Math Research of Victor PortonAbstract In this work I introduce and study in details the concepts...

Algebraic General Topology�Volume 1

Victor Porton

Web: http://www.mathematics21.org

June 6, 2015

�. This document has been written using the GNU TEXMACS text editor (see www.texmacs.org).

Abstract

In this work I introduce and study in details the concepts of funcoids which generalize proximityspaces and reloids which generalize uniform spaces, and generalizations thereof. The concept offuncoid is generalized concept of proximity, the concept of reloid is cleared from super�uous details(generalized) concept of uniformity. Also funcoids and reloids are generalizations of binary relationswhose domains and ranges are �lters (instead of sets).

Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this pro-vides us with a common generalization of calculus and discrete mathematics.

The concept of continuity is de�ned by an algebraic formula (instead of old messy epsilon-deltanotation) for arbitrary morphisms (including funcoids and reloids) of a partially ordered category.In one formula continuity, proximity continuity, and uniform continuity are generalized.

Also I de�ne connectedness for funcoids and reloids.Then I consider generalizations of funcoids: pointfree funcoids and generalization of pointfree

funcoids: multifuncoids. Also I de�ne several kinds of products of funcoids and other morphisms.Before going to topology, this book studies properties of co-brouwerian lattices and �lters.

Keywords: algebraic general topology, quasi-uniform spaces, generalizations of proximity spaces,generalizations of nearness spaces, generalizations of uniform spacesA.M.S. subject classi�cation: 54J05, 54A05, 54D99, 54E05, 54E15, 54E17, 54E99

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Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

1.1 Draft status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.2 Intended audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.3 Reading Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.4 Our topic and rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.5 Earlier works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.6 Kinds of continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.7 Structure of this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.8 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

1.8.1 Grothendieck universes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?1.8.2 Misc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

1.9 Unusual notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

2 Common knowledge, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

2.1 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.1 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

2.1.1.1 Intersecting and joining elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.2 Linear order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.3 Meets and joins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.4 Semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.5 Lattices and complete lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.6 Distributivity of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.7 Di�erence and complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.8 Boolean lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.9 Center of a lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.10 Atoms of posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.11 Kuratowski's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.12 Homomorphisms of posets and lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.13 Galois connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.14 Co-Brouwerian lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.1.15 Dual pseudocomplement on co-Heyting lattices . . . . . . . . . . . . . . . . . . . . . . . ?

2.2 Intro to category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?2.3 Intro to group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3 More on order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.1 Straight maps and separation subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.1.1 Straight maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.1.2 Separation subsets and full stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.1.3 Atomically Separable Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.2 Free Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.2.1 Starrish posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.3 Quasidi�erence and Quasicomplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.4 Several equal ways to express pseudodi�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.5 Partially ordered categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.5.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5

3.5.2 Dagger categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.5.2.1 Some special classes of morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.6 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.7 A proposition about binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8 In�nite associativity and ordinated product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.2 Used notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.8.2.1 Currying and uncurrying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?The customary de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?Currying and uncurrying with a dependent variable . . . . . . . . . . . . . . . . . . ?

3.8.2.2 Functions with ordinal numbers of arguments . . . . . . . . . . . . . . . . . . . . . ?3.8.3 On sums of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4 Ordinated product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

3.8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.2 Concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.3 Finite example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.4 The de�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.5 De�nition with composition for every multiplier . . . . . . . . . . . . . . . . . . . . ?3.8.4.6 De�nition with shifting arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?3.8.4.7 Associativity of ordinated product . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

In�nite associativity implies associativity . . . . . . . . . . . . . . . . . . . . . . . . . ?Concatenation is associative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4 Filters and �ltrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1 Introduction to �lters and �ltrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.1.1 Filters on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1.2 Intro to �lters on a meet-semilattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1.3 Intro to �lters on a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.1.4 Intro to �ltrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.2 Filtrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.1 Core Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.2 Filtrators with Separable Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.3 Intersection and Joining with an Element of the Core . . . . . . . . . . . . . . . . . . . ?4.2.4 Characterization of Finitely Meet-Closed Filtrators . . . . . . . . . . . . . . . . . . . . . ?4.2.5 Stars of Elements of Filtrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.6 Atomic Elements of a Filtrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.7 Prime Filtrator Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.8 Some Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.9 Complements and Core Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.10 Core Part and Atomic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.11 Distributivity of Core Part over Lattice Operations . . . . . . . . . . . . . . . . . . . . ?4.2.12 Co-Separability of Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.2.13 Filtrators over Boolean Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.2.13.1 Distributivity for an Element of Boolean Core . . . . . . . . . . . . . . . . . . . . ?4.3 Filters on a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.1 Filters on posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.2 Filters on meet-semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.3 Order of �lters. Principal �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.3.1 Minimal and maximal �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.4 Primary �ltrator is �ltered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.5 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.6 Co-separability of Core for Primary Filtrators . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.7 Core Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.8 Intersecting and Joining with an Element of the Core . . . . . . . . . . . . . . . . . . . ?4.3.9 Formulas for Meets and Joins of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.10 Separability of Core for Primary Filtrators . . . . . . . . . . . . . . . . . . . . . . . . . ?

6 Table of contents

4.3.11 Distributivity of the Lattice of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.12 Filters over Boolean Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.12.1 Distributivity for an Element of Boolean Core . . . . . . . . . . . . . . . . . . . . ?4.3.13 Generalized Filter Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.14 Stars for �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.3.14.1 Stars of Filters on Boolean Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.15 More about the Lattice of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.16 Atomic Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.17 Some Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.18 Filters and a Special Sublattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.19 Core Part and Atomic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.20 Complements and Core Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.3.21 Complementive Filters and Factoring by a Filter . . . . . . . . . . . . . . . . . . . . . . ?4.3.22 Pseudodi�erence of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.4 Filters on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.4.1 Fréchet Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.4.2 Number of Filters on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.5 Some Counter-Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.5.1 Weak and Strong Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

4.6 Open problems about �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.6.1 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.6.2 Quasidi�erence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?4.6.3 Non-Formal Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5 Common knowledge, part 2 (topology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.1.1 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.1.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.2 Pretopological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.2.1 Pretopology induced by a metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?5.3.1 Relationships between pretopologies and topologies . . . . . . . . . . . . . . . . . . . . . ?

5.3.1.1 Topological space induced by preclosure space . . . . . . . . . . . . . . . . . . . . . ?5.3.1.2 Preclosure space induced by topological space . . . . . . . . . . . . . . . . . . . . . ?5.3.1.3 Topology induced by a metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

5.4 Proximity spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6 Funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.1 Informal introduction into funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.2 Basic de�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.2.1 Composition of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.3 Funcoid as continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.4 Lattices of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.5 More on composition of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.6 Domain and range of a funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.7 Categories of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.8 Specifying funcoids by functions or relations on atomic �lters . . . . . . . . . . . . . . . . . . ?6.9 Direct product of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.10 Atomic funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.11 Complete funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.12 Funcoids corresponding to pretopologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.13 Completion of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.13.1 More on completion of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.13.1.1 Open maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

6.14 Monovalued and injective funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6.15 T0-, T1-, T2-, and T3-separable funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

Table of contents 7

6.16 Filters closed regarding a funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

7 Reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

7.1 Basic de�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.2 Composition of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.3 Direct product of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.4 Restricting reloid to a �lter. Domain and image . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.5 Categories of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.6 Monovalued and injective reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?7.7 Complete reloids and completion of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

8 Relationships between funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . ?

8.1 Funcoid induced by a reloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?8.2 Reloids induced by a funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?8.3 Galois connections between funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . ?8.4 Funcoidal reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

9 On distributivity of composition with a principal reloid . . . . . . . . . . . . . . . . . ?

9.1 Decomposition of composition of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . ?9.2 Decomposition of composition of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?9.3 Lemmas for the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?9.4 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?9.5 Embedding reloids into funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

10 Continuous morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

10.1 Traditional de�nitions of continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.1.1 Pretopology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.1.2 Proximity spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.1.3 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

10.2 Our three de�nitions of continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?10.3 Continuity of a restricted morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

11 Connectedness regarding funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . ?

11.1 Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.2 Endomorphism series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.3 Connectedness regarding binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.4 Connectedness regarding funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . ?11.5 Algebraic properties of S and S� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

12 Total boundness of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

12.1 Thick binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.2 Totally bounded endoreloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.3 Special case of uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.4 Relationships with other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?12.5 Additional predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

13 Orderings of �lters in terms of reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

13.1 Equivalent �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.2 Ordering of �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

13.2.1 Existence of no more than one monovalued injective reloid for a given pair of ultra�lters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.2.1.1 The lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.2.1.2 The main theorem and its consequences . . . . . . . . . . . . . . . . . . . . . . . . ?

