Funcoids and Reloids a generalization of proximities and uniformities

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1 Funcoids and Reloids a generalization of proximities and uniformities by Victor Porton October 26, 2010 Abstract It is a part of my Algebraic General Topology research. In this article I introduce the concepts of funcoids which generalize proximity spaces and reloids which generalize uniform spaces. 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 analysis and discrete mathematics. 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 are generalized continuity, proximity continuity, and uniform continuity. Keywords: algebraic general topology, quasi-uniform spaces, generalizations of proximity spaces, generalizations of nearness spaces, generalizations of uniform spaces A.M.S. subject classification: 54J05, 54A05, 54D99, 54E05, 54E15, 54E17, 54E99 Table of contents 1 Common Draft status Earlier works Used concepts, notation and statements Filters Partially ordered dagger categories Partially ordered categories Dagger categories Monovalued and entirely defined morphisms Funcoids Informal introduction into funcoids Basic definitions Composition of funcoids Funcoid as continuation Lattice of funcoids More on composition of funcoids Domain and range of a funcoid Category of funcoids Specifying funcoids by functions or relations on atomic filter objects Direct product of filter objects Atomic funcoids Complete funcoids Completion of funcoids Monovalued funcoids This document has been written using the GNU TEX MAC S text editor (see 1

2 2 Section T 0 -, T 1 - and T 2 -separable funcoids Filter objects closed regarding a funcoid Reloids Composition of reloids Direct product of filter objects Restricting reloid to a filter object. Domain and image Category of reloids Monovalued reloids Complete reloids and completion of reloids Relationships of funcoids and reloids Funcoid induced by a reloid Reloids induced by funcoid Galois connections of funcoids and reloids Continuous morphisms Traditional definitions of continuity Pre-topology Proximity spaces Uniform spaces Our three definitions of continuity Continuousness of a restricted morphism Connectedness regarding funcoids and reloids Some lemmas Endomorphism series Connectedness regarding binary relations Connectedness regarding funcoids and reloids Algebraic properties of S and S Postface Misc Pointfree funcoids and reloids Appendix A Some counter-examples Bibliography Common 1.1 Draft status This article is a draft. This text refers to a preprint edition of [14]. Theorem number clashes may appear due editing both of these manuscripts. 1.2 Earlier works Some mathematicians were researching generalizations of proximities and uniformities before me but they have failed to reach the right degree of generalization which is presented in this work allowing to represent properties of spaces with algebraic (or categorical) formulas. Proximity structures were introduced by Smirnov in [4].

3 Common 3 Some references to predecessors: In [5], [6], [11], [2], [17] are studied generalized uniformities and proximities. Proximities and uniformities are also studied in [9], [10], [16], [18], [19]. [7] and [8] contains recent progress in quasi-uniform spaces. [8] has a very long list of related literature. Some works ([15]) about proximity spaces consider relationships of proximities and compact topological spaces. In this work is not done the attempt to define or research their generalization, compactness of funcoids or reloids. 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 article. [3] studies mappings between proximity structures. (In this work no attempt to research mappings between funcoids is done.) [12] 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. 1.3 Used concepts, notation and statements The set of functions from a set A to a set B is denoted as B A. 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={fα α X} for a set X. For simplicity I will assume that all sets in consideration are subsets of universal set Filters In this work the word filter will refer to a filter on a set (in contrast to [14] where are considered filters on arbitrary posets). Note that I do not require filters to be proper. I will call the set of filters ordered reverse to set-theoretic inclusion of filters the set of filter objects F and its element filter objects (f.o. for short). I will denote upf the filter corresponding to a filter object F. So we have A B upa upb for every filter objects A and B. We also will equate filter objects corresponding to principal filters with corresponding sets. (Thus we have P F.) See [14] for formal definition of filter objects in the framework of ZF. Filters (and filter objects) are studied in the work [14]. Prior reading of [14] is needed to fully understand this work. Filter objects corresponding to ultrafilters are atoms of the lattice F and will be called atomic filter objects. Also we will need to introduce the concept of generalized filter base. Definition 1. Generalized filter base is a set S PF\{ } such that A,B S C S:C A F B. Proposition 2. Let S is a generalized filter base. If A 1,,A n S (n N), then C S:C A 1 F F A n. Can be easily proved by induction. Theorem 3. If S is a generalized filter base, then up F S= up S. Obviously up F S up S. Reversely, let K up F S; then K =A 1 A n where A i upa i where A i S, i=1,,n, n N; so exists C S such that C A 1 A n A 1 A n =K, K upc, K up S. Corollary 4. If S is a generalized filter base, then F S= S. F S= up F S up S X S: upx S.

