860
119905 119910 isin 119866 05 lt
119905 le 119860(119909) le max119860(119899(119910 + 119909) minus 119899119910) 05 = 119860(119899(119910 + 119909) minus 119899119910)Hence 119899(119910 + 119909) minus 119899119910 isin 119860
119905 Then 119860
119905is an ideal of 119866
A definition for the previous kind of fuzzy subset wasgiven for the case of near-rings in [17] Now we give thedefinition for 119873-groups
Definition 28 A fuzzy subset of 119866 is called an (120598 120598 or 119902)-fuzzysubgroup of 119866 if for all 119905 119903 isin (0 1] and for all 119909 119910 isin 119866 119899 isin 119873
(i) (a) (119909 + 119910)min(119905119903)isin119860 implies 119909119905isinV119902119860 or 119910
119903isinV119902119860
(b) (minus119909)119905isin119860 implies 119909
119905isinV119902119860
Algebra 5
(ii) (119899119909)119905isin119860 implies 119909
119905isinV119902119860
Moreover 119860 is called an (120598 120598 or 119902)-fuzzy ideal of 119866 if119860 is (120598 120598 or 119902)-fuzzy subgroup of 119866 and
(iii) (119910 + 119909 minus 119910)119905isin119860 implies 119909
119905isinV119902119860
(iv) (119899(119909 + 119910) minus 119899119910)119905isin119860 implies 119909
119905isinV119902119860
Theorem 29 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 if and only if
(1) (a) max119860(119909 + 119910) 05 ge min119860(119909) 119860(119910)(b) max119860(minus119909) 05 ge 119860(119909)
(2) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)(3) max119860(119899119909) 05 ge 119860(119909)(4) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
Proof (i)a hArr (1)a Let 119909 119910 isin 119866 be such that max119860(119909 +
119910) 05 lt min119860(119909) 119860(119910) Let 119905 = min119860(119909) 119860(119910) then05 lt 119905 le 1 (119909 + 119910)
119905isin119860 So we must have 119909
119905isinV119902119860 or 119910
119905isinV119902119860
But 119909119905isin 119860 and 119910
119905isin 119860 Here 119909
119905119902119860 or 119910
119905119902119860 then 119905 le 119860(119909) and
119860(119909) + 119905 le 1 or 119905 le 119860(119910) and 119860(119910) + 119905 le 1 then 119905 le 05 acontradiction
Conversely let (119909 + 119910)min(119905119903)isin119860 Then 119860(119909 + 119910) lt min(119905119903) If 119860(119909 + 119910) ge min119860(119909) 119860(119910) then min119860(119909) 119860(119910) lt
min(119905 119903) Hence either 119860(119909) lt 119905 or 119860(119910) lt 119903 which implies119909119905isin119860 or 119910
119903isin119860 Thus 119909
119905isinV119902119860 or 119910
119903isinV119902119860
Again if 119860(119909 + 119910) lt min119860(119909) 119860(119910) then by (1)a
05 ge min 119860 (119909) 119860 (119910) gt 119860 (119909 + 119910) (4)
Suppose that 119909119905
isin 119860 or 119910119903
isin 119860 then 119905 le 119860(119909) le 05 or119903 le 119860(119910) le 05 It follows that either 119909
119905119902119860 or 119910
119905119902119860 and thus
119909119905isinV119902119860 or 119910
119903isinV119902119860
(i)b hArr (1)b Suppose that there exists 119909 isin 119866 such thatmax119860(minus119909) 05 lt 119860(119909) If 119860(119909) = 119905 then 05 lt 119905 le 1 and119860(minus119909) lt 119905 so that (minus119909)
119905isin119860 But then we must have either
(119909)119905isin119860 or 119909
119905119902119860 Also we have (119909)
119905isin 119860 So 119860(119909) + 1 le 1
which means that 119905 le 05 a contradictionConversely suppose that (119909)
119905isin119860 then 119860(minus119909) lt 119905 If
119860(minus119909) ge 119860(119909) then 119860(119909) lt 119905 which gives 119909119905isinV119902119860 Again if
119860(minus119909) lt 119860(119909) by (1)b we have 05 ge 119860(119909) Putting (119909)119905isin 119860
then 119905 le 119860(119909) le 05 so that 119909119905119902119860 which means that 119909
119905isinV119902119860
Similarly we can prove the remaining parts
Theorem 30 A fuzzy subset 119860 of 119866 is an (120598 120598V119902)-fuzzy idealif and only if 119860
119905( = 120601) is an ideal of 119866 for all 119905 isin (05 1]
3 Fuzzy Cosets and Isomorphism Theorem
In this section we first study the properties of (120598 120598 or 119902)-fuzzyideals under a homomorphismThen we introduce the fuzzycosets and prove the fundamental isomorphism theorem on119873-groupswith respect to the structure induced by these fuzzycosets
Theorem 31 Let 119866 and 1198661015840 be two 119873-groups and let 119891 119866 rarr
1198661015840 be an119873-homomorphism If119891 is surjective and119860 is an (120598 120598or
119902)-fuzzy ideal of119866 then so is119891(119860) If119861 is a (120598 120598or119902)-fuzzy idealof 1198661015840 then 119891
minus1(119861) is a fuzzy ideal of 119866
Proof We assume that 119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 Forany 119909 119910 isin 119866
1015840 it follows that
119891 (119860) (119909 + 119910)
= sup119909+119910=119891(119911)
119860 (119911)
ge sup119891(119906)=119909119891(V)=119910
119860 (119906 + V)
ge sup119891(119906)=119909119891(V)=119910
min 119860 (119906) 119860 (V) 05
= min sup119891(119906)=119909
119860 (119906) sup119891(V)=119910
119860 (V) 05
= min 119891 (119860) (119909) 119891 (119860) (119910) 05
(5)
Also
119891 (119860) (minus119909)
= sup119891(119911)=minus119909
119860 (119911) = sup119891(minus119911)=119909
119860 (119911)
= sup119891(119911)=119909
119860 (minus119911)
ge sup119891(119911)=119909
min 119860 (119906) 05
= min sup119891(119911)=119909
119860 (119911) 05
= min 119891 (119860) (119909) 05
(6)
Again
119891 (119860) (119899119909)
= sup119891(119911)=119899119909
119860 (119911) ge sup119891(119906)=119909
119860 (119899119906)
ge sup119891(119906)=119909
min 119860 (119906) 05
= min sup119891(119906)=119909
119860 (119906) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119910 + 119909 minus 119910)
= sup119891(119911)=119910+119909minus119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119906 + V minus 119906)
6 Algebra
ge sup119891(V)=119909
min 119860 (V) 05
= min sup119891(V)=119909
119860 (V) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119899 (119910 + 119909) minus 119899119910)
= sup119891(119911)=119899(119910+119909)minus119899119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119899 (119906 + V) minus 119899119906)
ge sup119891(V)=119909119891(119906)=119910
min 119860 (119906) 05
= min 119891 (119860) (119909) 05
(7)
Therefore 119891(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866 Similarly wecan show that 119891minus1(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866
Definition 32 Let119860 be (isin isinor119902)-fuzzy subgroup of119866 For any119909 isin 119866 let 119860
119909be defined by 119860
119909(119892) = min119860(119892 minus 119909) 05 for
all 119892 isin 119866 This fuzzy subset 119860119909is called the (isin isin or119902)-fuzzy
left coset of 119866 determined by 119860 and 119909
Remark 33 Let 119860 be an (120598 120598 or 119902)-fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy normal if and only if min119860(119892 minus
119909) 05 = min119860(minus119909 + 119892) 05 for all 119909 119892 isin 119866 If 119860 is an(120598 120598 or 119902)-fuzzy ideal we simply denote fuzzy coset by 119860
119909
Lemma 34 Let 119860 be an (120598 120598 or 119902)-fuzzy ideal of 119866 Then119860119909
= 119860119910if an only if 119860(119909 minus 119910) ge 05
Proof Assume that 119860(119909 minus 119910) ge 05 and 119892 isin 119866 119860119909(119892) =
min119860(119892minus119909) 05 ge min119860(119892minus119910) 119860(119910minus119909) 05 ge min119860(119892minus
119910) 119860(119909 minus 119910) 05 = min119860(119892 minus 119910) 05 = 119860119910(119892) which
implies that 119860119909
ge 119860119910 Similarly we can verify that 119860
119909le
119860119910 Conversely we assume that 119860
119909= 119860119910 Then 119860
119910(119909) =
119860119909(119909) rArr min119860(119909 minus 119910) 05 = 05 rArr 119860(119909 minus 119910) ge 05
Proposition 35 Every fuzzy coset 119860119909is constant on every
coset of 1198660= 119909 isin 119866 | 119860(119909) = 119860(0)
Proof Let 119910 + 1199100
isin 119910 + 1198660 Now we have 119860
119909(119910 + 119910
0) =
min119860(119910+1199100minus119909) 05 ge min119860(119910minus119909) 119860(119909+119910
0minus119909) 05 ge
min119860(119910minus119909) 119860(1199100) 05 = min119860(119910minus119909) 05 = 119860
119909(119910) Also
119860119909(119910) = min119860(119910minus119909) 05 ge min119860(minus119909+119910+119910
0minus1199100) 05 ge
min119860(minus119909+119910+1199100) 119860(minus119910
0) 05 = min119860(minus119909+119910+119910
0) 05 ge
min119860(119910+1199100minus119909) 05 = 119860
119909(119910+1199100)Thus119860
119909(119910+1199100) = 119860
119909(119910)
for all 1199100isin 1198660
Theorem 36 For any (120598 120598 or 119902)-fuzzy ideal 119860 of 119866 119866119860the set
of all fuzzy cosets of 119860 in 119866 is an 119873-group under the additionand scalar multiplication defined by119860
119909+119860119910
= 119860119909+119910
119899(119860119909) =
119860119899119909
for all 119909 119910 isin 119866 119899 isin 119873 The function 119860(119860119909) 119866119860
rarr
[0 1] defined by 119860(119860119909) = 119860(119909) for all 119860
119909isin 119866119860is (120598 120598 or 119902)-
fuzzy ideal of 119866119860
Proof First we show that the compositions are well definedLet 119909 119910 119888 119889 isin 119866 119899 isin 119873 such that 119860
119909= 119860119910and 119860
119888= 119860119889
Then119860(119909minus119910) ge 05119860(119888minus119889) ge 05 Now we havemin119860(119909+
119888minus119889minus119910) 05 ge min119860(minus119910+119909+119888minus119889) 05 = 05which impliesthat119860(119909+ 119888minus119889minus119910) ge 05 So by Lemma 34 we have119860
119909+119888=
119860119910+119889
Again 119860119899119909
(119892) = min119860(119892 minus 119899119909) 05 ge min119860(119892 minus
119899119910) 119860(119899119909 minus 119899119910) 05 = min119860(119892 minus 119899119910) 119860[119899(119910 minus 119910 + 119909) minus
119899119910] 05 ge min119860(119892minus119899119910) 119860(minus119910+119909) 05 = 119860119899119910
(119892) Similarlywe show that 119860
119899119910(119892) ge 119860
119899119909(119892) Thus for 119860
119899119910(119892) = 119860
119899119909(119892)
for all 119892 isin 119866 Hence the compositions are well defined It isnow easy to verify that119866
119860is an119873-groupwith null element119860
0
and negative element119860minus119909 Next we check that119860 is (120598 120598 or 119902)-
fuzzy ideal of 119866119860 Let 119860
119909 119860119910
isin 119866119860 We have
(i) 119860(119860119909minus119860119910) = 119860(119860
119909minus119910) = 119860(119909minus119910) ge min119860(119909) 119860(119910)
05 = min119860(119860119909) 119860(119860
119910) 05
(ii) 119860(119860119899119909
) = 119860(119899119909) ge min119860(119909) 05 = min119860(119860119909)
05(iii) 119860(119860
119910+ 119860119909
minus 119860119910) = 119860(119860119910+119909minus119910
) = 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05 = min119860(119860119909) 05
(iv) 119860[119899(119860119910+119860119909)minus119899119860
119910] = 119860[119860
119899(119910+119909)minus119899119910] = 119860[119899(119910+119909)minus
119899119910] ge min119860(119909) 05 = min119860(119860119909) 05
Hence the proof is completed
Lemma 37 Let 119867 = 119909 isin 119866 | 119860119909
ge 1198600 119870 = 119909 isin 119866 | 119860
119909=
1198600 Then 119870 = 119867 = 119860
05
Proof Let 119909 isin 119867 Then for all 119892 isin 119866 119860119909(119892) ge 119860
0(119892) rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr min119860(minus119909) 05 ge
min119860(0) 05 = 05 rArr 119860(119909) ge 05 Also for 119892 isin 1198661198600(119892) = min119860(119892) 05 = min119860(119892 minus 119909 + 119909) 05 ge
min119860(119892 minus 119909) 119860(119909) 05 = min119860(119892 minus 119909) 05 = 119860119909(119892) rArr
1198600
ge 119860119909 So 119867 = 119870 As we have seen if 119909 isin 119867 then
119860(119909) ge 05 rArr 119909 isin 11986005 Conversely if 119909 isin 119860
05 for any
119892 isin 119866 119860(119892 minus 119909) ge min119860(119892) 119860(119909) 05 = min119860(119892) 05 rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr 119860119909
ge 1198600 Thus
119867 = 11986005 which means that 119867 is an ideal of 119866
Theorem38 If119860 is an (120598 120598or119902)-fuzzy ideal of119866 then themap119891 119866 rarr 119866
119860given by 119891(119909) = 119860
119909is an 119873-homomorphism
with kernel 119891 = 11986005
and so 11986611986005
is isomorphic to 119866119860
Proof It is clear that 119891 is an onto 119873-homomorphism from 119866
to 119866119860with kernel 119891 = 119909 isin 119866 | 119891(119909) = 119891(0) = 119909 isin 119866 |
119860119909
= 1198600 = 119860
05
Corollary 39 Let119860 be an (120598 120598or119902)-fuzzy ideal of119866 and let 119861be an (120598 120598or119902)-fuzzy ideal of119866
119860Then 119861 119866 rarr [0 1] defined
by 119861(119909) = 119861(119860119909) is an (120598 120598 or 119902)-fuzzy ideal of 119866 containing 119860
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
Algebra 7
[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971
[3] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982
[4] S Abou-Zaid ldquoOn fuzzy subnear-rings and idealsrdquo Fuzzy Setsand Systems vol 44 no 1 pp 139ndash146 1991
[5] B Davvaz ldquoFuzzy ideals of near-rings with interval valuedmembership functionsrdquo Journal of Sciences Islamic Republic ofIran vol 12 no 2 pp 171ndash175 2001
[6] K H Kim ldquoOn vague119877-subgroups of near-ringsrdquo InternationalMathematical Forum vol 4 no 33ndash36 pp 1655ndash1661 2009
[7] S K Bhakat and P Das ldquo(120598 120598 or 119902)-fuzzy subgrouprdquo Fuzzy Setsand Systems vol 80 no 3 pp 359ndash368 1996
[8] K H Kim Y B Jun and Y H Yon ldquoOn anti fuzzy ideals innear-ringsrdquo Iranian Journal of Fuzzy Systems vol 2 no 2 pp71ndash80 2005
[9] K H Kim and Y B Jun ldquoA note on fuzzy 119877-subgroups of near-ringsrdquo Soochow Journal of Mathematics vol 28 no 4 pp 339ndash346 2002
[10] SDKim andH S Kim ldquoOn fuzzy ideals of near-ringsrdquoBulletinof the Korean Mathematical Society vol 33 no 4 pp 593ndash6011996
[11] O R Devi ldquoIntuitionistic fuzzy near-rings and its propertiesrdquoThe Journal of the Indian Mathematical Society vol 74 no 3-4pp 203ndash219 2007
[12] S K Bhakat and P Das ldquoOn the definition of a fuzzy subgrouprdquoFuzzy Sets and Systems vol 51 no 2 pp 235ndash241 1992
[13] S K Bhakat and P Das ldquoFuzzy subrings and ideals redefinedrdquoFuzzy Sets and Systems vol 81 no 3 pp 383ndash393 1996
[14] B Davvaz ldquo(120598 120598 or 119902)-fuzzy subnear-rings and idealsrdquo SoftComputing vol 10 no 3 pp 206ndash211 2006
[15] B Davvaz ldquoFuzzy 119877-subgroups with thresholds of near-ringsand implication operatorsrdquo Soft Computing vol 12 no 9 pp875ndash879 2008
[16] A Narayanan and T Manikantan ldquo(120598 120598 or 119902)-fuzzy subnear-rings and (120598 120598or119902)-fuzzy ideals of near-ringsrdquo Journal of AppliedMathematics amp Computing vol 18 no 1-2 pp 419ndash430 2005
[17] J Zhan and B Davvaz ldquoGeneralized fuzzy ideals of near-ringsrdquoApplied Mathematics vol 24 no 3 pp 343ndash349 2009
[18] O R Devi and M R Singh ldquoOn fuzzy ideals of 119873-groupsrdquoBulletin of Pure amp Applied Sciences E vol 26 no 1 pp 11ndash232007
[19] T K Dutta and B K Biswas ldquoOn fuzzy congruence of a near-ring modulerdquo Fuzzy Sets and Systems vol 112 no 2 pp 343ndash348 2000
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Complex AnalysisJournal of
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![Page 2: Research Article A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf · of a ring was introduced by Liu [ ]. e notion of fuzzy subnear-ring and fuzzy ideals](https://reader036.fdocuments.net/reader036/viewer/2022071404/60f839985e18e36f2e647f1c/html5/thumbnails/2.jpg)
2 Algebra
By an 119873-group we mean a nonempty set 119866 together witha map Φ 119873 times 119866 rarr 119866 written as Φ(119899 119892) = 119899119892 satisfying thefollowing conditions
(i) (119866 +) is a group (not necessarily abelian)
(ii) (1198991+ 1198992)119892 = 119899
1119892 + 1198992119892
(iii) (11989911198992)119892 = 119899
1(1198992119892) for all 119899
1 1198992isin 119873 119892 isin 119866
Example 2 Let 119873 be a subnear-ring of 119872(G) Then G is an119873-group via function application as operation
Example 3 The additive group (119873 +) of a near-ring (119873 + sdot)
is an 119873-group via the near-ring multiplication
An ideal 119868 of 119873-group 119866 is an additive normal subgroupof 119866 such that 119873119868 sube 119868 and 119899(119892 + ℎ) minus 119899119892 isin 119868 for all ℎ isin 119868119892 isin 119866 119899 isin 119873 A mapping between two 119873-groups 119866 and 119866
