# Algebraic General Topology

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Издательство:
НИЦ ИНФРА-М

Автор:
Портон Виктор Львович

Год издания: 2019

Кол-во страниц: 395

Дополнительно

Вид издания:
Монография

Уровень образования:
Дополнительное профессиональное образование

ISBN-онлайн: 978-5-16-108176-1

Артикул: 720653.01.99

ABSTRACT. In this work I introduce and study in details the concepts of funcoids which generalize proximity spaces and reloids which generalize uniform spaces, and generalizations thereof. The concept of funcoid is generalized concept of proximity, the concept of reloid is cleared from superfluous details (generalized) concept of uniformity.
Also funcoids and reloids are generalizations of binary relations whose domains and ranges are filters (instead of sets). Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this provides us with a common generalization of calculus and discrete mathematics.
It is defined a generalization of limit for arbitrary (including discontinuous and multivalued) functions, what allows to define for example derivative of an arbitrary real function.
The concept of continuity is defined by an algebraic formula (instead of old messy epsilon-delta notation) 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 define connectedness for funcoids and reloids.
Then I consider generalizations of funcoids: pointfree funcoids and generalization of pointfree funcoids: staroids and multifuncoids. Also I define several kinds of products of funcoids and other morphisms.
Before going to topology, this book studies properties of co-brouwerian lattices and filters.

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В.Л. ПОРТОН ALGEBRAIC GENERAL TOPOLOGY Монография Москва ИНФРА-М 2019

УДК 512(075.4) ББК 22.152 П60 Портон В.Л. П60 Algebraic General Topology : монография / В.Л. Портон. — Москва : ИНФРА-М, 2019. — 395 с. ISBN 978-5-16-108176-1 (online) ABSTRACT. In this work I introduce and study in details the concepts of funcoids which generalize proximity spaces and reloids which generalize uniform spaces, and generalizations thereof. The concept of funcoid is generalized concept of proximity, the concept of reloid is cleared from superfluous details (generalized) concept of uniformity. Also funcoids and reloids are generalizations of binary relations whose domains and ranges are filters (instead of sets). Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this provides us with a common generalization of calculus and discrete mathematics. It is defined a generalization of limit for arbitrary (including discontinuous and multivalued) functions, what allows to define for example derivative of an arbitrary real function. The concept of continuity is defined by an algebraic formula (instead of old messy epsilon-delta notation) 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 define connectedness for funcoids and reloids. Then I consider generalizations of funcoids: pointfree funcoids and generalization of pointfree funcoids: staroids and multifuncoids. Also I define several kinds of products of funcoids and other morphisms. Before going to topology, this book studies properties of co brouwerian lattices and filters. УДК 512(075.4) ББК 22.152 ISBN 978-5-16-108176-1 (online) © Портон В.Л., 2019 ФЗ № 436-ФЗ Издание не подлежит маркировке в соответствии с п. 1 ч. 2 ст. 1

Contents Part 1. Introductory chapters 8 Chapter 1. Introduction 9 1.1. License and editing 9 1.2. Intended audience 9 1.3. Reading Order 9 1.4. Our topic and rationale 10 1.5. Earlier works 10 1.6. Kinds of continuity 11 1.7. Responses to some accusations against style of my exposition 11 1.8. Structure of this book 12 1.9. Basic notation 12 1.10. Implicit arguments 13 1.11. Unusual notation 13 Chapter 2. Common knowledge, part 1 15 2.1. Order theory 15 2.2. Intro to category theory 31 2.3. Intro to group theory 35 Chapter 3. More on order theory 36 3.1. Straight maps and separation subsets 36 3.2. Quasidifference and Quasicomplement 39 3.3. Several equal ways to express pseudodifference 42 3.4. Partially ordered categories 43 3.5. Partitioning 46 3.6. A proposition about binary relations 47 3.7. Infinite associativity and ordinated product 47 3.8. Galois surjections 55 3.9. Some properties of frames 55 Chapter 4. Typed sets and category Rel 61 4.1. Relational structures 61 4.2. Typed elements and typed sets 61 4.3. Category Rel 62 4.4. Product of typed sets 66 Chapter 5. Filters and filtrators 68 5.1. Implication tuples 68 5.2. Introduction to filters and filtrators 68 5.3. Filters on a poset 69 5.4. Filters on a Set 71 5.5. Filtrators 72 5.6. Alternative primary filtrators 74 5.7. Basic properties of filters 80 3