4 4 Section 2 2 Partially ordered dagger categories 2.1 Partially ordered categories Definition 5. I will call a partially ordered (pre)category a (pre)category together with partial order on each of its Hom-sets with the additional requirement that f 1 f 2 g 1 g 2 g 1 f 1 g 2 f 2 for every morphisms f 1, g 1, f 2, g 2 such that Src f 1 = Src f 2 Dst f 1 = Dst f 2 = Src g 1 = Src g 2 Dstg 1 = Dstg Dagger categories Definition 6. I will call a dagger precategory a precategory together with an involutive contravariant identity-on-objects prefunctor x x. In other words, a dagger precategory is a precategory equipped with a function x x on its set of morphisms which reverses the source and the destination and is subject to the following identities for every morphisms f and g: 1. f =f; 2. (g f) =f g. Definition 7. I will call a dagger category a category together with an involutive contravariant identity-on-objects functor x x. In other words, a dagger category is a category equipped with a function x x on its set of morphisms which reverses the source and the destination and is subject to the following identities for every morphisms f and g and object A: 1. f =f; 2. (g f) =f g ; 3. (1 A ) =1 A. Theorem 8. If a category is a dagger precategory then it is a dagger category. We need to prove only that (1 A ) =1 A. Really (1 A ) =(1 A ) 1 A =(1 A ) (1 A ) =((1 A ) 1 A ) =(1 A ) =1 A. For a partially ordered dagger (pre)category I will additionally require (for every morphisms f and g) f g f g. An example of dagger category is the category Rel whose objects are sets and whose morphisms are binary relations between these sets with usual composition of binary relations and with f = f 1. Definition 9. A morphism f of a dagger category is called unitary when it is an isomorphism and f =f 1. Definition 10. Symmetric (endo)morphism of a dagger precategory is such a morphism f that f =f. Definition 11. Transitive (endo)morphism of a precategory is such a morphism f that f = f f.

5 Partially ordered dagger categories 5 Theorem 12. The following conditions are equivalent for a morphism f of a dagger precategory: 1. f is symmetric and transitive. 2. f =f f. (1) (2). If f is symmetric and transitive then f f =f f =f. (2) (1). f =(f f) = f f = f f = f, so f is symmetric. f = f f = f f, so f is transitive Some special classes of morphisms Definition 13. For a partially ordered dagger category I will call monovalued morphism such a morphism f that f f 1 Dst f. Definition 14. For a partially ordered dagger category I will call entirely defined morphism such a morphism f that f f 1 Src f. Definition 15. For a partially ordered dagger category I will call injective morphism such a morphism f that f f 1 Src f. Definition 16. For a partially ordered dagger category I will call surjective morphism such a morphism f that f f 1 Dst f. Remark 17. Easy to show that this is a generalization of monovalued, entirely defined, injective, and surjective binary relations as morphisms of the category Rel. Obvious 18. Injective morphism is a dual of monovalued morphism and surjective morphism is a dual of entirely defined morphism. Definition 19. For a given partially ordered dagger category C the category of monovalued (entirely defined, injective, surjective) morphisms of C is the category with the same set of objects as of C and the set of morphisms being the set of monovalued (entirely defined, injective, surjective) morphisms of C with the composition of morphisms the same as in C. We need to prove that these are really categories, that is that composition of monovalued (entirely defined) morphisms is monovalued (entirely defined) and that identity morphisms are monovalued and entirely defined. We will prove only for monovalued morphisms and entirely defined morphisms, as injective and surjective morphisms are their duals. Monovalued. Let f and g are monovalued morphisms, Dstf = Src g. (g f) (g f) = g f f g g 1 Dst f g = g 1 Src g g = g g 1 Dst g =1 Dst(g f). So g f is monovalued. That identity morphisms are monovalued follows from the following: 1 A (1 A ) =1 A 1 A =1 A =1 Dst1A 1 Dst 1A. Entirely defined. Let f and g are entirely defined morphisms, Dstf = Src g. (g f) (g f)= f g g f f 1 Src g f = f 1 Dst f f = f f 1 Src f =1 Src(g f). So g f is entirely defined. That identity morphisms are entirely defined follows from the following: (1 A ) 1 A = 1 A 1 A =1 A =1 Src 1A 1 Src1A. Definition 20. I will call a bijective morphism a morphism which is entirely defined, monovalued, injective, and surjective. Obvious 21. Bijective morphisms form a full subcategory.