13.3 Rudin-Keisler equivalence and Rudin-Keisler order . . . . . . . . . . . . . . . . . . . . . . . . ?

8 Table of contents

13.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?13.4.1 Metamonovalued reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

14 Counter-examples about funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . ?

14.1 Second product. Oblique product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

15 Pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

15.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.2 Composition of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.3 Pointfree funcoid as continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.4 The order of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.5 Domain and range of a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.6 Category of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.7 Specifying funcoids by functions or relations on atomic �lters . . . . . . . . . . . . . . . . . ?15.8 More on composition of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.9 Direct product of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.10 Atomic pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.11 Complete pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.12 Completion and co-completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.13 Monovalued and injective pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.14 Elements closed regarding a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . ?15.15 Connectedness regarding a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

16 Convergence of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

16.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?16.2 Relationships between convergence and continuity . . . . . . . . . . . . . . . . . . . . . . . . ?16.3 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?16.4 Generalized limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

16.4.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17 Multifuncoids and staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.1 Product of two funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.1.1 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.1.2 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.2 Function spaces of posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.3 De�nition of staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.4 Upgrading and downgrading a set regarding a �ltrator . . . . . . . . . . . . . . . . . . . . . ?

17.4.1 Upgrading and downgrading staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.5 Principal staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.6 Multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.7 Join of multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.8 In�nite product of poset elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.9 On products of staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.10 Star categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.10.1 Abrupt of quasi-invertible categories with star-morphisms . . . . . . . . . . . . . . . ?17.11 Product of an arbitrary number of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.11.1 Mapping a morphism into a pointfree funcoid . . . . . . . . . . . . . . . . . . . . . . . ?17.11.2 General cross-composition product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.3 Star composition of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.4 Star composition of Rel-morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.5 Cross-composition product of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.11.6 Simple product of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.12 Multireloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.12.1 Starred reloidal product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.13 Subatomic product of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

Table of contents 9

17.14 On products and projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.14.1 Staroidal product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.14.2 Cross-composition product of pointfree funcoids . . . . . . . . . . . . . . . . . . . . . ?17.14.3 Subatomic product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.14.4 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?

17.15 Relationships between cross-composition and subatomic products . . . . . . . . . . . . . ?17.16 Coordinate-wise continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?17.17 Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.18 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

17.18.1 Informal questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

18 Identity staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

18.1 Additional propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.2 On pseudofuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318.3 Complete staroids and multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

18.3.1 Complete free stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?18.3.1.1 Completely starrish posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

18.3.2 More on free stars and complete free stars . . . . . . . . . . . . . . . . . . . . . . . . . 1418.3.3 Complete staroids and multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

18.4 Identity staroids and multifuncoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.1 Identity relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.2 General de�nitions of identity staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.3 Identities are staroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.4 Special case of sets and �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.5 Relationships between big and small identity staroids . . . . . . . . . . . . . . . . . . 1718.4.6 Identity staroids on principal �lters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4.7 Identity staroids represented as meets and joins . . . . . . . . . . . . . . . . . . . . . 17

18.5 Finite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.6 Counter-examples and conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

19 Postface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

19.1 Formalizing this theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

10 Table of contents

Chapter 1IntroductionFor related materials, articles, research questions, and erratum consult the Web page of the authorof the book:

http://www.mathematics21.org/algebraic-general-topology.html

1.1 Draft statusThis is a draft.

1.2 Intended audienceThis book is suitable for any math student as well as for researchers.

To make this book be understandable even for �rst grade students, I made a chapter aboutbasic concepts (posets, lattices, topological spaces, etc.), which an already knowledgeable personmay skip reading. It is assumed that the reader knows basic set theory.

But it is also valuable for mature researchers, as it contains much original research which youcould not �nd in any other source except of my work.

Knowledge of the basic set theory is expected from the reader.Despite that this book presents new research, it is well structured and is suitable to be used as

a textbook for a college course.Your comments about this book are welcome to the email [email protected].

1.3 Reading OrderIf you know basic order and lattice theory (including Galois connections and brouwerian lattices)and basics of category theory, you may skip reading the chapter 2 (�Common knowledge, part 1�).

You are recommended to read the rest of this book by the order.

1.4 Our topic and rationaleFrom [38]: Point-set topology, also called set-theoretic topology or general topology, is the studyof the general abstract nature of continuity or "closeness" on spaces. Basic point-set topologicalnotions are ones like continuity, dimension, compactness, and connectedness.

In this work we study a new approach to point-set topology (and pointfree topology).Traditionally general topology is studied using topological spaces (de�ned below in the section

5.3). I however argue that the theory of topological spaces is not the best method of studyinggeneral topology and introduce an alternative theory, the theory of funcoids . Despite of popularityof the theory of topological spaces it has some drawbacks and is in my opinion not the mostappropriate formalism to study most of general topology. Because topological spaces are tailoredfor study of special sets, so called open and closed sets, studying general topology with topologicalspaces is a little anti-natural and ugly. In my opinion the theory of funcoids is more elegant thanthe theory of topological spaces, and it is better to study funcoids than topological spaces. Oneof the main purposes of this work is to present an alternative General Topology based on funcoidsinstead of being based on topological spaces as it is customary. In order to study funcoids theprior knowledge of topological spaces is not necessary. Nevertheless in this work I will considertopological spaces and the topic of interrelation of funcoids with topological spaces.

11

In fact funcoids are a generalization of topological spaces, so the well known theory of topolog-ical spaces is a special case of the below presented theory of funcoids.

But probably the most important reason to study funcoids is that funcoids are a generalizationof proximity spaces (see section 2.2 for the de�nition of proximity spaces). Before this work it waswritten that the theory of proximity spaces was an example of a stalled research, almost nothinginteresting was discovered about this theory. It was so because the proper way to research proximityspaces is to research their generalization, funcoids. And so it was stalled until discovery of funcoids.That generalized theory of proximity spaces will bring us yet many interesting results.

In addition to funcoids I research reloids. Using below de�ned terminology it may be said thatreloids are (basically) �lters on Cartesian product of sets, and this is a special case of uniformspaces. We don't need to de�ne uniform spaces in this work, it is enough for the reader just toknow that uniform spaces are certain �lters on direct product of sets.

Afterward we study some generalizations.Somebody might ask, why to study it? My approach relates to traditional general topology like

complex numbers to real numbers theory. Be sure this will �nd applications.This book has a de�ciency: It does not properly relate my theory with previous research in

general topology and does not consider deeper category theory properties. It is however OK fornow, as I am going to do this study in later volumes (continuation of this book).

Many proofs in this book may seem too easy and thus this theory not sophisticated enough.But it is largely a result of a well structured digraph of proofs, where more di�cult results aremade easy by reducing them to easier lemmas and propositions.

1.5 Earlier works

Some mathematicians were researching generalizations of proximities and uniformities before mebut they have failed to reach the right degree of generalization which is presented in this workallowing to represent properties of spaces with algebraic (or categorical) formulas.

Proximity structures were introduced by Smirnov in [11].Some references to predecessors:

� In [14], [15], [24], [2], [33] generalized uniformities and proximities are studied.

� Proximities and uniformities are also studied in [21], [22], [32], [34], [35].

� [19] and [20] contains recent progress in quasi-uniform spaces. [20] has a very long list ofrelated literature.

Some works ([31]) about proximity spaces consider relationships of proximities and compact topo-logical spaces. In this work the attempt to de�ne or research their generalization, compactness offuncoids or reloids is not done. It seems potentially productive to attempt to borrow the de�nitionsand procedures from the above mentioned works. I hope to do this study in a separate volume.

[10] studies mappings between proximity structures. (In this volume no attempt to researchmappings between funcoids is done.) [25] researches relationships of quasi-uniform spaces andtopological spaces. [1] studies how proximity structures can be treated as uniform structures andcompacti�cation regarding proximity and uniform spaces.

This book is based partially on my articles [29], [27], [28]. [TODO: Add more references to myarticles.]