CONTENTS 4 5.8. More advanced properties of filters 81 5.9. Misc filtrator properties 86 5.10. Characterization of Binarily Meet-Closed Filtrators 86 5.11. Core Part 87 5.12. Intersection and Joining with an Element of the Core 88 5.13. Stars of Elements of Filtrators 89 5.14. Atomic Elements of a Filtrator 90 5.15. Prime Filtrator Elements 92 5.16. Stars for filters 93 5.17. Generalized Filter Base 94 5.18. Separability of filters 95 5.19. Some Criteria 96 5.20. Co-Separability of Core 98 5.21. Complements and Core Parts 99 5.22. Core Part and Atomic Elements 101 5.23. Distributivity of Core Part over Lattice Operations 102 5.24. Separability criteria 103 5.25. Filtrators over Boolean Lattices 104 5.26. Distributivity for an Element of Boolean Core 105 5.27. More about the Lattice of Filters 106 5.28. More Criteria 106 5.29. Filters and a Special Sublattice 107 5.30. Distributivity of quasicomplements 108 5.31. Complementive Filters and Factoring by a Filter 110 5.32. Pseudodifference of filters 112 5.33. Function spaces of posets 113 5.34. Filters on a Set 118 5.35. Bases on filtrators 121 5.36. Some Counter-Examples 122 5.37. Open problems about filters 126 5.38. Further notation 126 5.39. Equivalent filters and rebase of filters 126 Chapter 6. Common knowledge, part 2 (topology) 135 6.1. Metric spaces 135 6.2. Pretopological spaces 136 6.3. Topological spaces 138 6.4. Proximity spaces 141 6.5. Definition of uniform spaces 142 Part 2. Funcoids and reloids 143 Chapter 7. Funcoids 144 7.1. Informal introduction into funcoids 144 7.2. Basic definitions 145 7.3. Funcoid as continuation 147 7.4. Another way to represent funcoids as binary relations 152 7.5. Lattices of funcoids 153 7.6. More on composition of funcoids 155 7.7. Domain and range of a funcoid 157 7.8. Categories of funcoids 159 7.9. Specifying funcoids by functions or relations on atomic filters 160 7.10. Funcoidal product of filters 164

CONTENTS 5 7.11. Atomic funcoids 167 7.12. Complete funcoids 169 7.13. Funcoids corresponding to pretopologies 174 7.14. Completion of funcoids 174 7.15. Monovalued and injective funcoids 177 7.16. Open maps 179 7.17. T0-, T1-, T2-, T3-, and T4-separable funcoids 180 7.18. Filters closed regarding a funcoid 181 7.19. Proximity spaces 182 Chapter 8. Reloids 183 8.1. Basic definitions 183 8.2. Composition of reloids 184 8.3. Reloidal product of filters 186 8.4. Restricting reloid to a filter. Domain and image 188 8.5. Categories of reloids 190 8.6. Monovalued and injective reloids 191 8.7. Complete reloids and completion of reloids 192 8.8. What uniform spaces are 196 Chapter 9. Relationships between funcoids and reloids 197 9.1. Funcoid induced by a reloid 197 9.2. Reloids induced by a funcoid 201 9.3. Galois connections between funcoids and reloids 204 9.4. Funcoidal reloids 207 9.5. Complete funcoids and reloids 210 9.6. Properties preserved by relationships 212 9.7. Some sub-posets of funcoids and reloids 213 9.8. Double filtrators 214 Chapter 10. On distributivity of composition with a principal reloid 215 10.1. Decomposition of composition of binary relations 215 10.2. Decomposition of composition of reloids 215 10.3. Lemmas for the main result 216 10.4. Proof of the main result 217 10.5. Embedding reloids into funcoids 217 Chapter 11. Continuous morphisms 220 11.1. Traditional definitions of continuity 220 11.2. Our three definitions of continuity 221 11.3. Continuity for topological spaces 222 11.4. C(µ ◦ µ−1, ν ◦ ν−1) 223 11.5. Continuity of a restricted morphism 223 Chapter 12. Connectedness regarding funcoids and reloids 225 12.1. Some lemmas 225 12.2. Endomorphism series 225 12.3. Connectedness regarding binary relations 226 12.4. Connectedness regarding funcoids and reloids 227 12.5. Algebraic properties of S and S∗ 229 12.6. Irreflexive reloids 230 12.7. Micronization 231 Chapter 13. Total boundness of reloids 232