6 6 Section 3 Proposition 22. If a morphism is bijective then it is an isomorphism. Let f is bijective. Then f f 1 Dst f, f f 1 Src f, f f 1 Src f, f f 1 Dst f. Thus f f =1 Dst f and f f =1 Src f that is f is an inverse of f. 3 Funcoids 3.1 Informal introduction into funcoids Funcoids are a generalization of proximity spaces and a generalization of pretopological spaces. Also funcoids are a generalization of binary relations. That funcoids are a common generalization of spaces (proximity spaces, (pre)topological spaces) and binary relations (including monovalued functions) makes them smart for describing properties of functions in regard of spaces. For example the statement f is a continuous function from a space µ to a space ν can be described in terms of funcoids as the formula f µ ν f (see below for details). Most naturally funcoids appear as a generalization of proximity spaces. Let δ be a proximity that is certain binary relation so that AδB is defined for every sets A and B. We will extend it from sets to filter objects by the formula: Aδ B A upa,b upb:aδb. Then (as will be proved below) exist two functions α,β F F such that Aδ B B F αa A F βb. The pair (α; β) is called funcoid when B F αa A F βb. So funcoids are a generalization of proximity spaces. Funcoids consist of two components the first α and the second β. The first component of a funcoid f is denoted as f and the second component is denoted as f 1. (The similarity of this notation with the notation for the image of a set under a function is not a coincidence, we will see that in the case of discrete funcoids (see below) these coincide.) One of the most important properties of a funcoid is that it is uniquely determined by just one of its components. That is a funcoid f is uniquely determined by the function f. Moreover a funcoid f is uniquely determined by f P that is by values of function f on sets. Next we will consider some examples of funcoids determined by specified values of the first component on sets. Funcoids as a generalization of pretopological spaces: Let α be a pretopological space that is a map α F. Then we define α X= def F {αx x X} for every set X. We will prove that there exists a unique funcoid f such that α = f P. So funcoids are a generalization of pretopological spaces. Funcoids are also a generalization of preclosure operators: For every preclosure operator p exists unique funcoid such that f P =p; in this case f P P P. For every binary relation p exists unique funcoid f such that X P : f X = p X (where p is defined in the introduction), recall that a funcoid is uniquely determined by the values of its first component on sets. I will call such funcoids discrete. So funcoids are a generalization of binary relations. Composition of binary relations (i.e. of discrete funcoids) complies with the formulas: g f = g f and (g f) 1 = f 1 g 1. By the same formulas we can define composition of every two funcoids. Also funcoids can be reversed (like reversal of X and Y in a binary relation) by the formula (α; β) 1 =(β;α). In particular case if µ is a proximity we have µ 1 = µ because proximities are symmetric. Funcoids behave similarly to (multivalued) functions but acting on filter objects instead of acting on sets. Below will be defined domain and image of a funcoid (the domain and the image of a funcoid are filter objects).