In [29] I introduced the concept of �lter objects. This was probably not a very good idea. In thiswork I instead use plain �lters (not �lter objects) and t and u notation for joins and meets insteadof [ and \, which may be confused with set theoretic operations, for lattices in consideration (andfor the lattice of �lters the order is reverse to the set theoretic inclusion). Also this work di�ersfrom [29] in using in some formulations the lattice of principal �lters which is isomorphic to thebase poset instead of using the base poset itself (what was possible in [29] thanks to using �lterobjects). I've replaced (F;A) notation for primary �ltrators with (F;Z) for consistency of notationamong sections.

12 Introduction

1.6 Kinds of continuityA research result based on this book but not fully included in this book (and not yet published) isthat the following kinds of continuity are described by the same algebraic (or rather categorical)formulas for di�erent kinds of continuity and have common properties:

� discrete continuity (between digraphs);� (pre)topological continuity;� proximal continuity;� uniform continuity;� Cauchy continuity;� (probably other kinds of continuity).

Thus my research justi�es using the same word �continuity� for these diverse kinds of continuity.See http://www.mathematics21.org/algebraic-general-topology.html

1.7 Structure of this bookIn the chapter 2 �Common knowledge, part 1� some well known de�nitions and theories are con-sidered. You may skip its reading if you already know it. That chapter contains info about:

� posets;� lattices and complete lattices;� Galois connections;� co-brouwerian lattices;� a very short intro into category theory (It is very basic, I even don't de�ne functors as they

have no use in my theory);

� a very short introduction to group theory.Afterward there are my little additions to poset/lattice and category theory.

Afterward there is the theory of �lters and �ltrators.Then there is �Common knowledge, part 2 (topology)�, which considers brie�y:

� metric spaces;� topological spaces;� pretopological spaces;� proximity spaces.

Despite of the name �Common knowledge� this second common knowledge chapter is recommendedto be read completely even if you know topology well, because it contains some rare theorems notknown to most mathematicians and hard to �nd in literature.

Then the most interesting thing in this book, the theory of funcoids, starts.Afterwards there is the theory of reloids.Then I show relationships between funcoids and reloids.The last I research generalizations of funcoids, pointfree funcoids , staroids and multifuncoids

and some di�erent kinds of products of morphisms.

1.8 Basic notation

1.8.1 Grothendieck universesWe will work in ZFC with an in�nite and uncountable Grothendieck universe.

1.8 Basic notation 13

A Grothendieck universe is just a set big enough to make all usual set theory inside it. Forexample if f is a Grothendieck universe, and sets X; Y 2 f, then also X [ Y 2 f, X \ Y 2 f,X �Y 2f, etc.

A set which is a member of a Grothendieck universe is called a small set (regarding thisGrothendieck universe). We can restrict our consideration to small sets in order to get rid troubleswith proper classes.

De�nition 1.1. Grothendieck universe is a set f such that:

1. If x2f and y 2 x then y 2f.2. If x; y 2f, then fx; yg2f.3. If x2f then Px2f.4. If fxi j i2 I 2fg is a family of elements of f, then

Si2I xi2f.

One can deduce from this also:

1. If x2f, then fxg2f.2. If x is a subset of y 2f, then x2f.3. If x; y 2f then the ordered pair (x; y)= ffx; yg; xg2f.4. If x; y 2f then x[ y and x� y are in f.5. If fxi j i2 I 2fg is a family of elements of f, then the product

Qi2I xi2f.

6. If x2f, then the cardinality of x is strictly less than the cardinality of f.

1.8.2 MiscIn this book quanti�ers bind tightly. That is 8x 2A: P (x) ^ Q and 8x 2A: P (x)) Q should beread (8x2A:P (x))^Q and (8x2A:P (x)))Q not 8x2A: (P (x)^Q) and 8x2A: (P (x))Q).

The set of functions from a set A to a set B is denoted as BA.I will often skip parentheses and write fx instead of f(x) to denote the result of a function f

acting on the argument x.I will denote hf iX=ff� j �2Xg and X [f ]Y ,9x2X; y2Y :xf y for sets X , Y and a binary

relation f . (Note that functions are a special case of binary relations.)By just hf i and [f ] I will denote the corresponding function and relation on small sets.�x2D: f(x)= f(x; f(x)) j x2Dg for a set D and and a form f depending on the variable x.I will denote source and destination of a morphism f of any category (See �Common knowledge,

part 1� chapter for a de�nition of a category.) as Src f and Dst f correspondingly. Note that belowde�ned domain and image of a funcoid are not the same as it source and destination.

I will denote GR(A;B; f)= f for any morphism (A;B; f) of either Set or Rel.I will denote hf i= hGR f i and [f ]=[GR f ] for any morphism f of either Set or Rel.

1.9 Unusual notation

In the chapter 2 (which you may skip reading if you are already knowledgeable) some non-standardnotation is de�ned. I summarize here this notation for the case if you choose to skip reading thatchapter:

Partial order is denoted as v.Meets and joins are denoted as u, t,

d,F

.I call element b substractive from an elements a (of a distributive lattice A) when the di�erence

a n b exists. I call b complementive to a when there exist c2A such that bu c=0 and bt c= a. Wewill prove that b is complementive to a i� b is substractive from a and bv a.

De�nition 1.2. Call a and b of a poset A intersecting, denoted a�/ b, when there exists a non-least element c such that cv a^ cv b.

14 Introduction

De�nition 1.3. a� b=def:(a�/ b).

De�nition 1.4. I call elements a and b of a poset A joining and denote a� b when there are nonon-greatest element c such that cw a^ cw b.

De�nition 1.5. a�/ b=def:(a� b).

Obvious 1.6. a�/ b i� au b is non-least, for every elements a, b of a meet-semilattice.

Obvious 1.7. a� b if at b is the greatest element, for every elements a, b of a join-semilattice.

I extend the de�nitions of pseudocomplement and dual pseudocomplement to arbitrary posets(not just lattices as it is customary):

De�nition 1.8. Let A be a poset. Pseudocomplement of a is

max fc2A j c� ag:

If z is the pseudocomplement of a we will denote z= a�.

De�nition 1.9. Let A be a poset. Dual pseudocomplement of a is

min fc2A j c� ag:

If z is the dual pseudocomplement of a we will denote z= a+.

1.9 Unusual notation 15

Chapter 2

Common knowledge, part 1

In this chapter we will consider some well known mathematical theories. If you already know themyou may skip reading this chapter (or its parts).

2.1 Order theory

2.1.1 Posets

De�nition 2.1. The identity relation on a set A is idA= f(a; a) j a2Ag.

De�nition 2.2. A preorder on a set A is a binary relation v which is:

� re�exive on A ((v)� idA);

� transitive ((v) � (v)� (v)).

De�nition 2.3. A partial order on a set A is a preorder on A which is antisymmetric ((v) \(v)¡1� (=)).

The reverse relation is denoted w.

De�nition 2.4. a is a subelement of b (or what is the same a is contained in b or b contains a)i� av b.

Obvious 2.5. The reverse of a partial order is also a partial order.

De�nition 2.6. A poset is a set A together with a partial order on it is called a partially orderedset (poset for short).

De�nition 2.7. Strict partial order @ corresponding to the partial order v on a set A is de�nedby the formula (@) = (v) n idA.

De�nition 2.8. A partial order on a set A restricted to a set B �A is (v)\ (B �B).

Obvious 2.9. A partial order on a set A restricted to a set B �A is a partial order on B.

De�nition 2.10.

� The least element 0 of a poset A is de�ned by the formula 8a2A: 0v a.

� The greatest element 1 of a poset A is de�ned by the formula 8a2A: 1w a.

Proposition 2.11. There exist no more than one least element and no more than one greatestelement (for a given poset).

Proof. By antisymmetry. �

De�nition 2.12. The dual order for v is w.

17

Obvious 2.13. Dual of a partial order is a partial order.

De�nition 2.14. The dual poset for a poset (A;v) is the poset (A;w).

Below we will sometimes use duality that is replacement of the partial order and all relatedoperations and relations with their duals. In other words, it is enough to prove a theorem for anorder v and the similar theorem for w follows by duality.

2.1.1.1 Intersecting and joining elements

Let A be a poset.

De�nition 2.15. Call elements a and b of A intersecting, denoted a�/ b, when there exists a non-least element c such that cv a^ cv b.

De�nition 2.16. a� b=def:(a�/ b).

Obvious 2.17. a0�/ b0^ a1w a0^ b1w b0) a1�/ b1.