CONTENTS 6 13.1. Thick binary relations 232 13.2. Totally bounded endoreloids 233 13.3. Special case of uniform spaces 233 13.4. Relationships with other properties 234 13.5. Additional predicates 235 Chapter 14. Orderings of filters in terms of reloids 236 14.1. Ordering of filters 236 14.2. Rudin-Keisler equivalence and Rudin-Keisler order 246 14.3. Consequences 248 Chapter 15. Counter-examples about funcoids and reloids 252 15.1. Second product. Oblique product 256 Chapter 16. Funcoids are filters 258 16.1. Rearrangement of collections of sets 258 16.2. Finite unions of Cartesian products 259 16.3. Before the diagram 260 16.4. Associativity over composition 262 16.5. The diagram 264 16.6. Some additional properties 265 16.7. More on properties of funcoids 267 16.8. Funcoid bases 268 16.9. Some (example) values 270 Chapter 17. Generalized cofinite filters 272 Chapter 18. Convergence of funcoids 276 18.1. Convergence 276 18.2. Relationships between convergence and continuity 277 18.3. Convergence of join 277 18.4. Limit 278 18.5. Generalized limit 278 18.6. Expressing limits as implications 280 Part 3. Pointfree funcoids and reloids 282 Chapter 19. Pointfree funcoids 283 19.1. Definition 283 19.2. Composition of pointfree funcoids 285 19.3. Pointfree funcoid as continuation 286 19.4. The order of pointfree funcoids 289 19.5. Domain and range of a pointfree funcoid 292 19.6. Specifying funcoids by functions or relations on atomic filters 294 19.7. More on composition of pointfree funcoids 296 19.8. Funcoidal product of elements 298 19.9. Category of pointfree funcoids 301 19.10. Atomic pointfree funcoids 302 19.11. Complete pointfree funcoids 303 19.12. Completion and co-completion 306 19.13. Monovalued and injective pointfree funcoids 306 19.14. Elements closed regarding a pointfree funcoid 308 19.15. Connectedness regarding a pointfree funcoid 308 19.16. Boolean funcoids 309

CONTENTS 7 19.17. Binary relations are pointfree funcoids 310 Chapter 20. Alternative representations of binary relations 312 Part 4. Staroids and multifuncoids 317 Chapter 21. Multifuncoids and staroids 318 21.1. Product of two funcoids 318 21.2. Definition of staroids 319 21.3. Upgrading and downgrading a set regarding a filtrator 322 21.4. Principal staroids 323 21.5. Multifuncoids 325 21.6. Join of multifuncoids 327 21.7. Infinite product of poset elements 329 21.8. On products of staroids 332 21.9. Star categories 335 21.10. Product of an arbitrary number of funcoids 338 21.11. Multireloids 347 21.12. Subatomic product of funcoids 351 21.13. On products and projections 354 21.14. Relationships between cross-composition and subatomic products 357 21.15. Cross-inner and cross-outer product 360 21.16. Coordinate-wise continuity 361 21.17. Upgrading and downgrading multifuncoids 362 21.18. On pseudofuncoids 364 21.19. Identity staroids and multifuncoids 369 21.20. Counter-examples 376 21.21. Conjectures 379 Part 5. Postface 383 Chapter 22. Postface 384 22.1. Pointfree reloids 384 22.2. Formalizing this theory 384 Chapter 23. Story of the discovery 386 Appendix A. Using logic of generalizations 388 A.1. Logic of generalization 388 Appendix. Bibliography 390