7 Funcoids Basic definitions Definition 23. Let s call a funcoid a pair (α;β) where α,β F F such that X,Y F:(Y F αx X F βy ). Definition 24. (α;β) = def α for a funcoid (α;β). Definition 25. (α;β) 1 =(β;α) for a funcoid (α;β). Proposition 26. If f is a funcoid then f 1 is also a funcoid. Follows from symmetry in the definition of funcoid. Obvious 27. (f 1 ) 1 =f for a funcoid f. Definition 28. The relation [f] PF 2 is defined by the formula (for every filter objects X, Y and funcoid f) X[f]Y= def Y F f X. Obvious 29. X[f]Y Y F f X X F f 1 Y for every filter objects X, Y and funcoid f. Obvious 30. [ f 1] =[f] 1 for a funcoid f. Theorem For given value of f exists no more than one funcoid f. 2. For given value of [f] exists no more than one funcoid f. Let f and g are funcoids. Obviously f = g [f]=[g] and f 1 = g 1 [f]=[g]. So enough to prove that [f]= [g] f = g. Provided that [f]=[g] we have Y F f X X[f]Y X[g]Y Y F g X and consequently f X = g X for every f.o. X and Y because the set of filter objects is separable [14], thus f = g. Proposition 32. f (I F J)= f I F f J for every funcoid f and I,J F. f (I F J) = { Y F Y F f (I F J) } = { Y F (I F J) F f 1 Y } = (by corollary 10 in [14]) { Y F (I F f 1 Y) F (J F f 1 Y) } = { Y F I F f 1 Y J F f 1 Y } = { Y F Y F f I Y F f J } = { Y F (Y F f I) F (Y F f J) } = (by corollary 10 in [14]) { Y F Y F ( f I F f J) } = ( f I F f J). Thus f (I F J)= f I F f J because F is separable Composition of funcoids Definition 33. Composition of funcoids is defined by the formula (α 2 ;β 2 ) (α 1 ;β 1 )=(α 2 α 1 ;β 1 β 2 ).

8 8 Section 3 Proposition 34. If f, g are funcoids then g f is funcoid. Let f =(α 1 ;β 1 ), g=(α 2 ;β 2 ). For every X,Y F we have Y F (α 2 α 1 )X Y F α 2 α 1 X α 1 X F β 2 Y X F β 1 β 2 Y X F (β 1 β 2 )Y. So (α 2 α 1 ;β 1 β 2 ) is a funcoid. Obvious 35. g f = g f for every funcoids f and g. Proposition 36. (h g) f =h (g f) for every funcoids f, g, h. (h g) f = h g f =( h g ) f = h ( g f )= h g f = h (g f). Theorem 37. (g f) 1 =f 1 g 1 for every funcoids f and g. (g f) 1 = f 1 g 1 = f 1 g Funcoid as continuation Theorem 38. For every funcoid f and filter objects X and Y 1. f X = F f upx ; 2. X[f]Y X upx,y upy:x[f]y. 2. X[f]Y Y F f X Y upy:y F f X Y upy:x[f]y. Analogously X[f]Y X upx:x[f]y. Combining these two equivalences we get X[f]Y X upx,y upy:x[f]y. 1. Y F f X X[f]Y X upx:x[f]y X upx:y F f X. Let s denote W = { Y F f X X upx }. We will prove that W is a generalized filter base. To prove this enough to show that V ={ f X X upx} is a generalized filter base. Let P,Q V. Then P = f A, Q= f B where A,B upx ; A B upx and R P F Q for R= f (A B) V. So V is a generalized filter base and thus W is a generalized filter base. W F W by the corollary 4 of the theorem 3. That is X upx:y F f X Y F F f upx. Comparing with the above, Y F f X Y F F f up X. So f X = F f up X because the lattice of filter objects is separable. Theorem A function α F P conforming to the formulas (for every I,J P ) α =, α(i J)=αI F αj can be continued to the function f for a unique funcoid f; for every filter object X. f X = F α upx (1) 2. A relation δ P(P ) 2 conforming to the formulas (for every I,J,K P ) ( δi), I J δk IδK J δk, (Iδ ), KδI J KδI KδJ (2) can be continued to the relation [f] for a unique funcoid f; X[f]Y X upx,y upy:x δy (3)