1015840
is called an 119873-homomorphism if 119891(119892 + ℎ) = 119891(119892) + 119891(ℎ) and119891(119899119892) = 119899119891(119892) for all 119892 ℎ isin 119866 119899 isin 119873
Throughout this study we use 119873 to denote a zero-symmetric near-ring and 119866 to denote an 119873-group
For any fuzzy subset 119860 of 119866 119868119898119860 = 119860(119909) | 119909 isin 119866
denotes the image of 119860 For any subset 119868 of 119866 120594119868denotes the
characteristic function of 119868
Definition 4 (see [2]) A fuzzy subset 119860 of a group 119866 is calleda fuzzy subgroup of 119866 if it satisfies the following conditions
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910)
(ii) 119860(minus119909) ge 119860(119909)
for all 119909 119910 isin 119866
Definition 5 For a fuzzy subset 119860 of 119866 119905 isin (0 1] the subset119860119905
= 119909 isin 119866 | 119860(119909) ge 119905 is called a level subset of 119866 deter-mined by 119860 and 119905
The set 119909 isin 119866 | 119860(119909) gt 0 is called the support of 119860 andis denoted by Supp119860 A fuzzy subset 119860 of 119866 of the form
119860 (119910) =
119905 ( = 0) if 119910 = 119909
0 if 119910 = 119909
(2)
is said to be a fuzzy point denoted by 119909119905 Here 119909 is called the
support point and 119905 is called its value A fuzzy point 119909119905is
said to belong to (resp quasi coincident with) a fuzzy set 119860
written as 119909119905120598119860 (resp 119909
119905119902119860) if 119860(119909) ge 119905 (resp 119860(119909) + 119905 gt 1)
If 119909119905
isin 119860 or 119909119905119902119860 then we write 119909
119905isin or119902119860 The symbols
119909119905isin119860 119909
119905119902119860 119909119905isin or119902119860 mean that 119909
119905isin 119860 119909
119905119902119860 119909119905
isin or119902119860 donot hold respectively
Definition 6 (see [7 12]) A fuzzy subset of a group 119866 is saidto be an (120598 120598 or 119902)-fuzzy subgroup of 119866 if for all 119909 119910 isin 119866 and119905 119903 isin (0 1]
(i) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
Remark 7 (see [7]) The conditions (i) and (ii) of Definition 6are respectively equivalent to
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910) 05(ii) 119860(minus119909) ge min119860(119909) 05
for all 119909 119910 isin 119866
Remark 8 For any (120598 120598 or 119902)-fuzzy subgroup119860 of 119866 such that119860(119909) ge 05 for some 119909 isin 119866 then119860(0) ge 05 and if119860(0) lt 05then 119860(119909) lt 05 for all 119909 isin 119866 So 119860 is just the usual fuzzysubgroup in the sense of Rosenfeld
Remark 9 It is noted that if 119860 is a fuzzy subgroup then it isan (120598 120598 or 119902)-fuzzy subgroup of 119866 However the converse maynot be true
Here onwards we assume that 119860 is an (120598 120598 or 119902)-fuzzysubgroup in the nontrivial sense for which case we have119860(0) ge 05
Definition 10 (see [7]) An (120598 120598or119902)-fuzzy subgroup of a group119866 is said to be (120598 120598or119902)-fuzzy normal subgroup if for any 119909 119910 isin
119866 and 119905 isin (0 1]
119909119905isin 119860 997904rArr (119910 + 119909 minus 119910)
119905isin or119902119860 (3)
Remark 11 (see [7]) The condition of (120598 120598 or 119902)-fuzzy normalsubgroup is given in the equivalent forms as
(i) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05(ii) 119860(119909 + 119910) ge min119860(119910 + 119909) 05(iii) 119860([119909 119910]) ge min119860(119909) 05 for all 119909 119910 isin 119866
Here [119909 119910] denotes the commutator of 119909 119910 in 119866
In the light of this fact the condition of Definition 10 canbe replaced by any one of the above conditions in Remark 8
Definition 12 (see [18]) Let 119860 be a fuzzy subset of an 119873-group 119866 It is called a fuzzy 119873-subgroup of 119866 if it satisfies thefollowing conditions
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910)(ii) 119860(119899119909) ge 119860(119909)
for all 119909 119910 isin 119866 119899 isin 119873
Remark 13 If 119866 is a unitary 119873-group the above conditionsare equivalent to conditions119860(119909minus119910) ge min119860(119909) 119860(119910) and119860(119899119909) ge 119860(119909) for all 119909 119910 isin 119866 119899 isin 119873
Definition 14 (see [18 19]) A nonempty fuzzy subset 119860 of an119873-group 119866 is called a fuzzy ideal if it satisfies the followingconditions
(i) 119860(119909 minus 119910) ge min119860(119909) 119860(119910)(ii) 119860(119899119909) ge 119860(119909)(iii) 119860(119910 + 119909 minus 119910) ge 119860(119909)(iv) 119860(119899(119909 + 119910) minus 119899119909) ge 119860(119910)
for all 119909 119910 isin 119866 119899 isin 119873
Algebra 3
Definition 15 (see [14]) A fuzzy set 119860 of a near-ring 119873 iscalled an (120598 120598or119902)-fuzzy subnear-ring of119873 if for all 119905 119903 isin (0 1]and 119909 119910 isin 119873
(i) (a) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(b) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
(ii) 119909119905 119910119903isin 119860 rArr (119909119910)min119905119903 isin or119902119860
119860 is called an (120598 120598or119902)-fuzzy ideal of119873 if it is (120598 120598or119902)-fuzzy subnear-ring of 119873 and
(iii) 119909119905isin 119860 rArr (119910 + 119909 minus 119910) isin or119902119860
(iv) 119910119903isin 119860 119909 isin 119873 rArr (119910119909)
119903isin or119902119860
(v) 119886119905isin 119860 rArr (119910(119909 + 119886) minus 119910119909)
119905isin or119902119860
for all 119909 119910 119886 isin 119873
2 Generalized Fuzzy Ideals
In this section we give the definition of (120598 120598 or 119902)-fuzzysubgroup and ideal of an 119873-group 119866 based on Definitions 14and 15
Definition 16 A fuzzy subset 119860 of an 119873-group 119866 is said to bean (120598 120598 or 119902)-fuzzy subgroup of 119866 if 119909 119910 isin 119866 119899 isin 119873 119905 119903 isin
(0 1]
(i) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
(iii) 119909119905isin 119860 119899 isin 119873 rArr (119899119909)
119905isin or119902119860
Lemma 17 Let119860 be a fuzzy subset of119866 and 119905 119903 isin (0 1]Then
(i) 119909119905 119910119903
isin 119860 rArr (119909 + 119910)min119905119903120598 or 119902119860 hArr 119860(119909 + 119910) ge
min119860(119909) 119860(119910) 05(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860 hArr 119860(minus119909) ge min119860(119909) 05
for all 119909 isin 119866(iii) 119909
119905isin 119860 rArr (119899119909)
119905120598 or 119902119860 hArr 119860(119899119909) ge min119860(119909) 05
for all 119909 isin 119866 119899 isin 119873
Proof (i) Let 119909 119910 isin 119866 Consider the case (a) min119860(119909)
119860(119910) lt 05Assume that 119860(119909 + 119910) lt min119860(119909) 119860(119910) 05 =
min119860(119909) 119860(119910) Choose 119905 such that 119860(119909 + 119910) lt 119905 lt
min119860(119909) 119860(119910) which implies that 119909119905
isin 119860 119910119905
isin 119860 but(119909 + 119910)
119905isin or119902119860 [as 119860(119909 + 119910) + 119905 lt 1 and 119860(119909 + 119910) lt 119905]
Consider the case (b) min119860(119909) 119860(119910) ge 05 Assume that119860(119909 + 119910) lt min119860(119909) 119860(119910) 05 = 05 Choose 119905 such that119860(119909 + 119910) lt 119905 lt 05 so that 119909
119905 119910119905isin 119860 but (119909 + 119910)
119905isin or119902119860
Conversely let 119909119905 119910119903
isin 119860 rArr 119860(119909) ge 119905 119860(119910) ge 119903 Then119860(119909 + 119910) ge min119860(119909) 119860(119910) 05 ge min119905 119903 05 Thus 119860(119909 +
119910) ge min119905 119903 if either 119905 or 119903 le 05 and 119860(119909 + 119910) ge 05 if both119905 and 119903 gt 05 which means (119909 + 119910)min119905119903120598 or 119902119860
(ii) Let 119909 isin 119866 min119860(119909) 05 le 05 Suppose 119860(minus119909) lt
min119860(119909) 05 le 05 Choose 119903 such that 119860(minus119909) lt 119903 lt
min119860(119909) 05 le 05 Then 119909119903
isin 119860 but (minus119909)119903isin or119902119860 which
contradicts the hypothesis So 119860(minus119909) ge min119860(119909) 05 forall 119909 isin 119866
Conversely let 119909119905
isin 119860 Then 119860(119909) ge 119905 But we have119860(minus119909) ge min119860(119909) 05 ge min119905 05 rArr 119860(minus119909) ge 119905 or119860(minus119909) ge 05 according as 119905 le 05 or 119905 gt 05 rArr (minus119909)
119905isin or119902119860
(iii) Let119909 isin 119866 andmin119860(119909) 05 le 05 Suppose119860(119899119909) lt
min119860(119909) 05 le 05 Choose 119903 such that 119860(119899119909) lt 119903 lt
min119860(119909) 05 le 05 Then 119860(119909) gt 119903 that is 119909119903
isin 119860 but(119899119909)119903isin or119902119860 as 119860(119899119909) lt 119903 and 119860(119899119909) + 119903 le 1
Conversely let 119909119905isin 119860 119899 isin 119873 then119860(119909) ge 119905 But119860(119899119909) ge
min119860(119909) 05 ge min119905 05 rArr 119860(119899119909) ge 119905 or 119860(119899119909) ge 05
according as 119905 le 05 or 119903 gt 05 rArr 119860(119899119909) ge 119905 or 119860(119899119909) + 119905 gt
1 rArr (119899119909)119905isin or119902119860
Theorem 18 Let119860 be a fuzzy subset of119866 Then119860 is an (120598 120598or
119902)-fuzzy subgroup of 119866 if and only if the following conditionsare satisfied
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910) 05(ii) 119860(minus119909) ge min119860(119909) 05(iii) 119860(119899119909) ge min119860(119909) 05
for all 119909 119910 isin 119866 119899 isin 119873
Proof It follows from the previous lemma
Definition 19 A fuzzy subset 119860 of an 119873-group 119866 is said to be(120598 120598 or 119902)-fuzzy ideal of 119866 if it is an (120598 120598 or 119902)-fuzzy subgroupand satisfies the following conditions
(i) 119909119905isin 119860 rArr (119910 + 119909 minus 119910)
119905isin V119902119860
(ii) 119886119905isin 119860 rArr (119899(119909 + 119886) minus 119899119909)
119905isin V119902119860
for any 119899 isin 119873 119909 119886 isin 119866
Lemma20 Let119860 be a fuzzy subset of119866 and 119905 119903 isin (0 1]Then
(i) 119909119905
isin 119860 rArr (119910 + 119909 minus 119910)119905
isin V119902119860 hArr 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05(ii) 119886119905isin 119860 rArr (119899(119909+119886)minus119899119909)
119905isin V119902119860 hArr 119860(119899(119909+119886)minus119899119909) ge
min119860(119909) 05
Proof (i) Assume that 119860(119910 + 119909 minus 119910) lt min119860(119909) 05Choose 119905 such that 119860(119910 + 119909 minus 119910) lt 119905 lt min119860(119909) 05 Butmin119860(119909) 05 le 119860(119909) or 05 according as 119860(119909) lt 05 or119860(119909) ge 05 So 119860(119909) gt 119905 or 119860(119909) ge 05 rArr 119909
119905isin 119860 or 119909
05isin 119860
But (119910 + 119909 minus 119910)119905isin V119902119860 or (119910 + 119909 minus 119910)
05isin V119902119860 respectively
which contradicts the hypothesisConversely assume that 119909
119905isin 119860 then 119860(119909) ge 119905 For any
119910 isin 119866 we have119860(119910+119909minus119910) ge min119860(119909) 05 ge min119905 05 rArr
119860(119910 + 119909 minus 119910) ge 119905 or 05 according as 119905 lt 05 or 119905 ge 05 rArr
(119910+119909minus119910)119905isin 119860 or119860(119910+119909minus119910)+119905 gt 1 SorArr (119910+119909minus119910)
119905isin V119902119860
(ii) Assume that119860(119899(119909+119886)minus119899119909) lt min119860(119886) 05 = 119860(119886)
or 05 for some 119899 isin 119873 119909 119886 isin 119866 According as 119860(119886) lt 05 or119860(119886) ge 05 Choose 119905 isin (0 1] such that 119860(119899(119909 + 119886) minus 119899119909) lt
119905 lt min119860(119886) 05 le 05 In either case 119860(119899(119909 + 119886) minus 119899119909) lt 119905
and 119860(119899(119909 + 119886) minus 119899119909) + 119905 lt 1 So (119899(119909 + 119886) minus 119899119909)119905isin V119902119860 a
contradictionConversely assume that119860(119899(119909+119886)minus119899119909) ge min119860(119886) 05
for all 119886 119909 isin 119866 119899 isin 119873 Let 119886119905
isin 119860 Then 119860(119886) ge 119905 So119860(119899(119909 + 119886) minus 119899119909) ge min119905 05 le 05 = 119905 or 05 according as119905 le 05 or 119905 gt 05 So 119860(119899(119909 + 119886) minus 119899119909)
119905isin V119902119860
4 Algebra
Theorem 21 Let 119860 be an (120598 120598 or 119902) fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 if and only if
(i) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 for all 119909 119910 isin 119866
(ii) 119860(119899(119909 + 119886) minus 119899119909) ge min119860(119886) 05 for all 119899 isin 119873119909 119886 isin 119866
Proof It is immediate from Lemma 20
By definition a fuzzy ideal of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 But the converse is not true in general as shown by thefollowing example
Example 22 Consider 119866 = S3
= 119894 1205881 1205882 1205911 1205912 1205913 (written
additively) to be a Z-group Define a fuzzy subset 119860 of 119866
as 119860(119894) = 1 119860(1205881) = 119860(120588
2) = 119860(120591
2) = 119860(120591
3) = 06
119860(1205911) = 08 which is not fuzzy ideal as 119860[2(120591
1+ 1205912) minus 2120591
2] =
119860(1205881) = 119860(120588
2) = 06 lt 119860(120591
1) it contradicts the condition
(iv) of Definition 14 As 119860(119909 minus 119910) 119860(119899119909) 119860(119910 + 119909 minus 119910) and119860(119899(119909+119886)minus119899119909) = 06 or 08 ge min05 06 or 08 = 05 thusthe notion of (120598 120598or119902)-fuzzy ideal is a successful generalizationof fuzzy ideals of 119866 as introduced in [18]
Theorem 23 Let 119860119894 119894 isin 119869 be any family of (120598 120598 or 119902)-fuzzy
ideals of 119866 Then 119860 = ⋂119860119894is an (120598 120598 or 119902)-fuzzy ideal of 119866
Proof It is straightforward
Theorem 24 A nonempty subset 119868 of 119866 is an ideal of 119866 if andonly if 120594
119868is an (isin isin or119902)-fuzzy ideal of 119866
Proof If 119868 is an ideal of119866 it is clear from [18 Proposition 211]that 120594
119868is fuzzy ideal of 119866 Since every fuzzy ideal is (120598 120598 or 119902)-
fuzzy ideal 120594119868is (120598 120598 or 119902)-fuzzy ideal of 119866
Conversely let 120594119868be an (120598 120598 or 119902)-fuzzy ideal of 119866 Let
119909 119910 isin 119868 120594119868(119909 minus 119910) ge min120594
119868(119909) 120594119868(119910) 05 = 05 So
120594119868(119909 minus 119910) = 1 rArr 119909 minus 119910 isin 119868 Let 119899 isin 119873 119909 isin 119868 120594
119868(119899119909) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899119909) = 1 rArr 119899119909 isin 119868119910 isin 119866 119909 isin 119868
120594119868(119910 + 119909 minus 119910) ge min120594
119868(119909) 05 = 05 rArr 120594
119868(119910 + 119909 minus 119910) =
1 rArr 119910 + 119909 minus 119910 isin 119868119899 isin 119873 119909 isin 119868 119910 isin 119866 120594119868(119899(119910 + 119909) minus 119899119910) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899(119910 + 119909) minus 119899119910) = 1 rArr 119899(119910 + 119909) minus
119899119910 isin 119868 Then 119868 is an ideal of 119866
Theorem 25 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy(subgroup) ideal of 119866 if and only if the level subset 119860
119905is a
(subgroup) ideal for 0 lt 119905 le 05
Proof We prove the result for (120598 120598or119902)-fuzzy ideal 119878 Let119860 bean (120598 120598 or 119902)-fuzzy ideal of 119866 Let 119905 le 05 119909 119910 119894 isin 119860
119905 119899 isin 119873
(i) 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 ge min119905 05 = 119905 rArr
119909 minus 119910 isin 119860119905
(ii) 119860(119899119909) ge min119860(119886) 05 ge min119905 05 = 119905 rArr 119899119909 isin
119860119905
(iii) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 ge min119905 05 = 119905 rArr
119910 + 119909 minus 119910 isin 119860119905 119910 isin 119866
(iv) 119860(119899(119910 + 119909) minus 119899119910) ge min119860(119909) 05 ge min119905 05 =
119905 rArr 119899(119910 + 119909) minus 119899119910 isin 119860119905 119910 isin 119866 119899 isin 119873
Hence 119860119905is an ideal of 119866 Again let 119860
119905be an ideal of 119866 for
all 119905 le 05 If possible let there exist 119909 119910 isin 119866 such that 119860(119909 minus
119910) lt 119905 lt min119860(119909) 119860(119910) 05 Let 119905 be such that 119860(119909 minus 119910) lt
119905 lt min119860(119909) 119860(119910) 05 rArr 119909 119910 isin 119860119905and 119909 minus 119910 notin 119860
119905 a