Part 1 Introductory chapters

CHAPTER 1 Introduction The main purpose of this book is to record the current state of my research. The book is however written in such a way that it can be used as a textbook for studying my research. For the latest version of this file, related materials, articles, research questions, and erratum consult the Web page of the author of the book: http://www.mathematics21.org/algebraic-general-topology.html 1.1. License and editing This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of the license, visit http://creativecommons.org/licenses/by/4.0/. You can create your own copy of LATEX source of the book and edit it (to correct errors, add new results, generalize existing results, enhance readability). The editable source of the book is presented at https://bitbucket.org/portonv/algebraic-general-topology Please consider reviewing this book at http://www.euro-math-soc.eu/node/add/book-review If you find any error (or some improvement idea), please report in our bug tracker: https://bitbucket.org/portonv/algebraic-general-topology/issues 1.2. Intended audience This book is suitable for any math student as well as for researchers. To make this book be understandable even for first grade students, I made a chapter about basic concepts (posets, lattices, topological spaces, etc.), which an already knowledgeable person may 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 you could not find 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 porton@narod.ru. 1.3. Reading Order If you know basic order and lattice theory (including Galois connections and brouwerian lattices) and basics of category theory, you may skip reading the chapter “Common knowledge, part 1”. You are recommended to read the rest of this book by the order. 9

1.5. EARLIER WORKS 10 1.4. Our topic and rationale From [42]: Point-set topology, also called set-theoretic topology or general topology, is the study of the general abstract nature of continuity or “closeness” on spaces. Basic point-set topological notions 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 (defined below in the section “Topological spaces”). I however argue that the theory of topological spaces is not the best method of studying general topology and introduce an alternative theory, the theory of funcoids. Despite of popularity of the theory of topological spaces it has some drawbacks and is in my opinion not the most appropriate formalism to study most of general topology. Because topological spaces are tailored for study of special sets, so called open and closed sets, studying general topology with topological spaces is a little anti-natural and ugly. In my opinion the theory of funcoids is more elegant than the theory of topological spaces, and it is better to study funcoids than topological spaces. One of the main purposes of this work is to present an alternative General Topology based on funcoids instead of being based on topological spaces as it is customary. In order to study funcoids the prior knowledge of topological spaces is not necessary. Nevertheless in this work I will consider topological spaces and the topic of interrelation of funcoids with topological spaces. In fact funcoids are a generalization of topological spaces, so the well known theory of topological 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 generalization of proximity spaces (see section “Proximity spaces” for the definition of proximity spaces). Before this work it was written that the theory of proximity spaces was an example of a stalled research, almost nothing interesting was discovered about this theory. It was so because the proper way to research proximity spaces 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 defined terminology it may be said that reloids are (basically) filters on Cartesian product of sets, and this is a special case of uniform spaces. 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 find applications. This book has a deficiency: 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 for now, 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 difficult results are made easy by reducing them to easier lemmas and propositions. 1.5. Earlier works Some mathematicians were researching generalizations of proximities and uniformities before me but they have failed to reach the right degree of generalization

1.7. RESPONSES TO SOME ACCUSATIONS AGAINST STYLE OF MY EXPOSITION 11 which is presented in this work allowing to represent properties of spaces with algebraic (or categorical) formulas. Proximity structures were introduced by Smirnov in [11]. Some references to predecessors: • In [15, 16, 25, 2, 36] generalized uniformities and proximities are studied. • Proximities and uniformities are also studied in [22, 23, 35, 37, 38]. • [20, 21] contains recent progress in quasi-uniform spaces. [21] has a very long list of related literature. Some works ([34]) about proximity spaces consider relationships of proximities and compact topological spaces. In this work the attempt to define or research their generalization, compactness of funcoids or reloids is not done. It seems potentially productive to attempt to borrow the definitions and 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 research mappings between funcoids is done.) [26] researches relationships of quasi-uniform spaces and topological spaces. [1] studies how proximity structures can be treated as uniform structures and compactification regarding proximity and uniform spaces. This book is based partially on my articles [30, 28, 29]. 1.6. Kinds of continuity A research result based on this book but not fully included in this book (and not yet published) is that the following kinds of continuity are described by the same algebraic (or rather categorical) formulas for different 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 justifies using the same word “continuity” for these diverse kinds of continuity. See http://www.mathematics21.org/algebraic-general-topology.html 1.7. Responses to some accusations against style of my exposition The proofs are generally hard to follow and unpleasant to read as they are just a bunch of equations thrown at you, without motivation or underlying reasoning, etc. I don’t think this is essential. The proofs are not the most important thing in my book. The most essential thing are definitions. The proofs are just to fill the gaps. So I deem it not important whether proofs are motivated. Also, note “algebraic” in the title of my book. The proofs are meant to be just sequences of formulas for as much as possible :-) It is to be thought algebraically. The meaning are the formulas themselves. Maybe it makes sense to read my book skipping all the proofs? Some proofs contain important ideas, but most don’t. The important thing are definitions.