De�nition 2.18. I call elements a and b of A joining and denote a� b when there is no a non-greatest element c such that cw a^ cw b.

De�nition 2.19. a�/ b=def:(a� b).

Obvious 2.20. Intersecting is the dual of non-joining.

Obvious 2.21. a0� b0^ a1w a0^ b1w b0) a1� b1.

2.1.2 Linear order

De�nition 2.22. A poset A is called linearly ordered set (or what is the same, totally ordered set)if aw b_ bw a for every a; b2A.

Example 2.23. The set of real numbers with the customary order is a linearly ordered set.

De�nition 2.24. A set X 2PA where A is a poset is called a chain if A restricted to X is a totalorder.

2.1.3 Meets and joinsLet A be a poset.

De�nition 2.25. Given a setX 2PA the least element (also called minimum and denoted minX)of X is such a2X that 8x2X: avx.

Least element does not necessarily exists. But if it exists:

Proposition 2.26. For a given X 2PA there exist no more than one least element.

Proof. It follows from anti-symmetry. �

Greatest element is the dual of least element:

De�nition 2.27. Given a set X 2PA the greatest element (also called maximum and denotedmaxX) of X is such a2X that 8x2X : awx.

Remark 2.28. Least and greatest elements of a set X is a trivial generalization of the abovede�ned least and greatest element for the entire poset.

18 Common knowledge, part 1

De�nition 2.29.

� A minimal element of a set X 2PA is such a2A that @x2X : (awx^ x=/ a).� A maximal element of a set X 2PA is such a2A that @x2X : (avx^ x=/ a).

Remark 2.30. Minimal element is not the same as minimum, and maximal element is not thesame as maximum.

Obvious 2.31.

1. The least element (if it exists) is a minimal element.

2. The greatest element (if it exists) is a maximal element.

Exercise 2.1. Show that there may be more than one minimal and more than one maximal element for someposet.

De�nition 2.32. Upper bounds of a set X is the set fy 2A j 8x2X: y wxg.

The dual notion:

De�nition 2.33. Lower bounds of a set X is the set fy 2A j 8x2X: yvxg.

De�nition 2.34. JoinFX (also called supremum and denoted �supX�) of a set X is the least

element of its upper bounds (if it exists).

De�nition 2.35. MeetdX (also called in�mum and denoted �infX�) of a set X is the greatest

element of its lower bounds (if it exists).

We will write b=FX when b2A is the join of X or say that

FX does not exist if there are

no such b2A. (And dually for meets.)Exercise 2.2. Provide an example of

FX 2/ X for some set X on some poset.

I will denote meets and joins for a speci�c poset A asdA and FA .

Proposition 2.36.

1. If b is the greatest element of X thenFX = b.

2. If b is the least element of X thendX = b.

Proof. We will prove only the �rst as the second is dual.Let b be the greatest element of X . Then upper bounds of X are fy 2A j yw bg. Obviously b

is the least element of this set, that is the join. �

De�nition 2.37. Binary joins and meets are de�ned by the formulas

xt y=Gfx; yg and xu y=

lfx; yg:

Obvious 2.38. t and u are symmetric operations (whenever these are de�ned for given x and y).

Theorem 2.39.

1. IfFX exists then yw

FX,8x2X : ywx.

2. IfdX exists then yv

dX,8x2X : yvx.

Proof. I will prove only the �rst as the second follows by duality.y w

FX, y is an upper bound for X,8x2X: y wx. �

Corollary 2.40.

1. If at b exists then yw at b, y w a^ y w b.

2.1 Order theory 19

2. If au b exists then yv au b, y v a^ y v b.

2.1.4 Semilattices

De�nition 2.41.

1. A join-semilattice is a poset A such that at b is de�ned for every a; b2A.2. A meet-semilattice is a poset A such that au b is de�ned for every a; b2A.

Theorem 2.42.

1. The operation t is associative for any join-semilattice.2. The operation u is associative for any meet-semilattice.

Proof. I will prove only the �rst as the second follows by duality.We need to prove (at b)t c= at (bt c) for every a; b; c2A.Taking into account the de�nition of join, it is enough to prove that

xw (at b)t c,xw at (bt c)

for every x2A. Really, this follows from the chain of equivalences:

xw (at b)t c,xw at b^ xw c,xw a^ xw b^ xw c,xw a^ xw bt c,xw at (bt c): �

Obvious 2.43. a�/ b i� au b is non-least, for every elements a, b of a meet-semilattice.

Obvious 2.44. a� b if at b is the greatest element, for every elements a, b of a join-semilattice.

2.1.5 Lattices and complete lattices

De�nition 2.45. A bounded poset is a poset having both least and greatest elements.

De�nition 2.46. Lattice is a poset which is both join-semilattice and meet-semilattice.

De�nition 2.47. A complete lattice is a poset A such that for every X 2PA bothFX and

dX

exist.

Obvious 2.48. Every complete lattice is a lattice.

Proposition 2.49. Every complete lattice is a bounded poset.

Proof.F; is the least and

d; is the greatest element. �

Theorem 2.50. Let A be a poset.

1. IfFX is de�ned for every X 2PA, then A is a complete lattice.

2. IfdX is de�ned for every X 2PA, then A is a complete lattice.

Proof. See [26] or any lattice theory reference. �

Obvious 2.51. If X �Y for some X;Y 2PA where A is a complete lattice, then1.FX v

FY ;

2.dX w

dY .

Proposition 2.52. If S 2PPA then for every complete lattice A1.F S

S=FfFX j X 2Sg;

2.d S

S=dfdX j X 2Sg.


Proof. We will prove only the �rst as the second is dual.By de�nition of joins, it is enough to prove yw

F SS, yw

FfFX j X 2Sg.

Really, y wF S

S , 8x 2S

S: y w x, 8X 2 S8x 2 X: y w x , 8X 2 S: y wF

X ,y w

FfFX j X 2Sg. �

2.1.6 Distributivity of lattices

De�nition 2.53. A distributive lattice is such lattice A that for every x; y; z 2A

1. xu (yt z)= (xu y)t (xu z);

2. xt (yu z)= (xt y)u (xt z).

Theorem 2.54. For a lattice to be distributive it is enough just one of the conditions:

1. xu (yt z)= (xu y)t (xu z);

2. xt (yu z)= (xt y)u (xt z).

Proof. (xt y)u (xtz)=((xt y)ux)t((xt y)uz)=xt ((xuz)t (yuz))=(xt (xuz))t (yuz)=xt (yu z) (applied xu (yt z) = (xu y)t (xu z) twice). �

2.1.7 Di�erence and complement

De�nition 2.55. Let A be a distributive lattice with least element 0. The di�erence (denoteda n b) of elements a and b is such c2A that bu c=0 and at b= bt c. I will call b substractive froma when a n b exists.

Theorem 2.56. If A is a distributive lattice with least element 0, there exists no more than onedi�erence of elements a, b.

Proof. Let c and d be both di�erences a n b. Then bu c= bu d=0 and at b= bt c= bt d. So

c= cu (bt c) = cu (bt d) = (cu b)t (cu d)= 0t (cu d)= cu d:

Similarly d= du c. Consequently c= cu d= du c= d. �

De�nition 2.57. I will call b complementive to a i� there exists c 2 A such that b u c = 0 andbt c= a.

Proposition 2.58. b is complementive to a i� b is substractive from a and bv a.

Proof.

(. Obvious.

). We deduce bv a from bt c= a. Thus at b= a= bt c. �

Proposition 2.59. If b is complementive to a then (a n b)t b= a.

Proof. Because bv a by the previous proposition. �

De�nition 2.60. Let A be a bounded distributive lattice. The complement (denoted a�) of anelement a2A is such b2A that au b=0 and at b=1.

Proposition 2.61. If A is a bounded distributive lattice then a�=1 n a.

Proof. b= a�, bu a=0^ bt a=1, bu a=0^ 1t a= at b, b=1 n a. �

Corollary 2.62. If A is a bounded distributive lattice then exists no more than one complementof an element a2A.

2.1 Order theory 21

De�nition 2.63. An element of bounded distributive lattice is called complemented when itscomplement exists.

De�nition 2.64. A distributive lattice is a complemented lattice i� every its element is comple-mented.

Proposition 2.65. For a distributive lattice (a n b)n c=a n (bt c) if a n b and (a n b)n c are de�ned.