9 Funcoids 9 for every filter objects X, Y. Existence of no more than one such funcoids and formulas (1) and (3) follow from the previous theorem. 2. Let define α F P by the formula (αx) ={Y P X δy } for every X P. (It is obvious that {Y P X δy } is a free star.) Analogously can be defined β F P by the formula (βx) = {X P X δy }. Let s continue α and β to α F F and β F F by the formulas and δ to δ PF 2 by the formula α X = F α upx and β X = F β upx X δ Y X upx,y upy:xδy. Y F α X Y F F α upx F Y F α upx. Let s prove that W = Y F α upx is a generalized filter base: To prove it is enough to show that α up X is a generalized filter base. If A,B α upx then exist X 1,X 2 upx such that A=αX 1 and A=αX 2. Then α(x 1 X 2 ) α upx. So α upx is a generalized filter base and thus W is a generalized filter base. Accordingly the corollary 4 of the theorem 3, F Y F α upx is equivalent to X upx:y F αx, what is equivalent to X upx,y upy:y F αx X upx,y up Y:Y (αx) X upx,y upy:x δy. Combining the equivalencies we get Y F α X X δ Y. Analogously X F β Y X δ Y. So Y F α X X F β Y, that is (α ; β ) is a funcoid. From the formula Y F α X X δ Y follows that [(α ;β )] is a continuation of δ. 1. Let define the relation δ P(P ) 2 by the formula X δy Y F αx. That ( δi) and (Iδ ) is obvious. We have I J δk (I J) F αk (I F J) F αk (I F αk) F (J F αk) I F αk J F αk IδK J δk and KδI J K F α(i J) K F (αi F αj) (K F αi) F (K F αj) K F αi K F αj KδI KδJ. That is the formulas (2) are true. Accordingly the above δ can be continued to the relation [f] for some funcoid f. X,Y P :(Y F f X X[f]Y Y F αx ), consequently X P :αx = f X. So f is a continuation of α. Note that by the last theorem to every proximity δ corresponds a unique funcoid. So funcoids are a generalization of (quasi-)proximity structures. Reverse funcoids can be considered as a generalization of conjugate quasi-proximity. Definition 40. Any (multivalued) function f will be considered as a funcoid, where by definition f X = F f upx for every X F. Using the last theorem it is easy to show that this definition is monovalued and does not contradict to former stuff. Definition 41. Funcoids corresponding to a binary relation are called discrete funcoids. We may equate discrete funcoids with corresponding binary relations by the method of appendix B in [14]. This is useful for describing relationships of funcoids and binary relations, such as for the formulas of continuous functions and continuous funcoids (see below). For simplicity I will not dive here into formal definition of equating discrete funcoids with binary relations (by the method shown in appendix B in [14]) but we simply will (informally) assume that discrete funcoids can be equated with binary relations.