1.9. BASIC NOTATION 12 1.8. Structure of this book In the chapter “Common knowledge, part 1” some well known definitions and theories are considered. 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; • 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 filters and filtrators. Then there is “Common knowledge, part 2 (topology)”, which considers briefly: • metric spaces; • topological spaces; • pretopological spaces; • proximity spaces. Despite of the name “Common knowledge” this second common knowledge chapter is recommended to be read completely even if you know topology well, because it contains some rare theorems not known to most mathematicians and hard to find 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 different kinds of products of morphisms. 1.9. Basic notation I will denote a set definition like x∈A P (x) instead of customary {x ∈ A | P(x)}. I do this because otherwise formulas don’t fit horizontally into the available space. 1.9.1. Grothendieck universes. We will work in ZFC with an infinite and uncountable Grothendieck universe. A Grothendieck universe is just a set big enough to make all usual set theory inside it. For example if ℧ is a Grothendieck universe, and sets X, Y ∈ ℧, then also X ∪ Y ∈ ℧, X ∩ Y ∈ ℧, X × Y ∈ ℧, etc. A set which is a member of a Grothendieck universe is called a small set (regarding this Grothendieck universe). We can restrict our consideration to small sets in order to get rid troubles with proper classes. Definition 1. Grothendieck universe is a set ℧ such that: 1◦. If x ∈ ℧ and y ∈ x then y ∈ ℧. 2◦. If x, y ∈ ℧ then {x, y} ∈ ℧. 3◦. If x ∈ ℧ then Px ∈ ℧. 4◦. If xi i∈I∈℧ is a family of elements of ℧, then i∈I xi ∈ ℧. One can deduce from this also: 1◦. If x ∈ ℧, then {x} ∈ ℧. 2◦. If x is a subset of y ∈ ℧, then x ∈ ℧. 3◦. If x, y ∈ ℧ then the ordered pair (x, y) = {{x, y}, x} ∈ ℧. 4◦. If x, y ∈ ℧ then x ∪ y and x × y are in ℧. 5◦. If xi i∈I∈℧ is a family of elements of ℧, then the product i∈I xi ∈ ℧.

1.11. UNUSUAL NOTATION 13 6◦. If x ∈ ℧, then the cardinality of x is strictly less than the cardinality of ℧. 1.9.2. Misc. In this book quantifiers bind tightly. That is ∀x ∈ A : P(x) ∧ Q and ∀x ∈ A : P(x) ⇒ Q should be read (∀x ∈ A : P(x))∧Q and (∀x ∈ A : P(x)) ⇒ Q not ∀x ∈ A : (P(x) ∧ Q) and ∀x ∈ A : (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 ⟨f⟩∗X = β∈im f ∃α∈X:αfβ (in other words ⟨f⟩∗X is the image of a set X under a function or binary relation f) and X [f]∗ Y ⇔ ∃x ∈ X, y ∈ Y : x f y for sets X, Y and a binary relation f. (Note that functions are a special case of binary relations.) By just ⟨f⟩∗ and [f]∗ I will denote the corresponding function and relation on small sets. Obvious 2. For a function f we have ⟨f⟩∗X = f(x) x∈X . Definition 3. f −1∗X is called the preimage of a set X by a function (or, more generally, a binary relation) f. Obvious 4. {α} [f]∗ {β} ⇔ α f β for every α and β. λx ∈ D : f(x) = (x,f(x)) x∈D for a set D and and a form f depending on the variable x. In other words, λx ∈ D : f(x) is the function which maps elements x of a set D into f(x). I will denote source and destination of a morphism f of any category (See chapter 2 for a definition of a category.) as Src f and Dst f correspondingly. Note that below defined domain and image of a funcoid are not the same as its source and destination. I will denote GR(A, B, f) = f for any morphism (A, B, f) of either Set or Rel. (See definitions of Set and Rel below.) 1.10. Implicit arguments Some notation such that ⊥A, ⊤A, ⊔A, ⊓A have indexes (in these examples A). We will omit these indexes when they can be restored from the context. For example, having a function f : A → B where A, B are posets with least elements, we will concisely denote f⊥ = ⊥ for f⊥A = ⊥B. (See below for definitions of these operations.) Note 5. In the above formula f⊥ = ⊥ we have the first ⊥ and the second ⊥ denoting different objects. We will assume (skipping this in actual proofs) that all omitted indexes can be restored from context. (Note that so called dependent type theory computer proof assistants do this like we implicitly.) 1.11. Unusual notation In the chapter “Common knowledge, part 1” (which you may skip reading if you are already knowledgeable) some non-standard notation is defined. I summarize here this notation for the case if you choose to skip reading that chapter: Partial order is denoted as ⊑. Meets and joins are denoted as ⊓, ⊔, , . I call element b substractive from an element a (of a distributive lattice A) when the difference a \ b exists. I call b complementive to a when there exists c ∈ A such