Proof. ((a n b) n c)u c=0; ((a n b) n c)t c=(a n b)t c; (a n b)u b=0; (a n b)t b= at b.We need to prove ((a n b) n c)u (bt c)= 0 and ((a n b) n c)t (bt c)= at (bt c).In fact,

((a n b) n c)u (bt c) =(((a n b) n c)u b)t (((a n b) n c)u c) =

(((a n b) n c)u b)t 0 =((a n b) n c)u b v

(a n b)u b = 0;

so ((a n b) n c)u (bt c)= 0;

((a n b) n c)t (bt c) =(((a n b) n c)t c)t b =

(a n b)t ct b =((a n b)t b)t c =

at bt c:

�

2.1.8 Boolean lattices

De�nition 2.66. A boolean lattice is a complemented distributive lattice.

The most important example of a boolean lattice is PA where A is a set, ordered by setinclusion.

Theorem 2.67. (De Morgan's laws) For every elements a, b of a boolean lattice

1. at b= a�u b�;2. au b= a�t b�.

Proof. We will prove only the �rst as the second is dual.It is enough to prove that at b is a complement of a�u b�. Really:

(at b)u (a�u b�)v au (a�u b�)= (au a�)u b�=0u b�=0;(at b)t (a�u b�)= ((at b)t a�)u ((at b)t b�)w (at a�)u (bt b�)=1u 1=1:

Thus (at b)u (a�u b�)=0 and (at b)t (a�u b�)=1. �

De�nition 2.68. A complete lattice A is join in�nite distributive when x uFS =

Fhx u iS;

complete lattice is meet in�nite distributive when xtdS=

dhxt iS for all x2A and S 2PA.

De�nition 2.69. In�nite distributive complete lattice is a complete lattice which is both joinin�nite distributive and meet in�nite distributive.

Theorem 2.70. Every complete boolean lattice is both join in�nite distributive and meet in�nitedistributive.

Proof. We will prove only join in�nitely distributivity, as the other is dual.Let S be a subset of a complete boolean lattice.


xuFS w

Fhxu iS is obvious. Now let u be any upper bound of hxu iS, that is xu yvu for

all y 2S. Theny= y u (xt x�)= (yu x)t (yu x�)vutx�;

and soFS vutx�. Thus

xuG

S vxu (ut x�)= (xu u)t (xux�)= (xuu)t 0=xuuvu;

that is xuFS is the least upper bound of hxu iS. �

Theorem 2.71. (in�nite De Morgan's laws) For every subset S of a complete boolean lattice

1.FS=

dx2S x�;

2.dS=

Fx2S x�.

Proof. It's enough to prove thatFS is a complement of

dx2S x� (the second follows from duality).

Really, using the previous theorem:GS t

l

x2Sx�=

l

x2S

GS t

�x�=

l

x2S

�GS t x� j x2S

wl

x2Sfxt x� j x2Sg=1;

GS u

l

x2Sx�=

Gy2S

*lx2S

x�u+y=

Gy2S

(lx2S

x�u y j y 2S)vGy2S

fy�u y j y 2Sg=0:

SoFS t

dx2S x�=1 and

FS u

dx2S x�=0. �

2.1.9 Center of a lattice

De�nition 2.72. The center Z(A) of a bounded distributive lattice A is the set of its comple-mented elements.

Remark 2.73. For a de�nition of center of non-distributive lattices see [5].

Remark 2.74. In [23] the word center and the notation Z(A) are used in a di�erent sense.

De�nition 2.75. A sublattice K of a complete lattice L is a closed sublattice of L if K containsthe meet and the join of any its nonempty subset.

Theorem 2.76. Center of an in�nitely distributive lattice is its closed sublattice.

Proof. See [16]. �

Remark 2.77. See [17] for a more strong result.

Theorem 2.78. The center of a bounded distributive lattice constitutes its sublattice.

Proof. Let A be a bounded distributive lattice and Z(A) be its center. Let a; b 2 Z(A). Conse-quently a�; b�2Z(A). Then a�t b� is the complement of au b because

(au b)u (a�t b�)= (au bu a�)t (au bu b�)=0t 0=0 and(au b)t (a�t b�)= (at a�t b�)u (bt a�t b�)=1u 1=1:

So au b is complemented. Similarly at b is complemented. �

Theorem 2.79. The center of a bounded distributive lattice constitutes a boolean lattice.

Proof. Because it is a distributive complemented lattice. �

2.1.10 Atoms of posets

De�nition 2.80. An atom of a poset is an element which has no non-least subelements.

2.1 Order theory 23

Remark 2.81. This de�nition is valid even for posets without least element.

I will denote (atomsAa) or just (atomsa) the set of atoms contained in an element a of a posetA. I will denote atomsA the set of all atoms of a poset A.

De�nition 2.82. A poset A is called atomic i� atoms a=/ ; for every non-least element a of theposet A.

De�nition 2.83. Atomistic poset is such a poset that a=F

atoms a for every non-least elementa of this poset.

Obvious 2.84. Every atomistic poset is atomic.

Proposition 2.85. Let A be a poset. If a is an atom of A and B 2A then avB, a�/ B.

Proof.

). avB) av a^ avB, thus a�/ B because a is not least.

(. a�/ B implies existence of non-least element x such that xvB and xv a. Because a is anatom, we have x= a. So avB. �

Theorem 2.86. atomsdS=

ThatomsiS whenever

dS is de�ned for every S 2PA where A is

a poset.

Proof. For any atom c

c2 atomsl

S ,cv

lS ,

8a2S: cv a ,8a2S: c2 atoms a ,c2\hatomsiS:

�

Corollary 2.87. atoms(au b) = atoms a\ atoms b for an arbitrary meet-semilattice.

Theorem 2.88. A complete boolean lattice is atomic i� it is atomistic.

Proof.

(. Obvious.

). Let A be an atomic boolean lattice. Let a2A. Suppose b=F

atomsa@a. If x2atoms(a nb)then x v a n b and so x v a and hence x v b. But we have x= x u b v (a n b) u b= 0 whatcontradicts to our supposition. �

2.1.11 Kuratowski's lemma

Theorem 2.89. (Kuratowski lemma) Any chain in a poset is contained in a maximal chain (if weorder chains by inclusion).

I will skip the proof of Kuratowski lemma as this proof can be found in any set theory or ordertheory reference.

2.1.12 Homomorphisms of posets and lattices

De�nition 2.90. A monotone function (also called order homomorphism) from a poset A to aposet B is such a function f that xv y) fxv fy.


De�nition 2.91. Order embedding is a monotone injective function whose inverse is alsomonotone.

De�nition 2.92. Order isomorphism is a surjective order embedding.

Order isomorphism preserves properties of posets, such as order, joins and meets, etc.

De�nition 2.93.

1. Join semilattice homomorphism is a function f from a join semilattice A to a join semilatticeB, such that f(xt y)= fxt fy for every x; y 2A.

2. Meet semilattice homomorphism is a function f from a meet semilattice A to a meet semi-lattice B, such that f(xu y)= fxu fy for every x; y 2A.

Obvious 2.94.

1. Join semilattice homomorphisms are monotone.

2. Meet semilattice homomorphisms are monotone.

De�nition 2.95. A lattice homomorphism is a function from a lattice to a lattice, which is bothjoin semilattice homomorphism and meet semilattice homomorphism.

De�nition 2.96. Complete lattice homomorphism from a complete lattice A to a complete latticeB is a function f from A to B which preserves all meets and joins, that is f

FS =

Fhf iS and

fdS=

dhf iS for every S 2PA.

2.1.13 Galois connectionsSee [3] and [12] for more detailed treatment of Galois connections.

De�nition 2.97. Let A and B be two posets. A Galois connection between A and B is a pair offunctions f =(f�; f�) with f�:A!B and f�:B!A such that:

8x2A; y 2B: (f�xv y, xv f� y):

f� is called the upper adjoint of f� and f� is called the lower adjoint of f�.

Theorem 2.98. A pair (f�; f�) of functions f�:A!B and f�:B!A is a Galois connection i�both of the following:

1. f� and f� are monotone.

2. xv f� f�x and f� f� y v y for every x2A and y 2B.

Proof.

).2. xv f� f�x since f�xv f�x; f� f� y v y since f� yv f� y.1. Let a; b2A and av b. Then av bv f� f� b. So by de�nition f� av f� b that is f� is

monotone. Analogously f� in monotone.

(. f� xv y) f� f�xv f� y)xv f� y. The other direction is analogous. �

Theorem 2.99.

1. f� � f� � f�= f�.2. f� � f� � f�= f�.

Proof.