10 10 Section 3 I will denote FCD the set of funcoids or the category of funcoids (see below) dependently on context. 3.4 Lattice of funcoids Definition 42. f g= def [f] [g] for f,g FCD. Thus FCD is a poset. (Taken into account that [f] [g] if f g.) Definition 43. I will call the filtrator of funcoids (see [14] for the definition of filtrators) the filtrator (FCD; P 2 ). Conjecture 44. The filtrator of funcoids is: 1. with separable core; 2. with co-separable core. Theorem 45. The set of funcoids is a complete lattice. For every R PFCD and X,Y P 1. X [ FCD R ] Y f R:X[f]Y ; 2. FCD R X= F { f X f R}. Accordingly [13] to prove that it is a complete lattice enough to prove existence of all joins. 2. αx= def F { f X f R}. We have α = ; α(i J) = F { f (I J) f R} = F { f (I F J) f R } = F { f I F f J f R } = F { f I f R} F F { f J f R} = αi F αj. So α can be continued to h for a funcoid h. Obviously f R:h f. (4) And h is the least funcoid for which holds the condition (4). So h= FCD R. 1. X [ FCD R ] Y Y F FCD R X Y F F { f X f R} f R: Y F f X f R:X[f]Y (used the theorem 40 in [14]). In the next theorem, compared to the previous one, the class of infinite unions is replaced with lesser class of finite unions and simultaneously class of sets is changed to more wide class of filter objects. Theorem 46. For every funcoids f and g and a filter object X 1. f FCD g X = f X F g X ; 2. [ f FCD g ] =[f] [g]. 1. Let αx = def f X F g X ; βy= def f 1 Y F g 1 Y for every X,Y F. Then Y F αx Y F f X Y F g X X F f 1 Y X F g 1 Y X F βy.

11 Funcoids 11 So h=(α; β) is a funcoid. Obviously h f and h g. If p f and p g for some funcoid p then p X f X F g X = h X that is p h. So f FCD g=h. 2. X [ f FCD g ] Y Y F f FCD g X Y F ( f X F g X) Y F f X Y F g X X[f]Y X[g]Y for every X,Y F. 3.5 More on composition of funcoids Proposition 47. [g f]=[g] f = g 1 1 [f] for f,g FCD. X[g f]y Y F g f X Y F g f X f X[g]Y X([g] f )Y for every X, Y F. [g f] = [ (f 1 g 1 ) 1] = [ f 1 g 1] 1 = ( [ f 1] g 1 ) 1 = g 1 1 [f]. The following theorem is a variant for funcoids of the statement (which defines compositions of relations) that x(g f)z y(xfy ygz) for every x and z and every binary relations f and g. Theorem 48. For every X,Z F and f,g FCD X[g f]z y atoms F :(X[f]y y[g]z). y atoms F :(X[f]y y[g]z) y atoms F :(Z F g y y F f X ) y atoms F :(Z F g y y f X) Z F g f X X[g f]z. Reversely, if X[g f]z then f X[g]Z, consequently exists y atoms F f X such that y[g]z; we have X[f]y. Theorem 49. If f, g, h are funcoids then 1. f (g FCD h)=f g FCD f h; 2. (g FCD h) f =g f FCD h f. I will prove only the first equality because the other is analogous. For every X,Z F X [ f (g FCD h) ] Z y atoms F :(X [ g FCD h ] y y[f]z) y atoms F :((X[g]y X[h]y) y[f]z) y atoms F :(X[g]y y[f]z X[h]y y[f]z) y atoms F :(X[g]y y[f]z) y atoms F :(X[h]y y[f]z) X[f g]z X[f h]z X [ f g FCD f h ] Z. 3.6 Domain and range of a funcoid Definition 50. Let A F. The identity funcoid I A FCD =(A F ;A F ). Proposition 51. The identity funcoid is a funcoid. We need to prove that (A F X) F Y (A F Y) F X what is obvious.