contradiction So 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 for all119909 119910 isin 119866 For 119899 isin 119873 119909 isin 119866 let 119860(119899119909) lt min119860(119909) 05 Ifpossible let 119905 be such that 119860(119899119909) lt 119905 lt min119860(119909) 05 Thisimplies 119909 isin 119860
119905 but 119899119909 notin 119860
119905 a contradiction Similarly we
can prove that 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 119860[119899(119910 + 119909) minus
119899119910] ge min119860(119909) 05 119909 119910 isin 119866 119899 isin 119873
Remark 26 For 119905 isin (05 1) 119860 may be an (120598 120598 or 119902)-fuzzyideal of 119866 but 119860
119905may not be an ideal of 119866 Let 119905 = 08 in
Example 22 Then 119860119905= 119894 120591
1 119860119905is not an ideal of S
3as it is
not a normal subgroup of S3
We are looking for a corresponding result when 119860119905is an
ideal of 119866 for all 119905 isin (05 1]
Theorem 27 Let 119860 be a fuzzy subset of an 119873-group 119866 Then119860119905
= 120601 is an ideal of119866 for all 119905 isin (05 1] if and only if119860 satisfiesthe following conditions
(i) max119860(119909 minus 119910) 05 ge min119860(119909) 119860(119910)
(ii) max119860(119899119909) 05 ge 119860(119909)
(iii) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)
(iv) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
for all 119909 119910 isin 119866 119899 isin 119873
Proof Suppose that 119860119905
= 120601 is an ideal of 119866 for all 119905 isin
(05 1] In order to prove (i) suppose that for some 119909 119910 isin
119866 max119860(119909 minus 119910) 05 lt min119860(119909) 119860(119910) Let 119905 =
min119860(119909) 119860(119910) So 119909 119910 isin 119860119905and 119905 isin (05 1] Since 119860
119905is
an ideal 119909 minus 119910 isin 119860119905 So 119860(119909 minus 119910) ge 119905 gt max119860(119909 minus 119910) 05
a contradiction In order to prove (ii) suppose that 119909 isin 119866119899 isin 119873 and max119860(119899119909) 05 lt 119860(119909) = 119905 (say) Then119909 isin 119860
119905rArr 119899119909 isin 119860
119905rArr 119860(119899119909) ge 119905 gt max119860(119899119909) 05 a
contradiction Similarly we can prove (iii) and (iv)Conversely suppose that conditions (i) to (iv) hold We
show that 119860119905is an ideal of 119866 for all 119905 isin (05 1] Let 119909 119910 isin 119860
119905
Then 05 lt 119905 le min119860(119909) 119860(119910) le max119860(119909 minus 119910) 05 =
119860(119909 minus 119910) So 119909 minus 119910 isin 119860119905 Let 119899 isin 119873 119909 isin 119860
119905 Then 05 lt
119905 le 119860(119909) le max119860(119909) 05 = 119860(119909) so 119899119909 isin 119860119905 For 119909 isin 119860
119905
119910 isin 119866 05 lt 119905 le 119860(119909) le max119860(119910 + 119909 minus 119910) 05 = 119860(119910 + 119909 minus
119910) rArr 119910 + 119909 minus 119910 isin 119860119905 Also if 119899 isin 119873 119909 isin 119860
119905 119910 isin 119866 05 lt
119905 le 119860(119909) le max119860(119899(119910 + 119909) minus 119899119910) 05 = 119860(119899(119910 + 119909) minus 119899119910)Hence 119899(119910 + 119909) minus 119899119910 isin 119860
119905 Then 119860
119905is an ideal of 119866
A definition for the previous kind of fuzzy subset wasgiven for the case of near-rings in [17] Now we give thedefinition for 119873-groups
Definition 28 A fuzzy subset of 119866 is called an (120598 120598 or 119902)-fuzzysubgroup of 119866 if for all 119905 119903 isin (0 1] and for all 119909 119910 isin 119866 119899 isin 119873
(i) (a) (119909 + 119910)min(119905119903)isin119860 implies 119909119905isinV119902119860 or 119910
119903isinV119902119860
(b) (minus119909)119905isin119860 implies 119909
119905isinV119902119860
Algebra 5
(ii) (119899119909)119905isin119860 implies 119909
119905isinV119902119860
Moreover 119860 is called an (120598 120598 or 119902)-fuzzy ideal of 119866 if119860 is (120598 120598 or 119902)-fuzzy subgroup of 119866 and
(iii) (119910 + 119909 minus 119910)119905isin119860 implies 119909
119905isinV119902119860
(iv) (119899(119909 + 119910) minus 119899119910)119905isin119860 implies 119909
119905isinV119902119860
Theorem 29 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 if and only if
(1) (a) max119860(119909 + 119910) 05 ge min119860(119909) 119860(119910)(b) max119860(minus119909) 05 ge 119860(119909)
(2) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)(3) max119860(119899119909) 05 ge 119860(119909)(4) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
Proof (i)a hArr (1)a Let 119909 119910 isin 119866 be such that max119860(119909 +
119910) 05 lt min119860(119909) 119860(119910) Let 119905 = min119860(119909) 119860(119910) then05 lt 119905 le 1 (119909 + 119910)
119905isin119860 So we must have 119909
119905isinV119902119860 or 119910
119905isinV119902119860
But 119909119905isin 119860 and 119910
119905isin 119860 Here 119909
119905119902119860 or 119910
119905119902119860 then 119905 le 119860(119909) and
119860(119909) + 119905 le 1 or 119905 le 119860(119910) and 119860(119910) + 119905 le 1 then 119905 le 05 acontradiction
Conversely let (119909 + 119910)min(119905119903)isin119860 Then 119860(119909 + 119910) lt min(119905119903) If 119860(119909 + 119910) ge min119860(119909) 119860(119910) then min119860(119909) 119860(119910) lt
min(119905 119903) Hence either 119860(119909) lt 119905 or 119860(119910) lt 119903 which implies119909119905isin119860 or 119910
119903isin119860 Thus 119909
119905isinV119902119860 or 119910
119903isinV119902119860
Again if 119860(119909 + 119910) lt min119860(119909) 119860(119910) then by (1)a
05 ge min 119860 (119909) 119860 (119910) gt 119860 (119909 + 119910) (4)
Suppose that 119909119905
isin 119860 or 119910119903
isin 119860 then 119905 le 119860(119909) le 05 or119903 le 119860(119910) le 05 It follows that either 119909
119905119902119860 or 119910
119905119902119860 and thus
119909119905isinV119902119860 or 119910
119903isinV119902119860
(i)b hArr (1)b Suppose that there exists 119909 isin 119866 such thatmax119860(minus119909) 05 lt 119860(119909) If 119860(119909) = 119905 then 05 lt 119905 le 1 and119860(minus119909) lt 119905 so that (minus119909)
119905isin119860 But then we must have either
(119909)119905isin119860 or 119909
119905119902119860 Also we have (119909)
119905isin 119860 So 119860(119909) + 1 le 1
which means that 119905 le 05 a contradictionConversely suppose that (119909)
119905isin119860 then 119860(minus119909) lt 119905 If
119860(minus119909) ge 119860(119909) then 119860(119909) lt 119905 which gives 119909119905isinV119902119860 Again if
119860(minus119909) lt 119860(119909) by (1)b we have 05 ge 119860(119909) Putting (119909)119905isin 119860
then 119905 le 119860(119909) le 05 so that 119909119905119902119860 which means that 119909
119905isinV119902119860
Similarly we can prove the remaining parts
Theorem 30 A fuzzy subset 119860 of 119866 is an (120598 120598V119902)-fuzzy idealif and only if 119860
119905( = 120601) is an ideal of 119866 for all 119905 isin (05 1]
3 Fuzzy Cosets and Isomorphism Theorem
In this section we first study the properties of (120598 120598 or 119902)-fuzzyideals under a homomorphismThen we introduce the fuzzycosets and prove the fundamental isomorphism theorem on119873-groupswith respect to the structure induced by these fuzzycosets
Theorem 31 Let 119866 and 1198661015840 be two 119873-groups and let 119891 119866 rarr
1198661015840 be an119873-homomorphism If119891 is surjective and119860 is an (120598 120598or
119902)-fuzzy ideal of119866 then so is119891(119860) If119861 is a (120598 120598or119902)-fuzzy idealof 1198661015840 then 119891
minus1(119861) is a fuzzy ideal of 119866
Proof We assume that 119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 Forany 119909 119910 isin 119866
1015840 it follows that
119891 (119860) (119909 + 119910)
= sup119909+119910=119891(119911)
119860 (119911)
ge sup119891(119906)=119909119891(V)=119910
119860 (119906 + V)
ge sup119891(119906)=119909119891(V)=119910
min 119860 (119906) 119860 (V) 05
= min sup119891(119906)=119909
119860 (119906) sup119891(V)=119910
119860 (V) 05
= min 119891 (119860) (119909) 119891 (119860) (119910) 05
(5)
Also
119891 (119860) (minus119909)
= sup119891(119911)=minus119909
119860 (119911) = sup119891(minus119911)=119909
119860 (119911)
= sup119891(119911)=119909
119860 (minus119911)
ge sup119891(119911)=119909
min 119860 (119906) 05
= min sup119891(119911)=119909
119860 (119911) 05
= min 119891 (119860) (119909) 05
(6)
Again
119891 (119860) (119899119909)
= sup119891(119911)=119899119909
119860 (119911) ge sup119891(119906)=119909
119860 (119899119906)
ge sup119891(119906)=119909
min 119860 (119906) 05
= min sup119891(119906)=119909
119860 (119906) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119910 + 119909 minus 119910)
= sup119891(119911)=119910+119909minus119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119906 + V minus 119906)
6 Algebra
ge sup119891(V)=119909
min 119860 (V) 05
= min sup119891(V)=119909
119860 (V) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119899 (119910 + 119909) minus 119899119910)
= sup119891(119911)=119899(119910+119909)minus119899119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119899 (119906 + V) minus 119899119906)
ge sup119891(V)=119909119891(119906)=119910
min 119860 (119906) 05
= min 119891 (119860) (119909) 05
(7)
Therefore 119891(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866 Similarly wecan show that 119891minus1(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866
Definition 32 Let119860 be (isin isinor119902)-fuzzy subgroup of119866 For any119909 isin 119866 let 119860
119909be defined by 119860
119909(119892) = min119860(119892 minus 119909) 05 for
all 119892 isin 119866 This fuzzy subset 119860119909is called the (isin isin or119902)-fuzzy
left coset of 119866 determined by 119860 and 119909
Remark 33 Let 119860 be an (120598 120598 or 119902)-fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy normal if and only if min119860(119892 minus
119909) 05 = min119860(minus119909 + 119892) 05 for all 119909 119892 isin 119866 If 119860 is an(120598 120598 or 119902)-fuzzy ideal we simply denote fuzzy coset by 119860
119909
Lemma 34 Let 119860 be an (120598 120598 or 119902)-fuzzy ideal of 119866 Then119860119909
= 119860119910if an only if 119860(119909 minus 119910) ge 05
Proof Assume that 119860(119909 minus 119910) ge 05 and 119892 isin 119866 119860119909(119892) =
min119860(119892minus119909) 05 ge min119860(119892minus119910) 119860(119910minus119909) 05 ge min119860(119892minus
119910) 119860(119909 minus 119910) 05 = min119860(119892 minus 119910) 05 = 119860119910(119892) which
implies that 119860119909
ge 119860119910 Similarly we can verify that 119860
119909le
119860119910 Conversely we assume that 119860
119909= 119860119910 Then 119860
119910(119909) =
119860119909(119909) rArr min119860(119909 minus 119910) 05 = 05 rArr 119860(119909 minus 119910) ge 05
Proposition 35 Every fuzzy coset 119860119909is constant on every
coset of 1198660= 119909 isin 119866 | 119860(119909) = 119860(0)
Proof Let 119910 + 1199100
isin 119910 + 1198660 Now we have 119860
119909(119910 + 119910
0) =
min119860(119910+1199100minus119909) 05 ge min119860(119910minus119909) 119860(119909+119910
0minus119909) 05 ge
min119860(119910minus119909) 119860(1199100) 05 = min119860(119910minus119909) 05 = 119860
119909(119910) Also
119860119909(119910) = min119860(119910minus119909) 05 ge min119860(minus119909+119910+119910
0minus1199100) 05 ge
min119860(minus119909+119910+1199100) 119860(minus119910
0) 05 = min119860(minus119909+119910+119910
0) 05 ge
min119860(119910+1199100minus119909) 05 = 119860
119909(119910+1199100)Thus119860
119909(119910+1199100) = 119860
119909(119910)
for all 1199100isin 1198660
Theorem 36 For any (120598 120598 or 119902)-fuzzy ideal 119860 of 119866 119866119860the set
of all fuzzy cosets of 119860 in 119866 is an 119873-group under the additionand scalar multiplication defined by119860
119909+119860119910
= 119860119909+119910
119899(119860119909) =
119860119899119909
for all 119909 119910 isin 119866 119899 isin 119873 The function 119860(119860119909) 119866119860
rarr
[0 1] defined by 119860(119860119909) = 119860(119909) for all 119860
119909isin 119866119860is (120598 120598 or 119902)-
fuzzy ideal of 119866119860
Proof First we show that the compositions are well definedLet 119909 119910 119888 119889 isin 119866 119899 isin 119873 such that 119860
119909= 119860119910and 119860
119888= 119860119889
Then119860(119909minus119910) ge 05119860(119888minus119889) ge 05 Now we havemin119860(119909+
119888minus119889minus119910) 05 ge min119860(minus119910+119909+119888minus119889) 05 = 05which impliesthat119860(119909+ 119888minus119889minus119910) ge 05 So by Lemma 34 we have119860
119909+119888=
119860119910+119889
Again 119860119899119909
(119892) = min119860(119892 minus 119899119909) 05 ge min119860(119892 minus
119899119910) 119860(119899119909 minus 119899119910) 05 = min119860(119892 minus 119899119910) 119860[119899(119910 minus 119910 + 119909) minus
119899119910] 05 ge min119860(119892minus119899119910) 119860(minus119910+119909) 05 = 119860119899119910
(119892) Similarlywe show that 119860
119899119910(119892) ge 119860
119899119909(119892) Thus for 119860
119899119910(119892) = 119860
119899119909(119892)
for all 119892 isin 119866 Hence the compositions are well defined It isnow easy to verify that119866
119860is an119873-groupwith null element119860
0
and negative element119860minus119909 Next we check that119860 is (120598 120598 or 119902)-
fuzzy ideal of 119866119860 Let 119860
119909 119860119910
isin 119866119860 We have
(i) 119860(119860119909minus119860119910) = 119860(119860
119909minus119910) = 119860(119909minus119910) ge min119860(119909) 119860(119910)
05 = min119860(119860119909) 119860(119860
119910) 05
(ii) 119860(119860119899119909
) = 119860(119899119909) ge min119860(119909) 05 = min119860(119860119909)
05(iii) 119860(119860
119910+ 119860119909
minus 119860119910) = 119860(119860119910+119909minus119910
) = 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05 = min119860(119860119909) 05
(iv) 119860[119899(119860119910+119860119909)minus119899119860
119910] = 119860[119860
119899(119910+119909)minus119899119910] = 119860[119899(119910+119909)minus
119899119910] ge min119860(119909) 05 = min119860(119860119909) 05
Hence the proof is completed
Lemma 37 Let 119867 = 119909 isin 119866 | 119860119909
ge 1198600 119870 = 119909 isin 119866 | 119860
119909=
1198600 Then 119870 = 119867 = 119860
05
Proof Let 119909 isin 119867 Then for all 119892 isin 119866 119860119909(119892) ge 119860
0(119892) rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr min119860(minus119909) 05 ge
min119860(0) 05 = 05 rArr 119860(119909) ge 05 Also for 119892 isin 1198661198600(119892) = min119860(119892) 05 = min119860(119892 minus 119909 + 119909) 05 ge
min119860(119892 minus 119909) 119860(119909) 05 = min119860(119892 minus 119909) 05 = 119860119909(119892) rArr
1198600
ge 119860119909 So 119867 = 119870 As we have seen if 119909 isin 119867 then
119860(119909) ge 05 rArr 119909 isin 11986005 Conversely if 119909 isin 119860
05 for any
119892 isin 119866 119860(119892 minus 119909) ge min119860(119892) 119860(119909) 05 = min119860(119892) 05 rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr 119860119909
ge 1198600 Thus
119867 = 11986005 which means that 119867 is an ideal of 119866
Theorem38 If119860 is an (120598 120598or119902)-fuzzy ideal of119866 then themap119891 119866 rarr 119866
119860given by 119891(119909) = 119860
119909is an 119873-homomorphism
with kernel 119891 = 11986005
and so 11986611986005
is isomorphic to 119866119860
Proof It is clear that 119891 is an onto 119873-homomorphism from 119866
to 119866119860with kernel 119891 = 119909 isin 119866 | 119891(119909) = 119891(0) = 119909 isin 119866 |
119860119909
= 1198600 = 119860
05
Corollary 39 Let119860 be an (120598 120598or119902)-fuzzy ideal of119866 and let 119861be an (120598 120598or119902)-fuzzy ideal of119866
119860Then 119861 119866 rarr [0 1] defined
by 119861(119909) = 119861(119860119909) is an (120598 120598 or 119902)-fuzzy ideal of 119866 containing 119860
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
Algebra 7
[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971
[3] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982
[4] S Abou-Zaid ldquoOn fuzzy subnear-rings and idealsrdquo Fuzzy Setsand Systems vol 44 no 1 pp 139ndash146 1991
[5] B Davvaz ldquoFuzzy ideals of near-rings with interval valuedmembership functionsrdquo Journal of Sciences Islamic Republic ofIran vol 12 no 2 pp 171ndash175 2001
[6] K H Kim ldquoOn vague119877-subgroups of near-ringsrdquo InternationalMathematical Forum vol 4 no 33ndash36 pp 1655ndash1661 2009
[7] S K Bhakat and P Das ldquo(120598 120598 or 119902)-fuzzy subgrouprdquo Fuzzy Setsand Systems vol 80 no 3 pp 359ndash368 1996
[8] K H Kim Y B Jun and Y H Yon ldquoOn anti fuzzy ideals innear-ringsrdquo Iranian Journal of Fuzzy Systems vol 2 no 2 pp71ndash80 2005