1. Let x 2 A. We have x v f� f� x; consequently f� x v f� f� f� x. On the other hand,f� f� f

�xv f� x. So f� f� f�x= f�x.

2.1 Order theory 25

2. Similar. �

De�nition 2.100. A function f is called idempotent i� f(f(X))= f(X) for every argument X .

Proposition 2.101. f� � f� and f� � f� are idempotent.

Proof. f� � f� is idempotent because f� f� f� f� y= f� f� y. f� � f� is similar. �

Theorem 2.102. Each of two adjoints is uniquely determined by the other.

Proof. Let p and q be both upper adjoints of f . We have for all x2A and y 2B:

xv p(y), f(x)v y,xv q(y):

For x= p(y) we obtain p(y)v q(y) and for x= q(y) we obtain q(y)v p(y). So q(y)= p(y). �

Theorem 2.103. Let f be a function from a poset A to a poset B.

1. Both:

1. If f is monotone and g(b) =max fx 2A j fx v bg is de�ned for every b 2B then gis the upper adjoint of f .

2. If g:B!A is the upper adjoint of f then g(b)=maxfx2A j fxv bg for every b2B.2. Both:

1. If f is monotone and g(b)=min fx2A j fxw bg is de�ned for every b2B then g isthe lower adjoint of f .

2. If g:B!A is the lower adjoint of f then g(b)=minfx2A j fxw bg for every b2B.

Proof. We will prove only the �rst as the second is its dual.

1. Let g(b) =max fx2A j fxv bg for every b2B. Then

xv gy, xvmax fx2A j fxv yg) fxv y

(because f is monotone) and

xv gy, xvmax fx2A j fxv yg( fxv y:

So fxv y,xv gy that is f is the lower adjoint of g.2. We have

g(b)=max fx2A j fxv bg ,fgbv b^8x2A: (fxv b)xv gb)

what is true by properties of adjoints. �

Theorem 2.104. Let f be a function from a poset A to a poset B.

1. If f is an upper adjoint, f preserves all existing in�ma in A.

2. If A is a complete lattice and f preserves all in�ma, then f is an upper adjoint of a functionB!A.

3. If f is a lower adjoint, f preserves all existing suprema in A.

4. If A is a complete lattice and f preserves all suprema, then f is a lower adjoint of a functionB!A.

Proof. We will prove only �rst two items because the rest items are similar.

1. Let S 2PA anddS exists. f

dS is a lower bound for hf iS because f is order-preserving.

If a is a lower bound for hf iS then 8x2S:av fx that is 8x2S: gavx where g is the loweradjoint of f . Thus ga v

dS and hence f

dS w a. So f

dS is the greatest lower bound

for hf iS.


2. Let A be a complete lattice and f preserves all in�ma. Let

g(a) =lfx2A j fxw ag:

Since f preserves in�ma, we have

f(g(a))=lff(x) j x2A; fxw agw a:

g(f(b))=dfx2A j fxw fbgv b.

Obviously f is monotone and thus g is also monotone.So f is the upper adjoint of g. �

Corollary 2.105. Let f be a function from a complete lattice A to a poset B. Then:

1. f is an upper adjoint of a function B!A i� f preserves all in�ma in A.

2. f is an lower adjoint of a function B!A i� f preserves all suprema in A.

2.1.14 Co-Brouwerian lattices

De�nition 2.106. Let A be a poset. Pseudocomplement of a2A is

max fc2A j c� ag:

If z is the pseudocomplement of a we will denote z= a�.

De�nition 2.107. Let A be a poset. Dual pseudocomplement of a2A is

min fc2A j c� ag:

If z is the dual pseudocomplement of a we will denote z= a+.

Proposition 2.108. If a is a complemented element of a bounded distributive lattice, then a� isboth pseudocomplement and dual pseudocomplement of a.

Proof. Because of duality it is enough to prove that a� is pseudocomplement of a.We need to prove c�a)cva� for every element c of our poset, and a��a. The second is obvious.

Let's prove c� a) cv a�.Really, let c� a. Then cu a=0; a�t (cu a)= a�; (a�t c)u (a�t a) = a�; a�t c= a�; cv a�. �

De�nition 2.109. Let A be a join-semilattice. Let a; b2A. Pseudodi�erence of a and b is

min fz 2A j av bt zg:

If z is a pseudodi�erence of a and b we will denote z= a n� b.

Remark 2.110. I do not require that a� is unde�ned if there are no pseudocomplement of a andlikewise for dual pseudocomplement and pseudodi�erence. In fact below I will de�ne quasicomple-ment, dual quasicomplement, and quasidi�erence which generalize pseudo-* counterparts. I willdenote a� the more general case of quasicomplement than of pseudocomplement, and likewise forother notation.

Obvious 2.111. Dual pseudocomplement is the dual of pseudocomplement.

De�nition 2.112. Co-brouwerian lattice is a lattice for which pseudodi�erence of any two itselements is de�ned.

Proposition 2.113. Every non-empty co-brouwerian lattice A has least element.

Proof. Let a be an arbitrary lattice element. Then

a n� a=min fz 2A j av at zg=minA:

2.1 Order theory 27

So minA exists. �

De�nition 2.114. Co-Heyting lattice is co-brouwerian lattice with greatest element.

Theorem 2.115. For a co-brouwerian lattice at¡ is an upper adjoint of ¡n�a for every a2A.

Proof. g(b)=minfx2A j atxw bg= b n�a exists for every b2A and thus is the lower adjoint ofat¡. �

Corollary 2.116. 8a; x; y 2A: (x n� av y,xv at y) for a co-brouwerian lattice.

De�nition 2.117. Let a; b2A where A is a complete lattice. Quasidi�erence a n� b is de�ned bythe formula:

a n� b=lfz 2A j av bt zg:

Remark 2.118. A more detailed theory of quasidi�erence (as well as quasicomplement and dualquasicomplement) will be considered below.

Lemma 2.119. (a n� b)t b=at b for elements a, b of a meet in�nite distributive complete lattice.

Proof.

(a n� b)t b =lfz 2A j av bt zgt b =l

fz t b j z 2A; av bt zg =lft2A j tw b; av tg =

at b:�

Theorem 2.120. The following are equivalent for a complete lattice A:

1. A is meet in�nite distributive.

2. A is a co-brouwerian lattice.

3. A is a co-Heyting lattice.

4. at¡ has lower adjoint for every a2A.

Proof.

(2),(3). Obvious (taking into account completeness of A).

(4))(1). Let ¡n�a be the lower adjoint of a t ¡. Let S 2 PA. For every y 2 S we havey w (a t y) n� a by properties of Galois connections; consequently y w (

dha t iS) n� a;d

S w (dhat iS) n� a. So

atl

S w¡¡l

hat iS�n� a

�t aw

lhat iS:

But atdS v

dhat iS is obvious.

(1))(2). Let a n� b=dfz 2 A j a v b t zg. To prove that A is a co-brouwerian lattice it is

enough to prove av bt (a n� b). But it follows from the lemma.

(2))(4). a n� b=min fz 2A j av bt zg. So at¡ is the upper adjoint of ¡n�a.

(1))(4). Because at¡ preserves all meets. �

Corollary 2.121. Co-brouwerian lattices are distributive.

The following theorem is essentially borrowed from [18]:


Theorem 2.122. A lattice A with least element 0 is co-brouwerian with pseudodi�erence n� i�n� is a binary operation on A satisfying the following identities:

1. a n� a=0;

2. at (b n� a)= at b;

3. bt (b n� a)= b;

4. (bt c) n� a=(b n� a)t (c n� a).

Proof.

(. We havecw b n� a) ct aw at (b n� a)= at bw b;

ct aw b) c= ct (c n� a)w (a n� a)t (c n� a)= (at c) n� aw b n� a.So cw b n� a, ct aw b that is at¡ is an upper adjoint of ¡n�a. By a theorem above

our lattice is co-brouwerian. By another theorem above n� is a pseudodi�erence.

).

1. Obvious.

2.

at (b n� a) =atlfz 2A j bv at zg =l

fat z j z 2A; bv at zg =at b:

3. bt (b n� a)= btdfz 2A j bv at zg=

dfbt z j z 2A; bv at zg= b.