1.11. UNUSUAL NOTATION 14 that b ⊓ c = ⊥ and b ⊔ c = a. We will prove that b is complementive to a iff b is substractive from a and b ⊑ a. Definition 6. Call a and b of a poset A intersecting, denoted a ̸≍ b, when there exists a non-least element c such that c ⊑ a ∧ c ⊑ b. Definition 7. a ≍ b def = ¬(a ̸≍ b). Definition 8. I call elements a and b of a poset A joining and denote a ≡ b when there are no non-greatest element c such that c ⊒ a ∧ c ⊒ b. Definition 9. a ̸≡ b def = ¬(a ≡ b). Obvious 10. a ̸≍ b iff a ⊓ b is non-least, for every elements a, b of a meetsemilattice. Obvious 11. a ≡ b iff a ⊔ b is the greatest element, for every elements a, b of a join-semilattice. I extend the definitions of pseudocomplement and dual pseudocomplement to arbitrary posets (not just lattices as it is customary): Definition 12. Let A be a poset. Pseudocomplement of a is max c ∈ A c ≍ a . If z is the pseudocomplement of a we will denote z = a∗. Definition 13. Let A be a poset. Dual pseudocomplement of a is min c ∈ A c ≡ a . If z is the dual pseudocomplement of a we will denote z = a+.

CHAPTER 2 Common knowledge, part 1 In this chapter we will consider some well known mathematical theories. If you already know them you may skip reading this chapter (or its parts). 2.1. Order theory 2.1.1. Posets. Definition 14. The identity relation on a set A is idA = (a,a) a∈A . Definition 15. A preorder on a set A is a binary relation ⊑ on A which is: • reflexive on A that is (⊑) ⊇ idA or what is the same ∀x ∈ A : x ⊑ x; • transitive that is (⊑) ◦ (⊑) ⊆ (⊑) or what is the same ∀x, y, z : (x ⊑ y ∧ y ⊑ z ⇒ x ⊑ z). Definition 16. A partial order on a set A is a preorder on A which is antisymmetric that is (⊑) ∩ (⊑) ⊆ idA or what is the same ∀x, y ∈ A : (x ⊑ y ∧ y ⊑ x ⇒ x = y). The reverse relation is denoted ⊒. Definition 17. a is a subelement of b (or what is the same a is contained in b or b contains a) iff a ⊑ b. Obvious 18. The reverse of a partial order is also a partial order. Definition 19. A set A together with a partial order on it is called a partially ordered set (poset for short). An example of a poset is the set R of real numbers with ⊑ = ≤. Another example is the set PA of all subsets of an arbitrary fixed set A with ⊑ = ⊆. Note that this poset is (in general) not linear (see definition of linear poset below.) Definition 20. Strict partial order ⊏ corresponding to the partial order ⊑ on a set A is defined by the formula (⊏) = (⊑) \ idA. In other words, a ⊏ b ⇔ a ⊑ b ∧ a ̸= b. An example of strict partial order is < on the set R of real numbers. Definition 21. A partial order on a set A restricted to a set B ⊆ A is (⊑) ∩ (B × B). Obvious 22. A partial order on a set A restricted to a set B ⊆ A is a partial order on B. Definition 23. • The least element ⊥ of a poset A is defined by the formula ∀a ∈ A : ⊥ ⊑ a. • The greatest element ⊤ of a poset A is defined by the formula ∀a ∈ A : ⊤ ⊒ a. 15