[9] K H Kim and Y B Jun ldquoA note on fuzzy 119877-subgroups of near-ringsrdquo Soochow Journal of Mathematics vol 28 no 4 pp 339ndash346 2002
[10] SDKim andH S Kim ldquoOn fuzzy ideals of near-ringsrdquoBulletinof the Korean Mathematical Society vol 33 no 4 pp 593ndash6011996
[11] O R Devi ldquoIntuitionistic fuzzy near-rings and its propertiesrdquoThe Journal of the Indian Mathematical Society vol 74 no 3-4pp 203ndash219 2007
[12] S K Bhakat and P Das ldquoOn the definition of a fuzzy subgrouprdquoFuzzy Sets and Systems vol 51 no 2 pp 235ndash241 1992
[13] S K Bhakat and P Das ldquoFuzzy subrings and ideals redefinedrdquoFuzzy Sets and Systems vol 81 no 3 pp 383ndash393 1996
[14] B Davvaz ldquo(120598 120598 or 119902)-fuzzy subnear-rings and idealsrdquo SoftComputing vol 10 no 3 pp 206ndash211 2006
[15] B Davvaz ldquoFuzzy 119877-subgroups with thresholds of near-ringsand implication operatorsrdquo Soft Computing vol 12 no 9 pp875ndash879 2008
[16] A Narayanan and T Manikantan ldquo(120598 120598 or 119902)-fuzzy subnear-rings and (120598 120598or119902)-fuzzy ideals of near-ringsrdquo Journal of AppliedMathematics amp Computing vol 18 no 1-2 pp 419ndash430 2005
[17] J Zhan and B Davvaz ldquoGeneralized fuzzy ideals of near-ringsrdquoApplied Mathematics vol 24 no 3 pp 343ndash349 2009
[18] O R Devi and M R Singh ldquoOn fuzzy ideals of 119873-groupsrdquoBulletin of Pure amp Applied Sciences E vol 26 no 1 pp 11ndash232007
[19] T K Dutta and B K Biswas ldquoOn fuzzy congruence of a near-ring modulerdquo Fuzzy Sets and Systems vol 112 no 2 pp 343ndash348 2000
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 3: Research Article A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf · of a ring was introduced by Liu [ ]. e notion of fuzzy subnear-ring and fuzzy ideals](https://reader036.fdocuments.net/reader036/viewer/2022071404/60f839985e18e36f2e647f1c/html5/thumbnails/3.jpg)
Algebra 3
Definition 15 (see [14]) A fuzzy set 119860 of a near-ring 119873 iscalled an (120598 120598or119902)-fuzzy subnear-ring of119873 if for all 119905 119903 isin (0 1]and 119909 119910 isin 119873
(i) (a) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(b) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
(ii) 119909119905 119910119903isin 119860 rArr (119909119910)min119905119903 isin or119902119860
119860 is called an (120598 120598or119902)-fuzzy ideal of119873 if it is (120598 120598or119902)-fuzzy subnear-ring of 119873 and
(iii) 119909119905isin 119860 rArr (119910 + 119909 minus 119910) isin or119902119860
(iv) 119910119903isin 119860 119909 isin 119873 rArr (119910119909)
119903isin or119902119860
(v) 119886119905isin 119860 rArr (119910(119909 + 119886) minus 119910119909)
119905isin or119902119860
for all 119909 119910 119886 isin 119873
2 Generalized Fuzzy Ideals
In this section we give the definition of (120598 120598 or 119902)-fuzzysubgroup and ideal of an 119873-group 119866 based on Definitions 14and 15
Definition 16 A fuzzy subset 119860 of an 119873-group 119866 is said to bean (120598 120598 or 119902)-fuzzy subgroup of 119866 if 119909 119910 isin 119866 119899 isin 119873 119905 119903 isin
(0 1]
(i) 119909119905 119910119903isin 119860 rArr (119909 + 119910)min119905119903 isin or119902119860
(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860
(iii) 119909119905isin 119860 119899 isin 119873 rArr (119899119909)
119905isin or119902119860
Lemma 17 Let119860 be a fuzzy subset of119866 and 119905 119903 isin (0 1]Then
(i) 119909119905 119910119903
isin 119860 rArr (119909 + 119910)min119905119903120598 or 119902119860 hArr 119860(119909 + 119910) ge
min119860(119909) 119860(119910) 05(ii) 119909119905isin 119860 rArr (minus119909)
119905isin or119902119860 hArr 119860(minus119909) ge min119860(119909) 05
for all 119909 isin 119866(iii) 119909
119905isin 119860 rArr (119899119909)
119905120598 or 119902119860 hArr 119860(119899119909) ge min119860(119909) 05
for all 119909 isin 119866 119899 isin 119873
Proof (i) Let 119909 119910 isin 119866 Consider the case (a) min119860(119909)
119860(119910) lt 05Assume that 119860(119909 + 119910) lt min119860(119909) 119860(119910) 05 =
min119860(119909) 119860(119910) Choose 119905 such that 119860(119909 + 119910) lt 119905 lt
min119860(119909) 119860(119910) which implies that 119909119905
isin 119860 119910119905
isin 119860 but(119909 + 119910)
119905isin or119902119860 [as 119860(119909 + 119910) + 119905 lt 1 and 119860(119909 + 119910) lt 119905]
Consider the case (b) min119860(119909) 119860(119910) ge 05 Assume that119860(119909 + 119910) lt min119860(119909) 119860(119910) 05 = 05 Choose 119905 such that119860(119909 + 119910) lt 119905 lt 05 so that 119909
119905 119910119905isin 119860 but (119909 + 119910)
119905isin or119902119860
Conversely let 119909119905 119910119903
isin 119860 rArr 119860(119909) ge 119905 119860(119910) ge 119903 Then119860(119909 + 119910) ge min119860(119909) 119860(119910) 05 ge min119905 119903 05 Thus 119860(119909 +
119910) ge min119905 119903 if either 119905 or 119903 le 05 and 119860(119909 + 119910) ge 05 if both119905 and 119903 gt 05 which means (119909 + 119910)min119905119903120598 or 119902119860
(ii) Let 119909 isin 119866 min119860(119909) 05 le 05 Suppose 119860(minus119909) lt
min119860(119909) 05 le 05 Choose 119903 such that 119860(minus119909) lt 119903 lt
min119860(119909) 05 le 05 Then 119909119903
isin 119860 but (minus119909)119903isin or119902119860 which
contradicts the hypothesis So 119860(minus119909) ge min119860(119909) 05 forall 119909 isin 119866
Conversely let 119909119905
isin 119860 Then 119860(119909) ge 119905 But we have119860(minus119909) ge min119860(119909) 05 ge min119905 05 rArr 119860(minus119909) ge 119905 or119860(minus119909) ge 05 according as 119905 le 05 or 119905 gt 05 rArr (minus119909)
119905isin or119902119860
(iii) Let119909 isin 119866 andmin119860(119909) 05 le 05 Suppose119860(119899119909) lt
min119860(119909) 05 le 05 Choose 119903 such that 119860(119899119909) lt 119903 lt
min119860(119909) 05 le 05 Then 119860(119909) gt 119903 that is 119909119903
isin 119860 but(119899119909)119903isin or119902119860 as 119860(119899119909) lt 119903 and 119860(119899119909) + 119903 le 1
Conversely let 119909119905isin 119860 119899 isin 119873 then119860(119909) ge 119905 But119860(119899119909) ge
min119860(119909) 05 ge min119905 05 rArr 119860(119899119909) ge 119905 or 119860(119899119909) ge 05
according as 119905 le 05 or 119903 gt 05 rArr 119860(119899119909) ge 119905 or 119860(119899119909) + 119905 gt
1 rArr (119899119909)119905isin or119902119860
Theorem 18 Let119860 be a fuzzy subset of119866 Then119860 is an (120598 120598or
119902)-fuzzy subgroup of 119866 if and only if the following conditionsare satisfied
(i) 119860(119909 + 119910) ge min119860(119909) 119860(119910) 05(ii) 119860(minus119909) ge min119860(119909) 05(iii) 119860(119899119909) ge min119860(119909) 05
for all 119909 119910 isin 119866 119899 isin 119873
Proof It follows from the previous lemma
Definition 19 A fuzzy subset 119860 of an 119873-group 119866 is said to be(120598 120598 or 119902)-fuzzy ideal of 119866 if it is an (120598 120598 or 119902)-fuzzy subgroupand satisfies the following conditions
(i) 119909119905isin 119860 rArr (119910 + 119909 minus 119910)
119905isin V119902119860
(ii) 119886119905isin 119860 rArr (119899(119909 + 119886) minus 119899119909)
119905isin V119902119860
for any 119899 isin 119873 119909 119886 isin 119866
Lemma20 Let119860 be a fuzzy subset of119866 and 119905 119903 isin (0 1]Then
(i) 119909119905
isin 119860 rArr (119910 + 119909 minus 119910)119905
isin V119902119860 hArr 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05(ii) 119886119905isin 119860 rArr (119899(119909+119886)minus119899119909)
119905isin V119902119860 hArr 119860(119899(119909+119886)minus119899119909) ge
min119860(119909) 05
Proof (i) Assume that 119860(119910 + 119909 minus 119910) lt min119860(119909) 05Choose 119905 such that 119860(119910 + 119909 minus 119910) lt 119905 lt min119860(119909) 05 Butmin119860(119909) 05 le 119860(119909) or 05 according as 119860(119909) lt 05 or119860(119909) ge 05 So 119860(119909) gt 119905 or 119860(119909) ge 05 rArr 119909
119905isin 119860 or 119909
05isin 119860
But (119910 + 119909 minus 119910)119905isin V119902119860 or (119910 + 119909 minus 119910)
05isin V119902119860 respectively
which contradicts the hypothesisConversely assume that 119909
119905isin 119860 then 119860(119909) ge 119905 For any
119910 isin 119866 we have119860(119910+119909minus119910) ge min119860(119909) 05 ge min119905 05 rArr
119860(119910 + 119909 minus 119910) ge 119905 or 05 according as 119905 lt 05 or 119905 ge 05 rArr
(119910+119909minus119910)119905isin 119860 or119860(119910+119909minus119910)+119905 gt 1 SorArr (119910+119909minus119910)
119905isin V119902119860
(ii) Assume that119860(119899(119909+119886)minus119899119909) lt min119860(119886) 05 = 119860(119886)
or 05 for some 119899 isin 119873 119909 119886 isin 119866 According as 119860(119886) lt 05 or119860(119886) ge 05 Choose 119905 isin (0 1] such that 119860(119899(119909 + 119886) minus 119899119909) lt
119905 lt min119860(119886) 05 le 05 In either case 119860(119899(119909 + 119886) minus 119899119909) lt 119905
and 119860(119899(119909 + 119886) minus 119899119909) + 119905 lt 1 So (119899(119909 + 119886) minus 119899119909)119905isin V119902119860 a
contradictionConversely assume that119860(119899(119909+119886)minus119899119909) ge min119860(119886) 05
for all 119886 119909 isin 119866 119899 isin 119873 Let 119886119905
isin 119860 Then 119860(119886) ge 119905 So119860(119899(119909 + 119886) minus 119899119909) ge min119905 05 le 05 = 119905 or 05 according as119905 le 05 or 119905 gt 05 So 119860(119899(119909 + 119886) minus 119899119909)
119905isin V119902119860
4 Algebra
Theorem 21 Let 119860 be an (120598 120598 or 119902) fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 if and only if
(i) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 for all 119909 119910 isin 119866
(ii) 119860(119899(119909 + 119886) minus 119899119909) ge min119860(119886) 05 for all 119899 isin 119873119909 119886 isin 119866
Proof It is immediate from Lemma 20
By definition a fuzzy ideal of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 But the converse is not true in general as shown by thefollowing example
Example 22 Consider 119866 = S3
= 119894 1205881 1205882 1205911 1205912 1205913 (written
additively) to be a Z-group Define a fuzzy subset 119860 of 119866
as 119860(119894) = 1 119860(1205881) = 119860(120588
2) = 119860(120591
2) = 119860(120591
3) = 06
119860(1205911) = 08 which is not fuzzy ideal as 119860[2(120591
1+ 1205912) minus 2120591
2] =
119860(1205881) = 119860(120588
2) = 06 lt 119860(120591
1) it contradicts the condition
(iv) of Definition 14 As 119860(119909 minus 119910) 119860(119899119909) 119860(119910 + 119909 minus 119910) and119860(119899(119909+119886)minus119899119909) = 06 or 08 ge min05 06 or 08 = 05 thusthe notion of (120598 120598or119902)-fuzzy ideal is a successful generalizationof fuzzy ideals of 119866 as introduced in [18]
Theorem 23 Let 119860119894 119894 isin 119869 be any family of (120598 120598 or 119902)-fuzzy
ideals of 119866 Then 119860 = ⋂119860119894is an (120598 120598 or 119902)-fuzzy ideal of 119866
Proof It is straightforward
Theorem 24 A nonempty subset 119868 of 119866 is an ideal of 119866 if andonly if 120594
119868is an (isin isin or119902)-fuzzy ideal of 119866
Proof If 119868 is an ideal of119866 it is clear from [18 Proposition 211]that 120594
119868is fuzzy ideal of 119866 Since every fuzzy ideal is (120598 120598 or 119902)-
fuzzy ideal 120594119868is (120598 120598 or 119902)-fuzzy ideal of 119866
Conversely let 120594119868be an (120598 120598 or 119902)-fuzzy ideal of 119866 Let
119909 119910 isin 119868 120594119868(119909 minus 119910) ge min120594
119868(119909) 120594119868(119910) 05 = 05 So
120594119868(119909 minus 119910) = 1 rArr 119909 minus 119910 isin 119868 Let 119899 isin 119873 119909 isin 119868 120594
119868(119899119909) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899119909) = 1 rArr 119899119909 isin 119868119910 isin 119866 119909 isin 119868
120594119868(119910 + 119909 minus 119910) ge min120594
119868(119909) 05 = 05 rArr 120594
119868(119910 + 119909 minus 119910) =
1 rArr 119910 + 119909 minus 119910 isin 119868119899 isin 119873 119909 isin 119868 119910 isin 119866 120594119868(119899(119910 + 119909) minus 119899119910) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899(119910 + 119909) minus 119899119910) = 1 rArr 119899(119910 + 119909) minus
119899119910 isin 119868 Then 119868 is an ideal of 119866
Theorem 25 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy(subgroup) ideal of 119866 if and only if the level subset 119860
119905is a
(subgroup) ideal for 0 lt 119905 le 05
Proof We prove the result for (120598 120598or119902)-fuzzy ideal 119878 Let119860 bean (120598 120598 or 119902)-fuzzy ideal of 119866 Let 119905 le 05 119909 119910 119894 isin 119860
119905 119899 isin 119873
(i) 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 ge min119905 05 = 119905 rArr
119909 minus 119910 isin 119860119905
(ii) 119860(119899119909) ge min119860(119886) 05 ge min119905 05 = 119905 rArr 119899119909 isin
119860119905
(iii) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 ge min119905 05 = 119905 rArr
119910 + 119909 minus 119910 isin 119860119905 119910 isin 119866
(iv) 119860(119899(119910 + 119909) minus 119899119910) ge min119860(119909) 05 ge min119905 05 =
119905 rArr 119899(119910 + 119909) minus 119899119910 isin 119860119905 119910 isin 119866 119899 isin 119873
Hence 119860119905is an ideal of 119866 Again let 119860
119905be an ideal of 119866 for
all 119905 le 05 If possible let there exist 119909 119910 isin 119866 such that 119860(119909 minus
119910) lt 119905 lt min119860(119909) 119860(119910) 05 Let 119905 be such that 119860(119909 minus 119910) lt
119905 lt min119860(119909) 119860(119910) 05 rArr 119909 119910 isin 119860119905and 119909 minus 119910 notin 119860
119905 a
contradiction So 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 for all119909 119910 isin 119866 For 119899 isin 119873 119909 isin 119866 let 119860(119899119909) lt min119860(119909) 05 Ifpossible let 119905 be such that 119860(119899119909) lt 119905 lt min119860(119909) 05 Thisimplies 119909 isin 119860
119905 but 119899119909 notin 119860
119905 a contradiction Similarly we
can prove that 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 119860[119899(119910 + 119909) minus
119899119910] ge min119860(119909) 05 119909 119910 isin 119866 119899 isin 119873
Remark 26 For 119905 isin (05 1) 119860 may be an (120598 120598 or 119902)-fuzzyideal of 119866 but 119860
119905may not be an ideal of 119866 Let 119905 = 08 in
Example 22 Then 119860119905= 119894 120591
1 119860119905is not an ideal of S
3as it is
not a normal subgroup of S3
We are looking for a corresponding result when 119860119905is an
ideal of 119866 for all 119905 isin (05 1]
Theorem 27 Let 119860 be a fuzzy subset of an 119873-group 119866 Then119860119905
= 120601 is an ideal of119866 for all 119905 isin (05 1] if and only if119860 satisfiesthe following conditions
(i) max119860(119909 minus 119910) 05 ge min119860(119909) 119860(119910)
(ii) max119860(119899119909) 05 ge 119860(119909)
(iii) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)
(iv) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
for all 119909 119910 isin 119866 119899 isin 119873
Proof Suppose that 119860119905
= 120601 is an ideal of 119866 for all 119905 isin
(05 1] In order to prove (i) suppose that for some 119909 119910 isin
119866 max119860(119909 minus 119910) 05 lt min119860(119909) 119860(119910) Let 119905 =
min119860(119909) 119860(119910) So 119909 119910 isin 119860119905and 119905 isin (05 1] Since 119860
119905is
an ideal 119909 minus 119910 isin 119860119905 So 119860(119909 minus 119910) ge 119905 gt max119860(119909 minus 119910) 05
a contradiction In order to prove (ii) suppose that 119909 isin 119866119899 isin 119873 and max119860(119899119909) 05 lt 119860(119909) = 119905 (say) Then119909 isin 119860
119905rArr 119899119909 isin 119860
119905rArr 119860(119899119909) ge 119905 gt max119860(119899119909) 05 a
contradiction Similarly we can prove (iii) and (iv)Conversely suppose that conditions (i) to (iv) hold We
show that 119860119905is an ideal of 119866 for all 119905 isin (05 1] Let 119909 119910 isin 119860
119905
Then 05 lt 119905 le min119860(119909) 119860(119910) le max119860(119909 minus 119910) 05 =
119860(119909 minus 119910) So 119909 minus 119910 isin 119860119905 Let 119899 isin 119873 119909 isin 119860
119905 Then 05 lt
119905 le 119860(119909) le max119860(119909) 05 = 119860(119909) so 119899119909 isin 119860119905 For 119909 isin 119860
119905
119910 isin 119866 05 lt 119905 le 119860(119909) le max119860(119910 + 119909 minus 119910) 05 = 119860(119910 + 119909 minus