4. Obviously (b t c) n� a w b n� a and (b t c) n� a w c n� a. Thus (b t c) n� a w(b n� a)t (c n� a). We have

(b n� a)t (c n� a)t a =((b n� a)t a)t ((c n� a)t a) =

(bt a)t (ct a) =at bt c w

bt c:

From this by de�nition of adjoints: (b n� a)t (c n� a)w (bt c) n� a. �

Theorem 2.123. (FS) n� a =

Ffx n� a j x 2 Sg for all a 2 A and S 2 PA where A is a co-

brouwerian lattice andFS is de�ned.

Proof. Because lower adjoint preserves all suprema. �

Theorem 2.124. (a n� b)n� c=a n� (bt c) for elements a, b, c of a complete co-brouwerian lattice.

Proof. a n� b=dfz 2A j av bt zg.

(a n� b) n� c=dfz 2A j a n� bv ct zg.

a n� (bt c)=dfz 2A j av bt ct zg.

It is left to prove a n� bv ct z, av bt ct z.Let a n� bv ct z. Then at bv bt ct z by the lemma and consequently av bt ct z.Let av bt ct z. Then a n� bv (bt ct z) n� bv ct z by a theorem above. �

2.1.15 Dual pseudocomplement on co-Heyting lattices

Proposition 2.125. For co-Heyting algebras 1 n� b= b+.

2.1 Order theory 29

Proof. 1 n� b=min fz 2A j 1v bt zg=min fz 2A j 1= bt zg=min fz 2A j b� zg= b+. �

Theorem 2.126. (au b)+= a+t b+ for every elements a, b of a co-Heyting algebra.

Proof. at (au b)+w (au b)t (au b)+w 1. So at (au b)+w 1; (au b)+w 1 n� a= a+.We have (au b)+w a+. Similarly (au b)+w b+. Thus (au b)+w a+t b+.On the other hand, a+ t b+ t (a u b) = (a+ t b+ t a) u (a+ t b+ t b). Obviously a+ t b+ t a=

a+t b+t b=1. So a+t b+t (au b)w 1 and thus a+t b+w 1 n� (au b)= (au b)+.So (au b)+= a+t b+. �

2.2 Intro to category theory

I recall that this is a very basic introduction to category theory, I even do not de�ne functors asthey have no use in my theory.

De�nition 2.127. A directed multigraph is:

1. a set O (vertices);2. a set M (edges);3. functions Src and Dst (source and destination) from M to O.

Note that in category theory vertices are called objects and edges are called morphisms .

De�nition 2.128. A precategory is a directed multigraph together with a partial binary operation� on the setM such that g � f is de�ned i� Dst f =Src g (for every morphisms f and g) such that

1. Src(g � f) = Src f and Dst(g � f) =Dst g whenever the composition g � f of morphisms fand g is de�ned.

2. (h� g) � f =h � (g � f) whenever compositions in this equation are de�ned.

De�nition 2.129. The set Mor(A; B) (morphisms from an object A to an object B) is exactlymorphisms which have A as the source and B as the destination.

De�nition 2.130. Identity morphism is such a morphism e that e� f = f and g � e= g whenevercompositions in these formulas are de�ned.

De�nition 2.131. A category is a precategory with additional requirement that for every objectX there exists identity morphism 1X.

Proposition 2.132. For every object X there exist no more than one identity morphism.

Proof. Let p and q be both identity morphisms for a object X . Then p= p � q= q. �

De�nition 2.133. An isomorphism is such a morphism f of a category that there exists amorphism f¡1 (inverse of f) such that f � f¡1=1Dst f and f¡1 � f =1Src f.

Proposition 2.134. An isomorphism has exactly one inverse.

Proof. Let g and h be both inverses of f . Then h=h � 1Dst f =h � f � g=1Src f � g= g. �

De�nition 2.135. A groupoid is a category all of whose morphisms are isomorphisms.

Some important examples of categories:

Exercise 2.3. Prove that the below examples of categories are really categories.

De�nition 2.136. The category Set is:

� Objects are small sets.


� Morphisms from an object A to an object B are triples (A;B; f) where f is a function fromA to B.

� Composition of morphisms is de�ned by the formula: (B; C; g) � (A;B; f) = (A; C; g � f)where g � f is function composition.

De�nition 2.137. The category Rel is:

� Objects are small sets.� Morphisms from an object A to an object B are triples (A;B; f) where f is a binary relation

between A and B.

� Composition of morphisms is de�ned by the formula: (B; C; g) � (A;B; f) = (A; C; g � f)where g � f is relation composition.

I will denote GR(A;B; f)= f for any morphism (A;B; f) of either Set or Rel.I will denote hf i= hGR f i and [f ]=[GR f ] for any morphism f of either Set or Rel.

De�nition 2.138. A morphism whose source is the same as destination is called endomorphism.

De�nition 2.139. Wide subcategory of a category (O;M) is a category (O;M0) where M0�Mand the composition on (O;M0) is a restriction of composition of (O;M). (Similarly wide sub-precategory can be de�ned.)

2.3 Intro to group theoryDe�nition 2.140. A semigroup is a pair of a set G and an associative binary operation on G.

De�nition 2.141. A group is a pair of a set G and a binary operation � on G such that:1. (h � g) � f =h � (g � f) for every f ; g; h2G.2. There exists an element e (identity) of G such that f � e= e � f = f for every f 2G.3. For every element f there exists an element f¡1 such that f � f¡1= f¡1 � f = e.

Obvious 2.142. Every group is a semigroup.

Proposition 2.143. In every group there exist exactly one identity element.

Proof. If p and q are both identities, then p= p � q= q. �

Proposition 2.144. Every group element has exactly one inverse.

Proof. Let p and q be both inverses of f 2G. Then f � p= p � f = e and f � q = q � f = e. Thenp= p � e= p � f � q= e � q= q. �

Proposition 2.145. (g � f)¡1= f¡1 � g¡1 for every group elements f and g.

Proof. (f¡1 � g¡1) � (g � f)= f¡1 � g¡1 � g � f= f¡1 �e � f= f¡1 � f=e. Similarly (g � f) � (f¡1 � g¡1)=e.So f¡1 � g¡1 is the inverse of g � f . �

De�nition 2.146. A permutation group on a set D is a group whose elements are functions onD and whose composition is function composition.

Obvious 2.147. Elements of a permutation group are bijections.

De�nition 2.148. A transitive permutation group on a set D is such a permutation group G onD that for every x; y 2D there exists r 2G such that y= r(x).

A groupoid with single (arbitrarily chosen) object corresponds to every group. The morphismsof this category are elements of the group and the composition of morphisms is the group operation.

2.3 Intro to group theory 31

Chapter 3

More on order theory

3.1 Straight maps and separation subsets

3.1.1 Straight maps

De�nition 3.1. Let f be a monotone map from a meet-semilattice A to some poset B. I call fa straight map when

8a; b2A: (fav fb) fa= f(au b)):

Proposition 3.2. The following statements are equivalent for a monotone map f :

1. f is a straight map.

2. 8a; b2A: (fav fb) fav f(au b)).

3. 8a; b2A: (fav fb) faAf(au b)).4. 8a; b2A: (faA f(au b)) favfb).

Proof.

(1),(2),(3). Due faw f(au b).

(3),(4). Obvious. �

Remark 3.3. The de�nition of straight map can be generalized for any poset A by the formula

8a; b2A: (fav fb)9c2A: (cv a^ cv b^ fa= fc)):

This generalization is not yet researched however.

Proposition 3.4. Let f be a monotone map from a meet-semilattice A to a meet-semilattice B. If

8a; b2A: f(au b)= fau fbthen f is a straight map.

Proof. Let fav fb. Then f(au b)= fau fb= fa. �

Proposition 3.5. Let f be a monotone map from a meet-semilattice A to some poset B. If

8a; b2A: (fav fb) av b)then f is a straight map.

Proof. fav fb) av b) a= au b) fa= f(au b). �

Theorem 3.6. If f is a straight monotone map from a meet-semilattice A then the followingstatements are equivalent:

1. f is an injection.

33

2. 8a; b2A: (fav fb) av b).

3. 8a; b2A: (a@ b) fa@ fb).4. 8a; b2A: (a@ b) fa=/ fb).5. 8a; b2A: (a@ b) fawfb).6. 8a; b2A: (fav fb) aAb).

Proof.

(1))(3). Let a; b2A. Let fa= fb)a= b. Let a@ b. fa=/ fb because a=/ b. fav fb becauseav b. So fa@ fb.

(2))(1). Let a; b2A. Let fav fb)avb. Let fa= fb. Then av b and bva and consequentlya= b.

(3))(2). Let 8a; b2A: (a@ b) fa@ fb). Let avb. Then aAau b. So faA f(au b). If fav fbthen fav f (au b) what is a contradiction.