119910) rArr 119910 + 119909 minus 119910 isin 119860119905 Also if 119899 isin 119873 119909 isin 119860
119905 119910 isin 119866 05 lt
119905 le 119860(119909) le max119860(119899(119910 + 119909) minus 119899119910) 05 = 119860(119899(119910 + 119909) minus 119899119910)Hence 119899(119910 + 119909) minus 119899119910 isin 119860
119905 Then 119860
119905is an ideal of 119866
A definition for the previous kind of fuzzy subset wasgiven for the case of near-rings in [17] Now we give thedefinition for 119873-groups
Definition 28 A fuzzy subset of 119866 is called an (120598 120598 or 119902)-fuzzysubgroup of 119866 if for all 119905 119903 isin (0 1] and for all 119909 119910 isin 119866 119899 isin 119873
(i) (a) (119909 + 119910)min(119905119903)isin119860 implies 119909119905isinV119902119860 or 119910
119903isinV119902119860
(b) (minus119909)119905isin119860 implies 119909
119905isinV119902119860
Algebra 5
(ii) (119899119909)119905isin119860 implies 119909
119905isinV119902119860
Moreover 119860 is called an (120598 120598 or 119902)-fuzzy ideal of 119866 if119860 is (120598 120598 or 119902)-fuzzy subgroup of 119866 and
(iii) (119910 + 119909 minus 119910)119905isin119860 implies 119909
119905isinV119902119860
(iv) (119899(119909 + 119910) minus 119899119910)119905isin119860 implies 119909
119905isinV119902119860
Theorem 29 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 if and only if
(1) (a) max119860(119909 + 119910) 05 ge min119860(119909) 119860(119910)(b) max119860(minus119909) 05 ge 119860(119909)
(2) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)(3) max119860(119899119909) 05 ge 119860(119909)(4) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
Proof (i)a hArr (1)a Let 119909 119910 isin 119866 be such that max119860(119909 +
119910) 05 lt min119860(119909) 119860(119910) Let 119905 = min119860(119909) 119860(119910) then05 lt 119905 le 1 (119909 + 119910)
119905isin119860 So we must have 119909
119905isinV119902119860 or 119910
119905isinV119902119860
But 119909119905isin 119860 and 119910
119905isin 119860 Here 119909
119905119902119860 or 119910
119905119902119860 then 119905 le 119860(119909) and
119860(119909) + 119905 le 1 or 119905 le 119860(119910) and 119860(119910) + 119905 le 1 then 119905 le 05 acontradiction
Conversely let (119909 + 119910)min(119905119903)isin119860 Then 119860(119909 + 119910) lt min(119905119903) If 119860(119909 + 119910) ge min119860(119909) 119860(119910) then min119860(119909) 119860(119910) lt
min(119905 119903) Hence either 119860(119909) lt 119905 or 119860(119910) lt 119903 which implies119909119905isin119860 or 119910
119903isin119860 Thus 119909
119905isinV119902119860 or 119910
119903isinV119902119860
Again if 119860(119909 + 119910) lt min119860(119909) 119860(119910) then by (1)a
05 ge min 119860 (119909) 119860 (119910) gt 119860 (119909 + 119910) (4)
Suppose that 119909119905
isin 119860 or 119910119903
isin 119860 then 119905 le 119860(119909) le 05 or119903 le 119860(119910) le 05 It follows that either 119909
119905119902119860 or 119910
119905119902119860 and thus
119909119905isinV119902119860 or 119910
119903isinV119902119860
(i)b hArr (1)b Suppose that there exists 119909 isin 119866 such thatmax119860(minus119909) 05 lt 119860(119909) If 119860(119909) = 119905 then 05 lt 119905 le 1 and119860(minus119909) lt 119905 so that (minus119909)
119905isin119860 But then we must have either
(119909)119905isin119860 or 119909
119905119902119860 Also we have (119909)
119905isin 119860 So 119860(119909) + 1 le 1
which means that 119905 le 05 a contradictionConversely suppose that (119909)
119905isin119860 then 119860(minus119909) lt 119905 If
119860(minus119909) ge 119860(119909) then 119860(119909) lt 119905 which gives 119909119905isinV119902119860 Again if
119860(minus119909) lt 119860(119909) by (1)b we have 05 ge 119860(119909) Putting (119909)119905isin 119860
then 119905 le 119860(119909) le 05 so that 119909119905119902119860 which means that 119909
119905isinV119902119860
Similarly we can prove the remaining parts
Theorem 30 A fuzzy subset 119860 of 119866 is an (120598 120598V119902)-fuzzy idealif and only if 119860
119905( = 120601) is an ideal of 119866 for all 119905 isin (05 1]
3 Fuzzy Cosets and Isomorphism Theorem
In this section we first study the properties of (120598 120598 or 119902)-fuzzyideals under a homomorphismThen we introduce the fuzzycosets and prove the fundamental isomorphism theorem on119873-groupswith respect to the structure induced by these fuzzycosets
Theorem 31 Let 119866 and 1198661015840 be two 119873-groups and let 119891 119866 rarr
1198661015840 be an119873-homomorphism If119891 is surjective and119860 is an (120598 120598or
119902)-fuzzy ideal of119866 then so is119891(119860) If119861 is a (120598 120598or119902)-fuzzy idealof 1198661015840 then 119891
minus1(119861) is a fuzzy ideal of 119866
Proof We assume that 119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 Forany 119909 119910 isin 119866
1015840 it follows that
119891 (119860) (119909 + 119910)
= sup119909+119910=119891(119911)
119860 (119911)
ge sup119891(119906)=119909119891(V)=119910
119860 (119906 + V)
ge sup119891(119906)=119909119891(V)=119910
min 119860 (119906) 119860 (V) 05
= min sup119891(119906)=119909
119860 (119906) sup119891(V)=119910
119860 (V) 05
= min 119891 (119860) (119909) 119891 (119860) (119910) 05
(5)
Also
119891 (119860) (minus119909)
= sup119891(119911)=minus119909
119860 (119911) = sup119891(minus119911)=119909
119860 (119911)
= sup119891(119911)=119909
119860 (minus119911)
ge sup119891(119911)=119909
min 119860 (119906) 05
= min sup119891(119911)=119909
119860 (119911) 05
= min 119891 (119860) (119909) 05
(6)
Again
119891 (119860) (119899119909)
= sup119891(119911)=119899119909
119860 (119911) ge sup119891(119906)=119909
119860 (119899119906)
ge sup119891(119906)=119909
min 119860 (119906) 05
= min sup119891(119906)=119909
119860 (119906) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119910 + 119909 minus 119910)
= sup119891(119911)=119910+119909minus119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119906 + V minus 119906)
6 Algebra
ge sup119891(V)=119909
min 119860 (V) 05
= min sup119891(V)=119909
119860 (V) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119899 (119910 + 119909) minus 119899119910)
= sup119891(119911)=119899(119910+119909)minus119899119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119899 (119906 + V) minus 119899119906)
ge sup119891(V)=119909119891(119906)=119910
min 119860 (119906) 05
= min 119891 (119860) (119909) 05
(7)
Therefore 119891(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866 Similarly wecan show that 119891minus1(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866
Definition 32 Let119860 be (isin isinor119902)-fuzzy subgroup of119866 For any119909 isin 119866 let 119860
119909be defined by 119860
119909(119892) = min119860(119892 minus 119909) 05 for
all 119892 isin 119866 This fuzzy subset 119860119909is called the (isin isin or119902)-fuzzy
left coset of 119866 determined by 119860 and 119909
Remark 33 Let 119860 be an (120598 120598 or 119902)-fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy normal if and only if min119860(119892 minus
119909) 05 = min119860(minus119909 + 119892) 05 for all 119909 119892 isin 119866 If 119860 is an(120598 120598 or 119902)-fuzzy ideal we simply denote fuzzy coset by 119860
119909
Lemma 34 Let 119860 be an (120598 120598 or 119902)-fuzzy ideal of 119866 Then119860119909
= 119860119910if an only if 119860(119909 minus 119910) ge 05
Proof Assume that 119860(119909 minus 119910) ge 05 and 119892 isin 119866 119860119909(119892) =
min119860(119892minus119909) 05 ge min119860(119892minus119910) 119860(119910minus119909) 05 ge min119860(119892minus
119910) 119860(119909 minus 119910) 05 = min119860(119892 minus 119910) 05 = 119860119910(119892) which
implies that 119860119909
ge 119860119910 Similarly we can verify that 119860
119909le
119860119910 Conversely we assume that 119860
119909= 119860119910 Then 119860
119910(119909) =
119860119909(119909) rArr min119860(119909 minus 119910) 05 = 05 rArr 119860(119909 minus 119910) ge 05
Proposition 35 Every fuzzy coset 119860119909is constant on every
coset of 1198660= 119909 isin 119866 | 119860(119909) = 119860(0)
Proof Let 119910 + 1199100
isin 119910 + 1198660 Now we have 119860
119909(119910 + 119910
0) =
min119860(119910+1199100minus119909) 05 ge min119860(119910minus119909) 119860(119909+119910
0minus119909) 05 ge
min119860(119910minus119909) 119860(1199100) 05 = min119860(119910minus119909) 05 = 119860
119909(119910) Also
119860119909(119910) = min119860(119910minus119909) 05 ge min119860(minus119909+119910+119910
0minus1199100) 05 ge
min119860(minus119909+119910+1199100) 119860(minus119910
0) 05 = min119860(minus119909+119910+119910
0) 05 ge
min119860(119910+1199100minus119909) 05 = 119860
119909(119910+1199100)Thus119860
119909(119910+1199100) = 119860
119909(119910)
for all 1199100isin 1198660
Theorem 36 For any (120598 120598 or 119902)-fuzzy ideal 119860 of 119866 119866119860the set
of all fuzzy cosets of 119860 in 119866 is an 119873-group under the additionand scalar multiplication defined by119860
119909+119860119910
= 119860119909+119910
119899(119860119909) =
119860119899119909
for all 119909 119910 isin 119866 119899 isin 119873 The function 119860(119860119909) 119866119860
rarr
[0 1] defined by 119860(119860119909) = 119860(119909) for all 119860
119909isin 119866119860is (120598 120598 or 119902)-
fuzzy ideal of 119866119860
Proof First we show that the compositions are well definedLet 119909 119910 119888 119889 isin 119866 119899 isin 119873 such that 119860
119909= 119860119910and 119860
119888= 119860119889
Then119860(119909minus119910) ge 05119860(119888minus119889) ge 05 Now we havemin119860(119909+
119888minus119889minus119910) 05 ge min119860(minus119910+119909+119888minus119889) 05 = 05which impliesthat119860(119909+ 119888minus119889minus119910) ge 05 So by Lemma 34 we have119860
119909+119888=
119860119910+119889
Again 119860119899119909
(119892) = min119860(119892 minus 119899119909) 05 ge min119860(119892 minus
119899119910) 119860(119899119909 minus 119899119910) 05 = min119860(119892 minus 119899119910) 119860[119899(119910 minus 119910 + 119909) minus
119899119910] 05 ge min119860(119892minus119899119910) 119860(minus119910+119909) 05 = 119860119899119910
(119892) Similarlywe show that 119860
119899119910(119892) ge 119860
119899119909(119892) Thus for 119860
119899119910(119892) = 119860
119899119909(119892)
for all 119892 isin 119866 Hence the compositions are well defined It isnow easy to verify that119866
119860is an119873-groupwith null element119860
0
and negative element119860minus119909 Next we check that119860 is (120598 120598 or 119902)-
fuzzy ideal of 119866119860 Let 119860
119909 119860119910
isin 119866119860 We have
(i) 119860(119860119909minus119860119910) = 119860(119860
119909minus119910) = 119860(119909minus119910) ge min119860(119909) 119860(119910)
05 = min119860(119860119909) 119860(119860
119910) 05
(ii) 119860(119860119899119909
) = 119860(119899119909) ge min119860(119909) 05 = min119860(119860119909)
05(iii) 119860(119860
119910+ 119860119909
minus 119860119910) = 119860(119860119910+119909minus119910
) = 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05 = min119860(119860119909) 05
(iv) 119860[119899(119860119910+119860119909)minus119899119860
119910] = 119860[119860
119899(119910+119909)minus119899119910] = 119860[119899(119910+119909)minus
119899119910] ge min119860(119909) 05 = min119860(119860119909) 05
Hence the proof is completed
Lemma 37 Let 119867 = 119909 isin 119866 | 119860119909
ge 1198600 119870 = 119909 isin 119866 | 119860
119909=
1198600 Then 119870 = 119867 = 119860
05
Proof Let 119909 isin 119867 Then for all 119892 isin 119866 119860119909(119892) ge 119860
0(119892) rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr min119860(minus119909) 05 ge
min119860(0) 05 = 05 rArr 119860(119909) ge 05 Also for 119892 isin 1198661198600(119892) = min119860(119892) 05 = min119860(119892 minus 119909 + 119909) 05 ge
min119860(119892 minus 119909) 119860(119909) 05 = min119860(119892 minus 119909) 05 = 119860119909(119892) rArr
1198600
ge 119860119909 So 119867 = 119870 As we have seen if 119909 isin 119867 then
119860(119909) ge 05 rArr 119909 isin 11986005 Conversely if 119909 isin 119860
05 for any
119892 isin 119866 119860(119892 minus 119909) ge min119860(119892) 119860(119909) 05 = min119860(119892) 05 rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr 119860119909
ge 1198600 Thus
119867 = 11986005 which means that 119867 is an ideal of 119866
Theorem38 If119860 is an (120598 120598or119902)-fuzzy ideal of119866 then themap119891 119866 rarr 119866
119860given by 119891(119909) = 119860
119909is an 119873-homomorphism
with kernel 119891 = 11986005
and so 11986611986005
is isomorphic to 119866119860
Proof It is clear that 119891 is an onto 119873-homomorphism from 119866
to 119866119860with kernel 119891 = 119909 isin 119866 | 119891(119909) = 119891(0) = 119909 isin 119866 |
119860119909
= 1198600 = 119860
05
Corollary 39 Let119860 be an (120598 120598or119902)-fuzzy ideal of119866 and let 119861be an (120598 120598or119902)-fuzzy ideal of119866
119860Then 119861 119866 rarr [0 1] defined
by 119861(119909) = 119861(119860119909) is an (120598 120598 or 119902)-fuzzy ideal of 119866 containing 119860
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
Algebra 7
[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971
[3] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982
[4] S Abou-Zaid ldquoOn fuzzy subnear-rings and idealsrdquo Fuzzy Setsand Systems vol 44 no 1 pp 139ndash146 1991
[5] B Davvaz ldquoFuzzy ideals of near-rings with interval valuedmembership functionsrdquo Journal of Sciences Islamic Republic ofIran vol 12 no 2 pp 171ndash175 2001
[6] K H Kim ldquoOn vague119877-subgroups of near-ringsrdquo InternationalMathematical Forum vol 4 no 33ndash36 pp 1655ndash1661 2009
[7] S K Bhakat and P Das ldquo(120598 120598 or 119902)-fuzzy subgrouprdquo Fuzzy Setsand Systems vol 80 no 3 pp 359ndash368 1996
[8] K H Kim Y B Jun and Y H Yon ldquoOn anti fuzzy ideals innear-ringsrdquo Iranian Journal of Fuzzy Systems vol 2 no 2 pp71ndash80 2005
[9] K H Kim and Y B Jun ldquoA note on fuzzy 119877-subgroups of near-ringsrdquo Soochow Journal of Mathematics vol 28 no 4 pp 339ndash346 2002
[10] SDKim andH S Kim ldquoOn fuzzy ideals of near-ringsrdquoBulletinof the Korean Mathematical Society vol 33 no 4 pp 593ndash6011996
[11] O R Devi ldquoIntuitionistic fuzzy near-rings and its propertiesrdquoThe Journal of the Indian Mathematical Society vol 74 no 3-4pp 203ndash219 2007
[12] S K Bhakat and P Das ldquoOn the definition of a fuzzy subgrouprdquoFuzzy Sets and Systems vol 51 no 2 pp 235ndash241 1992
[13] S K Bhakat and P Das ldquoFuzzy subrings and ideals redefinedrdquoFuzzy Sets and Systems vol 81 no 3 pp 383ndash393 1996
[14] B Davvaz ldquo(120598 120598 or 119902)-fuzzy subnear-rings and idealsrdquo SoftComputing vol 10 no 3 pp 206ndash211 2006
[15] B Davvaz ldquoFuzzy 119877-subgroups with thresholds of near-ringsand implication operatorsrdquo Soft Computing vol 12 no 9 pp875ndash879 2008
[16] A Narayanan and T Manikantan ldquo(120598 120598 or 119902)-fuzzy subnear-rings and (120598 120598or119902)-fuzzy ideals of near-ringsrdquo Journal of AppliedMathematics amp Computing vol 18 no 1-2 pp 419ndash430 2005
[17] J Zhan and B Davvaz ldquoGeneralized fuzzy ideals of near-ringsrdquoApplied Mathematics vol 24 no 3 pp 343ndash349 2009
[18] O R Devi and M R Singh ldquoOn fuzzy ideals of 119873-groupsrdquoBulletin of Pure amp Applied Sciences E vol 26 no 1 pp 11ndash232007
[19] T K Dutta and B K Biswas ldquoOn fuzzy congruence of a near-ring modulerdquo Fuzzy Sets and Systems vol 112 no 2 pp 343ndash348 2000
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 4: Research Article A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf · of a ring was introduced by Liu [ ]. e notion of fuzzy subnear-ring and fuzzy ideals](https://reader036.fdocuments.net/reader036/viewer/2022071404/60f839985e18e36f2e647f1c/html5/thumbnails/4.jpg)
4 Algebra
Theorem 21 Let 119860 be an (120598 120598 or 119902) fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 if and only if
(i) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 for all 119909 119910 isin 119866
(ii) 119860(119899(119909 + 119886) minus 119899119909) ge min119860(119886) 05 for all 119899 isin 119873119909 119886 isin 119866
Proof It is immediate from Lemma 20
By definition a fuzzy ideal of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 But the converse is not true in general as shown by thefollowing example
Example 22 Consider 119866 = S3
= 119894 1205881 1205882 1205911 1205912 1205913 (written
additively) to be a Z-group Define a fuzzy subset 119860 of 119866
as 119860(119894) = 1 119860(1205881) = 119860(120588
2) = 119860(120591
2) = 119860(120591
3) = 06
119860(1205911) = 08 which is not fuzzy ideal as 119860[2(120591
1+ 1205912) minus 2120591
2] =
119860(1205881) = 119860(120588
2) = 06 lt 119860(120591
1) it contradicts the condition
(iv) of Definition 14 As 119860(119909 minus 119910) 119860(119899119909) 119860(119910 + 119909 minus 119910) and119860(119899(119909+119886)minus119899119909) = 06 or 08 ge min05 06 or 08 = 05 thusthe notion of (120598 120598or119902)-fuzzy ideal is a successful generalizationof fuzzy ideals of 119866 as introduced in [18]
Theorem 23 Let 119860119894 119894 isin 119869 be any family of (120598 120598 or 119902)-fuzzy
ideals of 119866 Then 119860 = ⋂119860119894is an (120598 120598 or 119902)-fuzzy ideal of 119866
Proof It is straightforward
Theorem 24 A nonempty subset 119868 of 119866 is an ideal of 119866 if andonly if 120594
119868is an (isin isin or119902)-fuzzy ideal of 119866
Proof If 119868 is an ideal of119866 it is clear from [18 Proposition 211]that 120594
119868is fuzzy ideal of 119866 Since every fuzzy ideal is (120598 120598 or 119902)-
fuzzy ideal 120594119868is (120598 120598 or 119902)-fuzzy ideal of 119866
Conversely let 120594119868be an (120598 120598 or 119902)-fuzzy ideal of 119866 Let
119909 119910 isin 119868 120594119868(119909 minus 119910) ge min120594
119868(119909) 120594119868(119910) 05 = 05 So
120594119868(119909 minus 119910) = 1 rArr 119909 minus 119910 isin 119868 Let 119899 isin 119873 119909 isin 119868 120594
119868(119899119909) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899119909) = 1 rArr 119899119909 isin 119868119910 isin 119866 119909 isin 119868
120594119868(119910 + 119909 minus 119910) ge min120594
119868(119909) 05 = 05 rArr 120594
119868(119910 + 119909 minus 119910) =
1 rArr 119910 + 119909 minus 119910 isin 119868119899 isin 119873 119909 isin 119868 119910 isin 119866 120594119868(119899(119910 + 119909) minus 119899119910) ge
min120594119868(119909) 05 = 05 rArr 120594
119868(119899(119910 + 119909) minus 119899119910) = 1 rArr 119899(119910 + 119909) minus
119899119910 isin 119868 Then 119868 is an ideal of 119866
Theorem 25 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy(subgroup) ideal of 119866 if and only if the level subset 119860
119905is a
(subgroup) ideal for 0 lt 119905 le 05
Proof We prove the result for (120598 120598or119902)-fuzzy ideal 119878 Let119860 bean (120598 120598 or 119902)-fuzzy ideal of 119866 Let 119905 le 05 119909 119910 119894 isin 119860
119905 119899 isin 119873
(i) 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 ge min119905 05 = 119905 rArr
119909 minus 119910 isin 119860119905
(ii) 119860(119899119909) ge min119860(119886) 05 ge min119905 05 = 119905 rArr 119899119909 isin
119860119905
(iii) 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 ge min119905 05 = 119905 rArr
119910 + 119909 minus 119910 isin 119860119905 119910 isin 119866
(iv) 119860(119899(119910 + 119909) minus 119899119910) ge min119860(119909) 05 ge min119905 05 =
119905 rArr 119899(119910 + 119909) minus 119899119910 isin 119860119905 119910 isin 119866 119899 isin 119873
Hence 119860119905is an ideal of 119866 Again let 119860
119905be an ideal of 119866 for
all 119905 le 05 If possible let there exist 119909 119910 isin 119866 such that 119860(119909 minus
119910) lt 119905 lt min119860(119909) 119860(119910) 05 Let 119905 be such that 119860(119909 minus 119910) lt
119905 lt min119860(119909) 119860(119910) 05 rArr 119909 119910 isin 119860119905and 119909 minus 119910 notin 119860
119905 a
contradiction So 119860(119909 minus 119910) ge min119860(119909) 119860(119910) 05 for all119909 119910 isin 119866 For 119899 isin 119873 119909 isin 119866 let 119860(119899119909) lt min119860(119909) 05 Ifpossible let 119905 be such that 119860(119899119909) lt 119905 lt min119860(119909) 05 Thisimplies 119909 isin 119860
119905 but 119899119909 notin 119860
119905 a contradiction Similarly we
can prove that 119860(119910 + 119909 minus 119910) ge min119860(119909) 05 119860[119899(119910 + 119909) minus
119899119910] ge min119860(119909) 05 119909 119910 isin 119866 119899 isin 119873
Remark 26 For 119905 isin (05 1) 119860 may be an (120598 120598 or 119902)-fuzzyideal of 119866 but 119860
119905may not be an ideal of 119866 Let 119905 = 08 in
Example 22 Then 119860119905= 119894 120591
1 119860119905is not an ideal of S
3as it is
not a normal subgroup of S3
We are looking for a corresponding result when 119860119905is an
ideal of 119866 for all 119905 isin (05 1]
Theorem 27 Let 119860 be a fuzzy subset of an 119873-group 119866 Then119860119905
= 120601 is an ideal of119866 for all 119905 isin (05 1] if and only if119860 satisfiesthe following conditions
(i) max119860(119909 minus 119910) 05 ge min119860(119909) 119860(119910)
(ii) max119860(119899119909) 05 ge 119860(119909)
(iii) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)
(iv) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
for all 119909 119910 isin 119866 119899 isin 119873
Proof Suppose that 119860119905
= 120601 is an ideal of 119866 for all 119905 isin
(05 1] In order to prove (i) suppose that for some 119909 119910 isin
119866 max119860(119909 minus 119910) 05 lt min119860(119909) 119860(119910) Let 119905 =
min119860(119909) 119860(119910) So 119909 119910 isin 119860119905and 119905 isin (05 1] Since 119860
119905is
an ideal 119909 minus 119910 isin 119860119905 So 119860(119909 minus 119910) ge 119905 gt max119860(119909 minus 119910) 05
a contradiction In order to prove (ii) suppose that 119909 isin 119866119899 isin 119873 and max119860(119899119909) 05 lt 119860(119909) = 119905 (say) Then119909 isin 119860
119905rArr 119899119909 isin 119860
119905rArr 119860(119899119909) ge 119905 gt max119860(119899119909) 05 a
contradiction Similarly we can prove (iii) and (iv)Conversely suppose that conditions (i) to (iv) hold We
show that 119860119905is an ideal of 119866 for all 119905 isin (05 1] Let 119909 119910 isin 119860
119905
Then 05 lt 119905 le min119860(119909) 119860(119910) le max119860(119909 minus 119910) 05 =
119860(119909 minus 119910) So 119909 minus 119910 isin 119860119905 Let 119899 isin 119873 119909 isin 119860
119905 Then 05 lt
119905 le 119860(119909) le max119860(119909) 05 = 119860(119909) so 119899119909 isin 119860119905 For 119909 isin 119860
119905
119910 isin 119866 05 lt 119905 le 119860(119909) le max119860(119910 + 119909 minus 119910) 05 = 119860(119910 + 119909 minus
119910) rArr 119910 + 119909 minus 119910 isin 119860119905 Also if 119899 isin 119873 119909 isin 119860
119905 119910 isin 119866 05 lt
119905 le 119860(119909) le max119860(119899(119910 + 119909) minus 119899119910) 05 = 119860(119899(119910 + 119909) minus 119899119910)Hence 119899(119910 + 119909) minus 119899119910 isin 119860
119905 Then 119860
119905is an ideal of 119866
A definition for the previous kind of fuzzy subset wasgiven for the case of near-rings in [17] Now we give thedefinition for 119873-groups
Definition 28 A fuzzy subset of 119866 is called an (120598 120598 or 119902)-fuzzysubgroup of 119866 if for all 119905 119903 isin (0 1] and for all 119909 119910 isin 119866 119899 isin 119873
(i) (a) (119909 + 119910)min(119905119903)isin119860 implies 119909119905isinV119902119860 or 119910
119903isinV119902119860
(b) (minus119909)119905isin119860 implies 119909
119905isinV119902119860
Algebra 5
(ii) (119899119909)119905isin119860 implies 119909
119905isinV119902119860
Moreover 119860 is called an (120598 120598 or 119902)-fuzzy ideal of 119866 if119860 is (120598 120598 or 119902)-fuzzy subgroup of 119866 and
(iii) (119910 + 119909 minus 119910)119905isin119860 implies 119909
119905isinV119902119860
(iv) (119899(119909 + 119910) minus 119899119910)119905isin119860 implies 119909
119905isinV119902119860
Theorem 29 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 if and only if
(1) (a) max119860(119909 + 119910) 05 ge min119860(119909) 119860(119910)(b) max119860(minus119909) 05 ge 119860(119909)
(2) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)(3) max119860(119899119909) 05 ge 119860(119909)(4) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
Proof (i)a hArr (1)a Let 119909 119910 isin 119866 be such that max119860(119909 +
119910) 05 lt min119860(119909) 119860(119910) Let 119905 = min119860(119909) 119860(119910) then05 lt 119905 le 1 (119909 + 119910)
119905isin119860 So we must have 119909
119905isinV119902119860 or 119910
119905isinV119902119860
But 119909119905isin 119860 and 119910
119905isin 119860 Here 119909
119905119902119860 or 119910
119905119902119860 then 119905 le 119860(119909) and
119860(119909) + 119905 le 1 or 119905 le 119860(119910) and 119860(119910) + 119905 le 1 then 119905 le 05 acontradiction
Conversely let (119909 + 119910)min(119905119903)isin119860 Then 119860(119909 + 119910) lt min(119905119903) If 119860(119909 + 119910) ge min119860(119909) 119860(119910) then min119860(119909) 119860(119910) lt
min(119905 119903) Hence either 119860(119909) lt 119905 or 119860(119910) lt 119903 which implies119909119905isin119860 or 119910
119903isin119860 Thus 119909
119905isinV119902119860 or 119910
119903isinV119902119860
Again if 119860(119909 + 119910) lt min119860(119909) 119860(119910) then by (1)a
05 ge min 119860 (119909) 119860 (119910) gt 119860 (119909 + 119910) (4)
Suppose that 119909119905
isin 119860 or 119910119903
isin 119860 then 119905 le 119860(119909) le 05 or119903 le 119860(119910) le 05 It follows that either 119909
119905119902119860 or 119910
119905119902119860 and thus
119909119905isinV119902119860 or 119910
119903isinV119902119860
(i)b hArr (1)b Suppose that there exists 119909 isin 119866 such thatmax119860(minus119909) 05 lt 119860(119909) If 119860(119909) = 119905 then 05 lt 119905 le 1 and119860(minus119909) lt 119905 so that (minus119909)
119905isin119860 But then we must have either
(119909)119905isin119860 or 119909
119905119902119860 Also we have (119909)
119905isin 119860 So 119860(119909) + 1 le 1
which means that 119905 le 05 a contradictionConversely suppose that (119909)
119905isin119860 then 119860(minus119909) lt 119905 If
119860(minus119909) ge 119860(119909) then 119860(119909) lt 119905 which gives 119909119905isinV119902119860 Again if
119860(minus119909) lt 119860(119909) by (1)b we have 05 ge 119860(119909) Putting (119909)119905isin 119860
then 119905 le 119860(119909) le 05 so that 119909119905119902119860 which means that 119909
119905isinV119902119860
Similarly we can prove the remaining parts
Theorem 30 A fuzzy subset 119860 of 119866 is an (120598 120598V119902)-fuzzy idealif and only if 119860
119905( = 120601) is an ideal of 119866 for all 119905 isin (05 1]
3 Fuzzy Cosets and Isomorphism Theorem
In this section we first study the properties of (120598 120598 or 119902)-fuzzyideals under a homomorphismThen we introduce the fuzzycosets and prove the fundamental isomorphism theorem on119873-groupswith respect to the structure induced by these fuzzycosets
Theorem 31 Let 119866 and 1198661015840 be two 119873-groups and let 119891 119866 rarr
1198661015840 be an119873-homomorphism If119891 is surjective and119860 is an (120598 120598or
119902)-fuzzy ideal of119866 then so is119891(119860) If119861 is a (120598 120598or119902)-fuzzy idealof 1198661015840 then 119891
minus1(119861) is a fuzzy ideal of 119866
Proof We assume that 119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 Forany 119909 119910 isin 119866
1015840 it follows that
119891 (119860) (119909 + 119910)
= sup119909+119910=119891(119911)
119860 (119911)
ge sup119891(119906)=119909119891(V)=119910
119860 (119906 + V)
ge sup119891(119906)=119909119891(V)=119910
min 119860 (119906) 119860 (V) 05
= min sup119891(119906)=119909
119860 (119906) sup119891(V)=119910
119860 (V) 05
= min 119891 (119860) (119909) 119891 (119860) (119910) 05
(5)
Also
119891 (119860) (minus119909)
= sup119891(119911)=minus119909
119860 (119911) = sup119891(minus119911)=119909
119860 (119911)
= sup119891(119911)=119909
119860 (minus119911)
ge sup119891(119911)=119909
min 119860 (119906) 05
= min sup119891(119911)=119909
119860 (119911) 05
= min 119891 (119860) (119909) 05
(6)
Again
119891 (119860) (119899119909)
= sup119891(119911)=119899119909
119860 (119911) ge sup119891(119906)=119909
119860 (119899119906)
ge sup119891(119906)=119909
min 119860 (119906) 05
= min sup119891(119906)=119909
119860 (119906) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119910 + 119909 minus 119910)
= sup119891(119911)=119910+119909minus119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119906 + V minus 119906)
6 Algebra
ge sup119891(V)=119909
min 119860 (V) 05
= min sup119891(V)=119909
119860 (V) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119899 (119910 + 119909) minus 119899119910)
= sup119891(119911)=119899(119910+119909)minus119899119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119899 (119906 + V) minus 119899119906)
ge sup119891(V)=119909119891(119906)=119910
min 119860 (119906) 05
= min 119891 (119860) (119909) 05
(7)
Therefore 119891(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866 Similarly wecan show that 119891minus1(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866
Definition 32 Let119860 be (isin isinor119902)-fuzzy subgroup of119866 For any119909 isin 119866 let 119860
119909be defined by 119860
119909(119892) = min119860(119892 minus 119909) 05 for
all 119892 isin 119866 This fuzzy subset 119860119909is called the (isin isin or119902)-fuzzy
left coset of 119866 determined by 119860 and 119909
Remark 33 Let 119860 be an (120598 120598 or 119902)-fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy normal if and only if min119860(119892 minus
119909) 05 = min119860(minus119909 + 119892) 05 for all 119909 119892 isin 119866 If 119860 is an(120598 120598 or 119902)-fuzzy ideal we simply denote fuzzy coset by 119860
119909
Lemma 34 Let 119860 be an (120598 120598 or 119902)-fuzzy ideal of 119866 Then119860119909
= 119860119910if an only if 119860(119909 minus 119910) ge 05
Proof Assume that 119860(119909 minus 119910) ge 05 and 119892 isin 119866 119860119909(119892) =
min119860(119892minus119909) 05 ge min119860(119892minus119910) 119860(119910minus119909) 05 ge min119860(119892minus
119910) 119860(119909 minus 119910) 05 = min119860(119892 minus 119910) 05 = 119860119910(119892) which
implies that 119860119909
ge 119860119910 Similarly we can verify that 119860
119909le
119860119910 Conversely we assume that 119860
119909= 119860119910 Then 119860
119910(119909) =
119860119909(119909) rArr min119860(119909 minus 119910) 05 = 05 rArr 119860(119909 minus 119910) ge 05
Proposition 35 Every fuzzy coset 119860119909is constant on every
coset of 1198660= 119909 isin 119866 | 119860(119909) = 119860(0)
Proof Let 119910 + 1199100
isin 119910 + 1198660 Now we have 119860
119909(119910 + 119910
0) =
min119860(119910+1199100minus119909) 05 ge min119860(119910minus119909) 119860(119909+119910
0minus119909) 05 ge
min119860(119910minus119909) 119860(1199100) 05 = min119860(119910minus119909) 05 = 119860
119909(119910) Also
119860119909(119910) = min119860(119910minus119909) 05 ge min119860(minus119909+119910+119910
0minus1199100) 05 ge
min119860(minus119909+119910+1199100) 119860(minus119910
0) 05 = min119860(minus119909+119910+119910
0) 05 ge
min119860(119910+1199100minus119909) 05 = 119860
119909(119910+1199100)Thus119860
119909(119910+1199100) = 119860
119909(119910)
for all 1199100isin 1198660
Theorem 36 For any (120598 120598 or 119902)-fuzzy ideal 119860 of 119866 119866119860the set
of all fuzzy cosets of 119860 in 119866 is an 119873-group under the additionand scalar multiplication defined by119860
119909+119860119910
= 119860119909+119910
119899(119860119909) =
119860119899119909
for all 119909 119910 isin 119866 119899 isin 119873 The function 119860(119860119909) 119866119860
rarr
[0 1] defined by 119860(119860119909) = 119860(119909) for all 119860
119909isin 119866119860is (120598 120598 or 119902)-
fuzzy ideal of 119866119860
Proof First we show that the compositions are well definedLet 119909 119910 119888 119889 isin 119866 119899 isin 119873 such that 119860
119909= 119860119910and 119860
119888= 119860119889
Then119860(119909minus119910) ge 05119860(119888minus119889) ge 05 Now we havemin119860(119909+
119888minus119889minus119910) 05 ge min119860(minus119910+119909+119888minus119889) 05 = 05which impliesthat119860(119909+ 119888minus119889minus119910) ge 05 So by Lemma 34 we have119860
119909+119888=
119860119910+119889
Again 119860119899119909
(119892) = min119860(119892 minus 119899119909) 05 ge min119860(119892 minus
119899119910) 119860(119899119909 minus 119899119910) 05 = min119860(119892 minus 119899119910) 119860[119899(119910 minus 119910 + 119909) minus
119899119910] 05 ge min119860(119892minus119899119910) 119860(minus119910+119909) 05 = 119860119899119910
(119892) Similarlywe show that 119860
119899119910(119892) ge 119860
119899119909(119892) Thus for 119860
119899119910(119892) = 119860
119899119909(119892)
for all 119892 isin 119866 Hence the compositions are well defined It isnow easy to verify that119866
119860is an119873-groupwith null element119860