(3))(5))(4). Obvious.

(4))(3). Because a@ b) av b) fav fb.(5),(6). Obvious. �

3.1.2 Separation subsets and full stars

De�nition 3.7. @Y a= fx2Y j x�/ ag for an element a of a poset A and Y 2PA.

De�nition 3.8. Full star of a2A is ?a= @A a.

Proposition 3.9. If A is a meet-semilattice, then ? is a straight monotone map.

Proof. Monotonicity is obvious. Let ?av ? (au b). Then it exists x2?a such that x2/ ?(au b). Soxu a2/ ?b but xu a2 ?a and consequently ?av ? b. �

De�nition 3.10. A separation subset of a poset A is such its subset Y that

8a; b2A: (@Y a=@Y b) a= b):

De�nition 3.11. I call separable such poset that ? is an injection.

Obvious 3.12. A poset is separable i� it has a separation subset.

De�nition 3.13. A poset A has disjunction property of Wallman i� for any a; b2A either bv aor there exists a non-least element cv b such that a� c.

Theorem 3.14. For a meet-semilattice with least element the following statements are equivalent:

1. A is separable.

2. 8a; b2A: (?av ?b) av b).

3. 8a; b2A: (a@ b) ?a@ ?b).4. 8a; b2A: (a@ b) ?a=/ ?b).5. 8a; b2A: (a@ b) ?aw ? b).6. 8a; b2A: (?av ?b) aAb).7. A conforms to Wallman's disjunction property.

8. 8a; b2A: (a@ b)9c2A n f0g: (c� a^ cv b)).

34 More on order theory

Proof.

(1),(2),(3),(4),(5),(6). By the above theorem.

(8))(4). Let property (8) hold. Let a@ b. Then it exists element c v b such that c=/ 0 andcu a=0. But cu b=/ 0. So ?a=/ ?b.

(2))(7). Let property (2) hold. Let avb. Then ?av ? b that is it there exists c 2 ?a suchthat c2/ ?b, in other words cu a=/ 0 and cu b=0. Let d= cu a. Then dv a and d=/ 0 anddu b=0. So disjunction property of Wallman holds.

(7))(8). Obvious.

(8))(7). Let bva. Then au b@ b that is a0@ b where a0= au b. Consequently 9c 2A n f0g:(c� a0 ^ c v b). We have c u a= c u b u a= c u a0. So c v b and c u a= 0. Thus Wallman'sdisjunction property holds. �

Proposition 3.15. Every boolean lattice is separable.

Proof. Let a; b 2 A where A is a boolean lattice an a =/ b. Then a u b� =/ 0 or a� u b =/ 0 becauseotherwise au b�=0 and at b�=1 and thus a= b. Without loss of generality assume au b�=/ 0. Thenau c=/ 0 and bu c=0 for c= au b�=/ 0. �

3.1.3 Atomically Separable Lattices

Proposition 3.16. �atoms� is a straight monotone map (for any meet-semilattice).

Proof. Monotonicity is obvious. The rest follows from the formula

atoms(au b)= atoms a\ atoms b

(the corollary 2.87). �

De�nition 3.17. I will call atomically separable such a poset that �atoms� is an injection.

Proposition 3.18. 8a; b2A: (a@b)atomsa�atoms b) i� A is atomically separable for a poset A.

Proof.

(. Obvious.

). Let a=/ b for example avb. Then au b@ a; atoms a� atoms(au b)= atomsa\ atoms b andthus atoms a=/ atoms b. �

Proposition 3.19. Any atomistic poset is atomically separable.

Proof. We need to prove that atoms a= atoms b) a= b. But it is obvious because

a=G

atoms a and b=G

atoms b: �

Theorem 3.20. If a lattice with least element is atomic and separable then it is atomistic.

Proof. Suppose the contrary that is aAF atoms a. Then, because our lattice is separable, thereexists c2A such that cu a=/ 0 and cu

Fatoms a=0. There exists atom dv c such that dv cu a.

duF

atoms av cuF

atoms a=0. But d2 atoms a. Contradiction. �

Theorem 3.21. Let A be an atomic meet-semilattice with least element. Then the followingstatements are equivalent:

1. A is separable.

2. A is atomically separable.

3. A conforms to Wallman's disjunction property.

3.1 Straight maps and separation subsets 35

4. 8a; b2A: (a@ b)9c2A n f0g: (c� a^ cv b)).

Proof.

(1),(3),(4). Proved above.(2))(4). Let our semilattice be atomically separable. Let a@ b. Then atomsa� atoms b and

so there exists c 2 atoms b such that c 2/ atoms a. c=/ 0 and c v b, from which (taking intoaccount that c is an atom) cvb and cua=0. So our semilattice conforms to the formula (4).

(4))(2). Let formula (4) hold. Then for any elements a@ b there exists c=/ 0 such that cv band cu a=0. Because A is atomic there exists atom dv c. d2 atoms b and d2/ atoms a. Soatoms a=/ atoms b and atoms a� atoms b. Consequently atoms a� atoms b. �

Theorem 3.22. Any atomistic meet-semilattice with least element is separable.

Proof. From the above. �

3.2 Free Stars

De�nition 3.23. An upper set is such a set F 2PZ that

8X 2F ; Y 2Z: (Y wX)Y 2F ):

De�nition 3.24. Let A be a poset. Free stars on A are such S 2PA that the least element (ifit exists) is not in S and for every X;Y 2A

8Z 2A: (Z wX ^Z wY )Z 2S),X 2S _Y 2S:

Proposition 3.25. S 2PA where A is a poset is a free star i� all of the following:1. The least element (if it exists) is not in S.

2. 8Z 2A: (Z wX ^Z wY )Z 2S))X 2S _Y 2S for every X;Y 2A.3. S is an upper set.

Proof.

). (1) and (2) are obvious. Let prove that S is an upper set. Let X 2S and X vY 2A. ThenX 2S _X 2S and thus 8Z 2A: (Z wX ^Z wX)Z 2S) that is 8Z 2A: (Z wX)Z 2S),and so Y 2S.

(. We need to prove that

8Z 2A: (Z wX ^Z wY )Z 2S)(X 2S _Y 2S:

LetX 2S_Y 2S. Then ZwX ^ZwY )Z2S for every Z 2A because S is an upper set. �

Proposition 3.26. Let A be a join-semilattice. S 2PA is a free star i� all of the following:1. The least element (if it exists) is not in S.

2. X tY 2S)X 2S _Y 2S for every X;Y 2A.3. S is an upper set.

Proof.

). We need to prove only X tY 2S)X 2S _Y 2S. Let X tY 2S. Because S is an upperset, we have 8Z 2A: (Z wX tY )Z 2S) and thus 8Z 2A: (Z wX ^Z wY )Z 2S) fromwhich we conclude X 2S _Y 2S.

(. We need to prove 8Z 2A: (Z wX ^Z wY )Z 2S)(X 2S _Y 2S.But it trivially follows from that S is an upper set. �

36 More on order theory

Proposition 3.27. Let A be a join-semilattice. S 2PA is a free star i� the least element (if itexists) is not in S and for every X;Y 2A

X tY 2S,X 2S _Y 2S:

Proof.

). We need to prove only X tY 2S(X 2S_Y 2S what follows from that S is an upper set.(. We need to prove only that S is an upper set. Let X 2 S and X v Y 2 A. Then

X 2S)X 2S _Y 2S,X tY 2S)Y 2S. So S is an upper set. �

3.2.1 Starrish posets

De�nition 3.28. I will call a poset starrish when the full star ?a is a free star for every elementa of this poset.

Proposition 3.29. Every distributive lattice is starrish.

Proof. Let A be a distributive lattice, a 2 A. Obviously 0 2/ ?a (if 0 exists); obviously ?a is anupper set. If xt y 2 ?a, then (x t y) u a is non-least that is (xu a) t (y u a) is non-least what isequivalent to xu a or yu a being non-least that is x2 ?a_ y 2 ?a. �

Theorem 3.30. If A is a starrish join-semilattice lattice then

atoms(at b)= atoms a[ atoms bfor every a; b2A.

Proof. For every atom c we have: c2atoms(at b), c�/ at b,at b2?c,a2?c_ b2?c, c�/ a_c�/ b, c2 atoms a_ c2 atoms b. �

3.3 Quasidi�erence and Quasicomplement

I've got quasidi�erence and quasicomplement (and dual quasicomplemen

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