0
and negative element119860minus119909 Next we check that119860 is (120598 120598 or 119902)-
fuzzy ideal of 119866119860 Let 119860
119909 119860119910
isin 119866119860 We have
(i) 119860(119860119909minus119860119910) = 119860(119860
119909minus119910) = 119860(119909minus119910) ge min119860(119909) 119860(119910)
05 = min119860(119860119909) 119860(119860
119910) 05
(ii) 119860(119860119899119909
) = 119860(119899119909) ge min119860(119909) 05 = min119860(119860119909)
05(iii) 119860(119860
119910+ 119860119909
minus 119860119910) = 119860(119860119910+119909minus119910
) = 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05 = min119860(119860119909) 05
(iv) 119860[119899(119860119910+119860119909)minus119899119860
119910] = 119860[119860
119899(119910+119909)minus119899119910] = 119860[119899(119910+119909)minus
119899119910] ge min119860(119909) 05 = min119860(119860119909) 05
Hence the proof is completed
Lemma 37 Let 119867 = 119909 isin 119866 | 119860119909
ge 1198600 119870 = 119909 isin 119866 | 119860
119909=
1198600 Then 119870 = 119867 = 119860
05
Proof Let 119909 isin 119867 Then for all 119892 isin 119866 119860119909(119892) ge 119860
0(119892) rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr min119860(minus119909) 05 ge
min119860(0) 05 = 05 rArr 119860(119909) ge 05 Also for 119892 isin 1198661198600(119892) = min119860(119892) 05 = min119860(119892 minus 119909 + 119909) 05 ge
min119860(119892 minus 119909) 119860(119909) 05 = min119860(119892 minus 119909) 05 = 119860119909(119892) rArr
1198600
ge 119860119909 So 119867 = 119870 As we have seen if 119909 isin 119867 then
119860(119909) ge 05 rArr 119909 isin 11986005 Conversely if 119909 isin 119860
05 for any
119892 isin 119866 119860(119892 minus 119909) ge min119860(119892) 119860(119909) 05 = min119860(119892) 05 rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr 119860119909
ge 1198600 Thus
119867 = 11986005 which means that 119867 is an ideal of 119866
Theorem38 If119860 is an (120598 120598or119902)-fuzzy ideal of119866 then themap119891 119866 rarr 119866
119860given by 119891(119909) = 119860
119909is an 119873-homomorphism
with kernel 119891 = 11986005
and so 11986611986005
is isomorphic to 119866119860
Proof It is clear that 119891 is an onto 119873-homomorphism from 119866
to 119866119860with kernel 119891 = 119909 isin 119866 | 119891(119909) = 119891(0) = 119909 isin 119866 |
119860119909
= 1198600 = 119860
05
Corollary 39 Let119860 be an (120598 120598or119902)-fuzzy ideal of119866 and let 119861be an (120598 120598or119902)-fuzzy ideal of119866
119860Then 119861 119866 rarr [0 1] defined
by 119861(119909) = 119861(119860119909) is an (120598 120598 or 119902)-fuzzy ideal of 119866 containing 119860
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
Algebra 7
[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971
[3] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982
[4] S Abou-Zaid ldquoOn fuzzy subnear-rings and idealsrdquo Fuzzy Setsand Systems vol 44 no 1 pp 139ndash146 1991
[5] B Davvaz ldquoFuzzy ideals of near-rings with interval valuedmembership functionsrdquo Journal of Sciences Islamic Republic ofIran vol 12 no 2 pp 171ndash175 2001
[6] K H Kim ldquoOn vague119877-subgroups of near-ringsrdquo InternationalMathematical Forum vol 4 no 33ndash36 pp 1655ndash1661 2009
[7] S K Bhakat and P Das ldquo(120598 120598 or 119902)-fuzzy subgrouprdquo Fuzzy Setsand Systems vol 80 no 3 pp 359ndash368 1996
[8] K H Kim Y B Jun and Y H Yon ldquoOn anti fuzzy ideals innear-ringsrdquo Iranian Journal of Fuzzy Systems vol 2 no 2 pp71ndash80 2005
[9] K H Kim and Y B Jun ldquoA note on fuzzy 119877-subgroups of near-ringsrdquo Soochow Journal of Mathematics vol 28 no 4 pp 339ndash346 2002
[10] SDKim andH S Kim ldquoOn fuzzy ideals of near-ringsrdquoBulletinof the Korean Mathematical Society vol 33 no 4 pp 593ndash6011996
[11] O R Devi ldquoIntuitionistic fuzzy near-rings and its propertiesrdquoThe Journal of the Indian Mathematical Society vol 74 no 3-4pp 203ndash219 2007
[12] S K Bhakat and P Das ldquoOn the definition of a fuzzy subgrouprdquoFuzzy Sets and Systems vol 51 no 2 pp 235ndash241 1992
[13] S K Bhakat and P Das ldquoFuzzy subrings and ideals redefinedrdquoFuzzy Sets and Systems vol 81 no 3 pp 383ndash393 1996
[14] B Davvaz ldquo(120598 120598 or 119902)-fuzzy subnear-rings and idealsrdquo SoftComputing vol 10 no 3 pp 206ndash211 2006
[15] B Davvaz ldquoFuzzy 119877-subgroups with thresholds of near-ringsand implication operatorsrdquo Soft Computing vol 12 no 9 pp875ndash879 2008
[16] A Narayanan and T Manikantan ldquo(120598 120598 or 119902)-fuzzy subnear-rings and (120598 120598or119902)-fuzzy ideals of near-ringsrdquo Journal of AppliedMathematics amp Computing vol 18 no 1-2 pp 419ndash430 2005
[17] J Zhan and B Davvaz ldquoGeneralized fuzzy ideals of near-ringsrdquoApplied Mathematics vol 24 no 3 pp 343ndash349 2009
[18] O R Devi and M R Singh ldquoOn fuzzy ideals of 119873-groupsrdquoBulletin of Pure amp Applied Sciences E vol 26 no 1 pp 11ndash232007
[19] T K Dutta and B K Biswas ldquoOn fuzzy congruence of a near-ring modulerdquo Fuzzy Sets and Systems vol 112 no 2 pp 343ndash348 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 5: Research Article A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf · of a ring was introduced by Liu [ ]. e notion of fuzzy subnear-ring and fuzzy ideals](https://reader036.fdocuments.net/reader036/viewer/2022071404/60f839985e18e36f2e647f1c/html5/thumbnails/5.jpg)
Algebra 5
(ii) (119899119909)119905isin119860 implies 119909
119905isinV119902119860
Moreover 119860 is called an (120598 120598 or 119902)-fuzzy ideal of 119866 if119860 is (120598 120598 or 119902)-fuzzy subgroup of 119866 and
(iii) (119910 + 119909 minus 119910)119905isin119860 implies 119909
119905isinV119902119860
(iv) (119899(119909 + 119910) minus 119899119910)119905isin119860 implies 119909
119905isinV119902119860
Theorem 29 A fuzzy subset 119860 of 119866 is an (120598 120598 or 119902)-fuzzy idealof 119866 if and only if
(1) (a) max119860(119909 + 119910) 05 ge min119860(119909) 119860(119910)(b) max119860(minus119909) 05 ge 119860(119909)
(2) max119860(119910 + 119909 minus 119910) 05 ge 119860(119909)(3) max119860(119899119909) 05 ge 119860(119909)(4) max119860(119899(119910 + 119909) minus 119899119910) 05 ge 119860(119909)
Proof (i)a hArr (1)a Let 119909 119910 isin 119866 be such that max119860(119909 +
119910) 05 lt min119860(119909) 119860(119910) Let 119905 = min119860(119909) 119860(119910) then05 lt 119905 le 1 (119909 + 119910)
119905isin119860 So we must have 119909
119905isinV119902119860 or 119910
119905isinV119902119860
But 119909119905isin 119860 and 119910
119905isin 119860 Here 119909
119905119902119860 or 119910
119905119902119860 then 119905 le 119860(119909) and
119860(119909) + 119905 le 1 or 119905 le 119860(119910) and 119860(119910) + 119905 le 1 then 119905 le 05 acontradiction
Conversely let (119909 + 119910)min(119905119903)isin119860 Then 119860(119909 + 119910) lt min(119905119903) If 119860(119909 + 119910) ge min119860(119909) 119860(119910) then min119860(119909) 119860(119910) lt
min(119905 119903) Hence either 119860(119909) lt 119905 or 119860(119910) lt 119903 which implies119909119905isin119860 or 119910
119903isin119860 Thus 119909
119905isinV119902119860 or 119910
119903isinV119902119860
Again if 119860(119909 + 119910) lt min119860(119909) 119860(119910) then by (1)a
05 ge min 119860 (119909) 119860 (119910) gt 119860 (119909 + 119910) (4)
Suppose that 119909119905
isin 119860 or 119910119903
isin 119860 then 119905 le 119860(119909) le 05 or119903 le 119860(119910) le 05 It follows that either 119909
119905119902119860 or 119910
119905119902119860 and thus
119909119905isinV119902119860 or 119910
119903isinV119902119860
(i)b hArr (1)b Suppose that there exists 119909 isin 119866 such thatmax119860(minus119909) 05 lt 119860(119909) If 119860(119909) = 119905 then 05 lt 119905 le 1 and119860(minus119909) lt 119905 so that (minus119909)
119905isin119860 But then we must have either
(119909)119905isin119860 or 119909
119905119902119860 Also we have (119909)
119905isin 119860 So 119860(119909) + 1 le 1
which means that 119905 le 05 a contradictionConversely suppose that (119909)
119905isin119860 then 119860(minus119909) lt 119905 If
119860(minus119909) ge 119860(119909) then 119860(119909) lt 119905 which gives 119909119905isinV119902119860 Again if
119860(minus119909) lt 119860(119909) by (1)b we have 05 ge 119860(119909) Putting (119909)119905isin 119860
then 119905 le 119860(119909) le 05 so that 119909119905119902119860 which means that 119909
119905isinV119902119860
Similarly we can prove the remaining parts
Theorem 30 A fuzzy subset 119860 of 119866 is an (120598 120598V119902)-fuzzy idealif and only if 119860
119905( = 120601) is an ideal of 119866 for all 119905 isin (05 1]
3 Fuzzy Cosets and Isomorphism Theorem
In this section we first study the properties of (120598 120598 or 119902)-fuzzyideals under a homomorphismThen we introduce the fuzzycosets and prove the fundamental isomorphism theorem on119873-groupswith respect to the structure induced by these fuzzycosets
Theorem 31 Let 119866 and 1198661015840 be two 119873-groups and let 119891 119866 rarr
1198661015840 be an119873-homomorphism If119891 is surjective and119860 is an (120598 120598or
119902)-fuzzy ideal of119866 then so is119891(119860) If119861 is a (120598 120598or119902)-fuzzy idealof 1198661015840 then 119891
minus1(119861) is a fuzzy ideal of 119866
Proof We assume that 119860 is an (120598 120598 or 119902)-fuzzy ideal of 119866 Forany 119909 119910 isin 119866
1015840 it follows that
119891 (119860) (119909 + 119910)
= sup119909+119910=119891(119911)
119860 (119911)
ge sup119891(119906)=119909119891(V)=119910
119860 (119906 + V)
ge sup119891(119906)=119909119891(V)=119910
min 119860 (119906) 119860 (V) 05
= min sup119891(119906)=119909
119860 (119906) sup119891(V)=119910
119860 (V) 05
= min 119891 (119860) (119909) 119891 (119860) (119910) 05
(5)
Also
119891 (119860) (minus119909)
= sup119891(119911)=minus119909
119860 (119911) = sup119891(minus119911)=119909
119860 (119911)
= sup119891(119911)=119909
119860 (minus119911)
ge sup119891(119911)=119909
min 119860 (119906) 05
= min sup119891(119911)=119909
119860 (119911) 05
= min 119891 (119860) (119909) 05
(6)
Again
119891 (119860) (119899119909)
= sup119891(119911)=119899119909
119860 (119911) ge sup119891(119906)=119909
119860 (119899119906)
ge sup119891(119906)=119909
min 119860 (119906) 05
= min sup119891(119906)=119909
119860 (119906) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119910 + 119909 minus 119910)
= sup119891(119911)=119910+119909minus119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119906 + V minus 119906)
6 Algebra
ge sup119891(V)=119909
min 119860 (V) 05
= min sup119891(V)=119909
119860 (V) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119899 (119910 + 119909) minus 119899119910)
= sup119891(119911)=119899(119910+119909)minus119899119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119899 (119906 + V) minus 119899119906)
ge sup119891(V)=119909119891(119906)=119910
min 119860 (119906) 05
= min 119891 (119860) (119909) 05
(7)
Therefore 119891(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866 Similarly wecan show that 119891minus1(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866
Definition 32 Let119860 be (isin isinor119902)-fuzzy subgroup of119866 For any119909 isin 119866 let 119860
119909be defined by 119860
119909(119892) = min119860(119892 minus 119909) 05 for
all 119892 isin 119866 This fuzzy subset 119860119909is called the (isin isin or119902)-fuzzy
left coset of 119866 determined by 119860 and 119909
Remark 33 Let 119860 be an (120598 120598 or 119902)-fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy normal if and only if min119860(119892 minus
119909) 05 = min119860(minus119909 + 119892) 05 for all 119909 119892 isin 119866 If 119860 is an(120598 120598 or 119902)-fuzzy ideal we simply denote fuzzy coset by 119860
119909
Lemma 34 Let 119860 be an (120598 120598 or 119902)-fuzzy ideal of 119866 Then119860119909
= 119860119910if an only if 119860(119909 minus 119910) ge 05
Proof Assume that 119860(119909 minus 119910) ge 05 and 119892 isin 119866 119860119909(119892) =
min119860(119892minus119909) 05 ge min119860(119892minus119910) 119860(119910minus119909) 05 ge min119860(119892minus
119910) 119860(119909 minus 119910) 05 = min119860(119892 minus 119910) 05 = 119860119910(119892) which
implies that 119860119909
ge 119860119910 Similarly we can verify that 119860
119909le
119860119910 Conversely we assume that 119860
119909= 119860119910 Then 119860
119910(119909) =
119860119909(119909) rArr min119860(119909 minus 119910) 05 = 05 rArr 119860(119909 minus 119910) ge 05
Proposition 35 Every fuzzy coset 119860119909is constant on every
coset of 1198660= 119909 isin 119866 | 119860(119909) = 119860(0)
Proof Let 119910 + 1199100
isin 119910 + 1198660 Now we have 119860
119909(119910 + 119910
0) =
min119860(119910+1199100minus119909) 05 ge min119860(119910minus119909) 119860(119909+119910
0minus119909) 05 ge
min119860(119910minus119909) 119860(1199100) 05 = min119860(119910minus119909) 05 = 119860
119909(119910) Also
119860119909(119910) = min119860(119910minus119909) 05 ge min119860(minus119909+119910+119910
0minus1199100) 05 ge
min119860(minus119909+119910+1199100) 119860(minus119910
0) 05 = min119860(minus119909+119910+119910
0) 05 ge
min119860(119910+1199100minus119909) 05 = 119860
119909(119910+1199100)Thus119860
119909(119910+1199100) = 119860
119909(119910)
for all 1199100isin 1198660
Theorem 36 For any (120598 120598 or 119902)-fuzzy ideal 119860 of 119866 119866119860the set
of all fuzzy cosets of 119860 in 119866 is an 119873-group under the additionand scalar multiplication defined by119860
119909+119860119910
= 119860119909+119910
119899(119860119909) =
119860119899119909
for all 119909 119910 isin 119866 119899 isin 119873 The function 119860(119860119909) 119866119860
rarr
[0 1] defined by 119860(119860119909) = 119860(119909) for all 119860
119909isin 119866119860is (120598 120598 or 119902)-
fuzzy ideal of 119866119860
Proof First we show that the compositions are well definedLet 119909 119910 119888 119889 isin 119866 119899 isin 119873 such that 119860
119909= 119860119910and 119860
119888= 119860119889
Then119860(119909minus119910) ge 05119860(119888minus119889) ge 05 Now we havemin119860(119909+
119888minus119889minus119910) 05 ge min119860(minus119910+119909+119888minus119889) 05 = 05which impliesthat119860(119909+ 119888minus119889minus119910) ge 05 So by Lemma 34 we have119860
119909+119888=
119860119910+119889
Again 119860119899119909
(119892) = min119860(119892 minus 119899119909) 05 ge min119860(119892 minus
119899119910) 119860(119899119909 minus 119899119910) 05 = min119860(119892 minus 119899119910) 119860[119899(119910 minus 119910 + 119909) minus
119899119910] 05 ge min119860(119892minus119899119910) 119860(minus119910+119909) 05 = 119860119899119910
(119892) Similarlywe show that 119860
119899119910(119892) ge 119860
119899119909(119892) Thus for 119860
119899119910(119892) = 119860
119899119909(119892)
for all 119892 isin 119866 Hence the compositions are well defined It isnow easy to verify that119866
119860is an119873-groupwith null element119860
0
and negative element119860minus119909 Next we check that119860 is (120598 120598 or 119902)-
fuzzy ideal of 119866119860 Let 119860
119909 119860119910
isin 119866119860 We have
(i) 119860(119860119909minus119860119910) = 119860(119860
119909minus119910) = 119860(119909minus119910) ge min119860(119909) 119860(119910)
05 = min119860(119860119909) 119860(119860
119910) 05
(ii) 119860(119860119899119909
) = 119860(119899119909) ge min119860(119909) 05 = min119860(119860119909)
05(iii) 119860(119860
119910+ 119860119909
minus 119860119910) = 119860(119860119910+119909minus119910
) = 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05 = min119860(119860119909) 05
(iv) 119860[119899(119860119910+119860119909)minus119899119860
119910] = 119860[119860
119899(119910+119909)minus119899119910] = 119860[119899(119910+119909)minus
119899119910] ge min119860(119909) 05 = min119860(119860119909) 05
Hence the proof is completed
Lemma 37 Let 119867 = 119909 isin 119866 | 119860119909
ge 1198600 119870 = 119909 isin 119866 | 119860
119909=
1198600 Then 119870 = 119867 = 119860
05
Proof Let 119909 isin 119867 Then for all 119892 isin 119866 119860119909(119892) ge 119860
0(119892) rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr min119860(minus119909) 05 ge
min119860(0) 05 = 05 rArr 119860(119909) ge 05 Also for 119892 isin 1198661198600(119892) = min119860(119892) 05 = min119860(119892 minus 119909 + 119909) 05 ge
min119860(119892 minus 119909) 119860(119909) 05 = min119860(119892 minus 119909) 05 = 119860119909(119892) rArr
1198600
ge 119860119909 So 119867 = 119870 As we have seen if 119909 isin 119867 then
119860(119909) ge 05 rArr 119909 isin 11986005 Conversely if 119909 isin 119860
05 for any
119892 isin 119866 119860(119892 minus 119909) ge min119860(119892) 119860(119909) 05 = min119860(119892) 05 rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr 119860119909
ge 1198600 Thus
119867 = 11986005 which means that 119867 is an ideal of 119866
Theorem38 If119860 is an (120598 120598or119902)-fuzzy ideal of119866 then themap119891 119866 rarr 119866
119860given by 119891(119909) = 119860
119909is an 119873-homomorphism
with kernel 119891 = 11986005
and so 11986611986005
is isomorphic to 119866119860
Proof It is clear that 119891 is an onto 119873-homomorphism from 119866
to 119866119860with kernel 119891 = 119909 isin 119866 | 119891(119909) = 119891(0) = 119909 isin 119866 |
119860119909
= 1198600 = 119860
05
Corollary 39 Let119860 be an (120598 120598or119902)-fuzzy ideal of119866 and let 119861be an (120598 120598or119902)-fuzzy ideal of119866
119860Then 119861 119866 rarr [0 1] defined
by 119861(119909) = 119861(119860119909) is an (120598 120598 or 119902)-fuzzy ideal of 119866 containing 119860
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
Algebra 7
[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971
[3] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982
[4] S Abou-Zaid ldquoOn fuzzy subnear-rings and idealsrdquo Fuzzy Setsand Systems vol 44 no 1 pp 139ndash146 1991
[5] B Davvaz ldquoFuzzy ideals of near-rings with interval valuedmembership functionsrdquo Journal of Sciences Islamic Republic ofIran vol 12 no 2 pp 171ndash175 2001
[6] K H Kim ldquoOn vague119877-subgroups of near-ringsrdquo InternationalMathematical Forum vol 4 no 33ndash36 pp 1655ndash1661 2009
[7] S K Bhakat and P Das ldquo(120598 120598 or 119902)-fuzzy subgrouprdquo Fuzzy Setsand Systems vol 80 no 3 pp 359ndash368 1996
[8] K H Kim Y B Jun and Y H Yon ldquoOn anti fuzzy ideals innear-ringsrdquo Iranian Journal of Fuzzy Systems vol 2 no 2 pp71ndash80 2005
[9] K H Kim and Y B Jun ldquoA note on fuzzy 119877-subgroups of near-ringsrdquo Soochow Journal of Mathematics vol 28 no 4 pp 339ndash346 2002
[10] SDKim andH S Kim ldquoOn fuzzy ideals of near-ringsrdquoBulletinof the Korean Mathematical Society vol 33 no 4 pp 593ndash6011996
[11] O R Devi ldquoIntuitionistic fuzzy near-rings and its propertiesrdquoThe Journal of the Indian Mathematical Society vol 74 no 3-4pp 203ndash219 2007
[12] S K Bhakat and P Das ldquoOn the definition of a fuzzy subgrouprdquoFuzzy Sets and Systems vol 51 no 2 pp 235ndash241 1992
[13] S K Bhakat and P Das ldquoFuzzy subrings and ideals redefinedrdquoFuzzy Sets and Systems vol 81 no 3 pp 383ndash393 1996
[14] B Davvaz ldquo(120598 120598 or 119902)-fuzzy subnear-rings and idealsrdquo SoftComputing vol 10 no 3 pp 206ndash211 2006
[15] B Davvaz ldquoFuzzy 119877-subgroups with thresholds of near-ringsand implication operatorsrdquo Soft Computing vol 12 no 9 pp875ndash879 2008
[16] A Narayanan and T Manikantan ldquo(120598 120598 or 119902)-fuzzy subnear-rings and (120598 120598or119902)-fuzzy ideals of near-ringsrdquo Journal of AppliedMathematics amp Computing vol 18 no 1-2 pp 419ndash430 2005
[17] J Zhan and B Davvaz ldquoGeneralized fuzzy ideals of near-ringsrdquoApplied Mathematics vol 24 no 3 pp 343ndash349 2009
[18] O R Devi and M R Singh ldquoOn fuzzy ideals of 119873-groupsrdquoBulletin of Pure amp Applied Sciences E vol 26 no 1 pp 11ndash232007
[19] T K Dutta and B K Biswas ldquoOn fuzzy congruence of a near-ring modulerdquo Fuzzy Sets and Systems vol 112 no 2 pp 343ndash348 2000
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 A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf · of a ring was introduced by Liu [ ]. e notion of fuzzy subnear-ring and fuzzy ideals](https://reader036.fdocuments.net/reader036/viewer/2022071404/60f839985e18e36f2e647f1c/html5/thumbnails/6.jpg)
6 Algebra
ge sup119891(V)=119909
min 119860 (V) 05
= min sup119891(V)=119909
119860 (V) 05
= min 119891 (119860) (119909) 05
119891 (119860) (119899 (119910 + 119909) minus 119899119910)
= sup119891(119911)=119899(119910+119909)minus119899119910
119860 (119911)
ge sup119891(V)=119909119891(119906)=119910
119860 (119899 (119906 + V) minus 119899119906)
ge sup119891(V)=119909119891(119906)=119910
min 119860 (119906) 05
= min 119891 (119860) (119909) 05
(7)
Therefore 119891(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866 Similarly wecan show that 119891minus1(119860) is an (120598 120598 or 119902)-fuzzy ideal of 119866
Definition 32 Let119860 be (isin isinor119902)-fuzzy subgroup of119866 For any119909 isin 119866 let 119860
119909be defined by 119860
119909(119892) = min119860(119892 minus 119909) 05 for
all 119892 isin 119866 This fuzzy subset 119860119909is called the (isin isin or119902)-fuzzy
left coset of 119866 determined by 119860 and 119909
Remark 33 Let 119860 be an (120598 120598 or 119902)-fuzzy subgroup of 119866 Then119860 is an (120598 120598 or 119902)-fuzzy normal if and only if min119860(119892 minus
119909) 05 = min119860(minus119909 + 119892) 05 for all 119909 119892 isin 119866 If 119860 is an(120598 120598 or 119902)-fuzzy ideal we simply denote fuzzy coset by 119860
119909
Lemma 34 Let 119860 be an (120598 120598 or 119902)-fuzzy ideal of 119866 Then119860119909
= 119860119910if an only if 119860(119909 minus 119910) ge 05
Proof Assume that 119860(119909 minus 119910) ge 05 and 119892 isin 119866 119860119909(119892) =
min119860(119892minus119909) 05 ge min119860(119892minus119910) 119860(119910minus119909) 05 ge min119860(119892minus
119910) 119860(119909 minus 119910) 05 = min119860(119892 minus 119910) 05 = 119860119910(119892) which
implies that 119860119909
ge 119860119910 Similarly we can verify that 119860
119909le
119860119910 Conversely we assume that 119860
119909= 119860119910 Then 119860
119910(119909) =
119860119909(119909) rArr min119860(119909 minus 119910) 05 = 05 rArr 119860(119909 minus 119910) ge 05
Proposition 35 Every fuzzy coset 119860119909is constant on every
coset of 1198660= 119909 isin 119866 | 119860(119909) = 119860(0)
Proof Let 119910 + 1199100
isin 119910 + 1198660 Now we have 119860
119909(119910 + 119910
0) =
min119860(119910+1199100minus119909) 05 ge min119860(119910minus119909) 119860(119909+119910
0minus119909) 05 ge
min119860(119910minus119909) 119860(1199100) 05 = min119860(119910minus119909) 05 = 119860
119909(119910) Also
119860119909(119910) = min119860(119910minus119909) 05 ge min119860(minus119909+119910+119910
0minus1199100) 05 ge
min119860(minus119909+119910+1199100) 119860(minus119910
0) 05 = min119860(minus119909+119910+119910
0) 05 ge
min119860(119910+1199100minus119909) 05 = 119860
119909(119910+1199100)Thus119860
119909(119910+1199100) = 119860
119909(119910)
for all 1199100isin 1198660
Theorem 36 For any (120598 120598 or 119902)-fuzzy ideal 119860 of 119866 119866119860the set
of all fuzzy cosets of 119860 in 119866 is an 119873-group under the additionand scalar multiplication defined by119860
119909+119860119910
= 119860119909+119910
119899(119860119909) =
119860119899119909
for all 119909 119910 isin 119866 119899 isin 119873 The function 119860(119860119909) 119866119860
rarr
[0 1] defined by 119860(119860119909) = 119860(119909) for all 119860
119909isin 119866119860is (120598 120598 or 119902)-
fuzzy ideal of 119866119860
Proof First we show that the compositions are well definedLet 119909 119910 119888 119889 isin 119866 119899 isin 119873 such that 119860
119909= 119860119910and 119860
119888= 119860119889
Then119860(119909minus119910) ge 05119860(119888minus119889) ge 05 Now we havemin119860(119909+
119888minus119889minus119910) 05 ge min119860(minus119910+119909+119888minus119889) 05 = 05which impliesthat119860(119909+ 119888minus119889minus119910) ge 05 So by Lemma 34 we have119860
119909+119888=
119860119910+119889
Again 119860119899119909
(119892) = min119860(119892 minus 119899119909) 05 ge min119860(119892 minus
119899119910) 119860(119899119909 minus 119899119910) 05 = min119860(119892 minus 119899119910) 119860[119899(119910 minus 119910 + 119909) minus
119899119910] 05 ge min119860(119892minus119899119910) 119860(minus119910+119909) 05 = 119860119899119910
(119892) Similarlywe show that 119860
119899119910(119892) ge 119860
119899119909(119892) Thus for 119860
119899119910(119892) = 119860
119899119909(119892)
for all 119892 isin 119866 Hence the compositions are well defined It isnow easy to verify that119866
119860is an119873-groupwith null element119860
0
and negative element119860minus119909 Next we check that119860 is (120598 120598 or 119902)-
fuzzy ideal of 119866119860 Let 119860
119909 119860119910
isin 119866119860 We have
(i) 119860(119860119909minus119860119910) = 119860(119860
119909minus119910) = 119860(119909minus119910) ge min119860(119909) 119860(119910)
05 = min119860(119860119909) 119860(119860
119910) 05
(ii) 119860(119860119899119909
) = 119860(119899119909) ge min119860(119909) 05 = min119860(119860119909)
05(iii) 119860(119860
119910+ 119860119909
minus 119860119910) = 119860(119860119910+119909minus119910
) = 119860(119910 + 119909 minus 119910) ge
min119860(119909) 05 = min119860(119860119909) 05
(iv) 119860[119899(119860119910+119860119909)minus119899119860
119910] = 119860[119860
119899(119910+119909)minus119899119910] = 119860[119899(119910+119909)minus
119899119910] ge min119860(119909) 05 = min119860(119860119909) 05
Hence the proof is completed
Lemma 37 Let 119867 = 119909 isin 119866 | 119860119909
ge 1198600 119870 = 119909 isin 119866 | 119860
119909=
1198600 Then 119870 = 119867 = 119860
05
Proof Let 119909 isin 119867 Then for all 119892 isin 119866 119860119909(119892) ge 119860
0(119892) rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr min119860(minus119909) 05 ge
min119860(0) 05 = 05 rArr 119860(119909) ge 05 Also for 119892 isin 1198661198600(119892) = min119860(119892) 05 = min119860(119892 minus 119909 + 119909) 05 ge
min119860(119892 minus 119909) 119860(119909) 05 = min119860(119892 minus 119909) 05 = 119860119909(119892) rArr
1198600
ge 119860119909 So 119867 = 119870 As we have seen if 119909 isin 119867 then
119860(119909) ge 05 rArr 119909 isin 11986005 Conversely if 119909 isin 119860
05 for any
119892 isin 119866 119860(119892 minus 119909) ge min119860(119892) 119860(119909) 05 = min119860(119892) 05 rArr
min119860(119892 minus 119909) 05 ge min119860(119892) 05 rArr 119860119909
ge 1198600 Thus
119867 = 11986005 which means that 119867 is an ideal of 119866
Theorem38 If119860 is an (120598 120598or119902)-fuzzy ideal of119866 then themap119891 119866 rarr 119866
119860given by 119891(119909) = 119860
119909is an 119873-homomorphism
with kernel 119891 = 11986005
and so 11986611986005
is isomorphic to 119866119860
Proof It is clear that 119891 is an onto 119873-homomorphism from 119866
to 119866119860with kernel 119891 = 119909 isin 119866 | 119891(119909) = 119891(0) = 119909 isin 119866 |
119860119909
= 1198600 = 119860
05
Corollary 39 Let119860 be an (120598 120598or119902)-fuzzy ideal of119866 and let 119861be an (120598 120598or119902)-fuzzy ideal of119866
119860Then 119861 119866 rarr [0 1] defined
by 119861(119909) = 119861(119860119909) is an (120598 120598 or 119902)-fuzzy ideal of 119866 containing 119860
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965
Algebra 7
[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971
[3] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982
[4] S Abou-Zaid ldquoOn fuzzy subnear-rings and idealsrdquo Fuzzy Setsand Systems vol 44 no 1 pp 139ndash146 1991
[5] B Davvaz ldquoFuzzy ideals of near-rings with interval valuedmembership functionsrdquo Journal of Sciences Islamic Republic ofIran vol 12 no 2 pp 171ndash175 2001
[6] K H Kim ldquoOn vague119877-subgroups of near-ringsrdquo InternationalMathematical Forum vol 4 no 33ndash36 pp 1655ndash1661 2009
[7] S K Bhakat and P Das ldquo(120598 120598 or 119902)-fuzzy subgrouprdquo Fuzzy Setsand Systems vol 80 no 3 pp 359ndash368 1996
[8] K H Kim Y B Jun and Y H Yon ldquoOn anti fuzzy ideals innear-ringsrdquo Iranian Journal of Fuzzy Systems vol 2 no 2 pp71ndash80 2005
[9] K H Kim and Y B Jun ldquoA note on fuzzy 119877-subgroups of near-ringsrdquo Soochow Journal of Mathematics vol 28 no 4 pp 339ndash346 2002
[10] SDKim andH S Kim ldquoOn fuzzy ideals of near-ringsrdquoBulletinof the Korean Mathematical Society vol 33 no 4 pp 593ndash6011996
[11] O R Devi ldquoIntuitionistic fuzzy near-rings and its propertiesrdquoThe Journal of the Indian Mathematical Society vol 74 no 3-4pp 203ndash219 2007
[12] S K Bhakat and P Das ldquoOn the definition of a fuzzy subgrouprdquoFuzzy Sets and Systems vol 51 no 2 pp 235ndash241 1992
[13] S K Bhakat and P Das ldquoFuzzy subrings and ideals redefinedrdquoFuzzy Sets and Systems vol 81 no 3 pp 383ndash393 1996
[14] B Davvaz ldquo(120598 120598 or 119902)-fuzzy subnear-rings and idealsrdquo SoftComputing vol 10 no 3 pp 206ndash211 2006
[15] B Davvaz ldquoFuzzy 119877-subgroups with thresholds of near-ringsand implication operatorsrdquo Soft Computing vol 12 no 9 pp875ndash879 2008
[16] A Narayanan and T Manikantan ldquo(120598 120598 or 119902)-fuzzy subnear-rings and (120598 120598or119902)-fuzzy ideals of near-ringsrdquo Journal of AppliedMathematics amp Computing vol 18 no 1-2 pp 419ndash430 2005
[17] J Zhan and B Davvaz ldquoGeneralized fuzzy ideals of near-ringsrdquoApplied Mathematics vol 24 no 3 pp 343ndash349 2009
[18] O R Devi and M R Singh ldquoOn fuzzy ideals of 119873-groupsrdquoBulletin of Pure amp Applied Sciences E vol 26 no 1 pp 11ndash232007
[19] T K Dutta and B K Biswas ldquoOn fuzzy congruence of a near-ring modulerdquo Fuzzy Sets and Systems vol 112 no 2 pp 343ndash348 2000
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 A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf · of a ring was introduced by Liu [ ]. e notion of fuzzy subnear-ring and fuzzy ideals](https://reader036.fdocuments.net/reader036/viewer/2022071404/60f839985e18e36f2e647f1c/html5/thumbnails/7.jpg)
Algebra 7
[2] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971
[3] W J Liu ldquoFuzzy invariant subgroups and fuzzy idealsrdquo FuzzySets and Systems vol 8 no 2 pp 133ndash139 1982
[4] S Abou-Zaid ldquoOn fuzzy subnear-rings and idealsrdquo Fuzzy Setsand Systems vol 44 no 1 pp 139ndash146 1991
[5] B Davvaz ldquoFuzzy ideals of near-rings with interval valuedmembership functionsrdquo Journal of Sciences Islamic Republic ofIran vol 12 no 2 pp 171ndash175 2001
[6] K H Kim ldquoOn vague119877-subgroups of near-ringsrdquo InternationalMathematical Forum vol 4 no 33ndash36 pp 1655ndash1661 2009
[7] S K Bhakat and P Das ldquo(120598 120598 or 119902)-fuzzy subgrouprdquo Fuzzy Setsand Systems vol 80 no 3 pp 359ndash368 1996
[8] K H Kim Y B Jun and Y H Yon ldquoOn anti fuzzy ideals innear-ringsrdquo Iranian Journal of Fuzzy Systems vol 2 no 2 pp71ndash80 2005
[9] K H Kim and Y B Jun ldquoA note on fuzzy 119877-subgroups of near-ringsrdquo Soochow Journal of Mathematics vol 28 no 4 pp 339ndash346 2002
[10] SDKim andH S Kim ldquoOn fuzzy ideals of near-ringsrdquoBulletinof the Korean Mathematical Society vol 33 no 4 pp 593ndash6011996
[11] O R Devi ldquoIntuitionistic fuzzy near-rings and its propertiesrdquoThe Journal of the Indian Mathematical Society vol 74 no 3-4pp 203ndash219 2007
[12] S K Bhakat and P Das ldquoOn the definition of a fuzzy subgrouprdquoFuzzy Sets and Systems vol 51 no 2 pp 235ndash241 1992
[13] S K Bhakat and P Das ldquoFuzzy subrings and ideals redefinedrdquoFuzzy Sets and Systems vol 81 no 3 pp 383ndash393 1996
[14] B Davvaz ldquo(120598 120598 or 119902)-fuzzy subnear-rings and idealsrdquo SoftComputing vol 10 no 3 pp 206ndash211 2006
[15] B Davvaz ldquoFuzzy 119877-subgroups with thresholds of near-ringsand implication operatorsrdquo Soft Computing vol 12 no 9 pp875ndash879 2008
[16] A Narayanan and T Manikantan ldquo(120598 120598 or 119902)-fuzzy subnear-rings and (120598 120598or119902)-fuzzy ideals of near-ringsrdquo Journal of AppliedMathematics amp Computing vol 18 no 1-2 pp 419ndash430 2005
[17] J Zhan and B Davvaz ldquoGeneralized fuzzy ideals of near-ringsrdquoApplied Mathematics vol 24 no 3 pp 343ndash349 2009
[18] O R Devi and M R Singh ldquoOn fuzzy ideals of 119873-groupsrdquoBulletin of Pure amp Applied Sciences E vol 26 no 1 pp 11ndash232007
[19] T K Dutta and B K Biswas ldquoOn fuzzy congruence of a near-ring modulerdquo Fuzzy Sets and Systems vol 112 no 2 pp 343ndash348 2000
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 A Study on Fuzzy Ideals of -Groupsdownloads.hindawi.com/archive/2013/594636.pdf · of a ring was introduced by Liu [ ]. e notion of fuzzy subnear-ring and fuzzy ideals](https://reader036.fdocuments.net/reader036/viewer/2022071404/60f839985e18e36f2e647f1c/html5/thumbnails/8.jpg)
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
![On T-Fuzzy Ideals in NearringsAbou-Zaid [13] introduced the notion of a fuzzy subnearring and studied fuzzy left (right) ideals of a nearring, and gave some properties of fuzzy prime](https://static.fdocuments.net/doc/165x107/60da4ef3558d4521cc7734bd/on-t-fuzzy-ideals-in-nearrings-abou-zaid-13-introduced-the-notion-of-a-fuzzy-subnearring.jpg)