12 12 Section 3 Obvious 52. (I A FCD ) 1 =I A FCD. Obvious 53. X [ I A FCD ] Y A F X F Y for anyx,y F. Definition 54. I will define restricting of a funcoid f to a filter object A by the formula f A FCD = def f I A FCD. Definition 55. Image of a funcoid f will be defined by the formula imf = f. Domain of a funcoid f is defined by the formula domf = imf 1. Proposition 56. f X = f (X F domf) for every f FCD, X F. For every filter object Y we have Y F f (X F domf) X F dom f F f 1 Y X F imf 1 F f 1 Y X F f 1 Y Y F f X. Thus f X = f (X F domf) because the lattice of filter objects is separable. Proposition 57. X F domf f X for every f FCD, X F. X F domf X F f 1 F f X f X. Corollary 58. domf = F { a atoms F f a }. This follows from the fact that F is an atomistic lattice. 3.7 Category of funcoids I will define the category FCD of funcoids: The set of objects is F. The set of morphisms from a filter object A to a filter object B is the set of triples (f;a; B) where f is a funcoid such that domf A, imf B. Composition of morphisms is defined in the natural way. Identity morphism of a filter object A is (I A FCD ;A;A). To prove that it is really a category is trivial. 3.8 Specifying funcoids by functions or relations on atomic filter objects Theorem 59. For every funcoid f and filter objects X and Y 1. f X = F f atoms F X ; 2. X[f]Y x atoms F X,y atoms F Y:x[f]y. 1. Y F f X X F f 1 Y x atoms F X:x F f 1 Y x atoms F X:Y F f x. f X = f atoms F X = F f atoms F X. 2. If X[f]Y, then Y F f X, consequently exists y atoms F Y such that y F f X, X[f]y. Repeating this second time we get that there exist x atoms F X such that x[f]y. From this follows The reverse is obvious. x atoms F X,y atoms F Y:x[f]y.

13 Funcoids 13 Theorem A function α F atomsf such that (for every a atoms F ) αa F F α atoms F up a (5) can be continued to the function f for a unique funcoid f; for every filter object X. f X = F α atoms F X (6) 2. A relation δ P(atoms F ) 2 such that (for every a,b atoms F ) X upa,y upb x atoms F X,y atoms F Y:xδ y aδb (7) can be continued to the relation [f] for a unique funcoid f; for every filter objects X, Y. X[f]Y x atoms F X,y atoms F Y:xδ y (8) Existence of no more than one such funcoids and formulas (6) and (8) follow from the previous theorem. 1. Consider the function α F defined by the formula (for every X P ) Obviously α =. For every I,J P α X= F α atoms F X. α (I J) = F α atoms F (I J) = F α (atoms F I atoms F J) = F ( α atoms F I α atoms F J ) = F α atoms F I F F α atoms F J. = α I F α J. Let continue α till a funcoid f (by the theorem 25): f X = F α upx. Let s prove the reverse of (5): F F α atoms F upa = F F atoms α F upa F F α {{a}} = ) } F{( F α {a} = } F{ F α {a} = } F{ F {αa} = F {αa}=αa. Finally, αa= F F α atoms F upa= F α upa= f a, so f is a continuation of α. 2. Consider the relation δ P(P ) 2 defined by the formula (for every X,Y P ) Obviously (Xδ ) and ( δ Y). X δ Y x atoms F X,y atoms F Y:xδ y. (I J)δ Y x atoms F (I J),y atoms F Y:xδ y x atoms F I atoms F J,y atoms F Y:xδ y x atoms F I,y atoms F Y:xδ y x atoms F J,y atoms F Y:xδ y Iδ Y J δ Y;

14 14 Section 3 analogously Xδ (I J) Xδ I Xδ J. Let s continue δ till a funcoid f (by the theorem 25): X[f]Y X upx,y upy:x δ Y The reverse of (7) implication is trivial, so X upa,y upb x atoms F X,y atoms F Y:xδ y aδb. X upa,y upb x atoms F X,y atoms F Y:xδ y X upa,y upb:xδ Y a[f]b. So aδb a[f]b, that is [f] is a continuation of δ. One of uses of the previous theorem is the proof of the following theorem: Theorem 61. If R is a set of funcoids, x,y atoms F, then 1. FCD R x= F { f x f R}; 2. x [ FCD R ] y f R:x[f]y. 2. Let denote xδ y f R:x[f]y. X upa,y upb x atoms F X,y atoms F Y:xδ y f R,X upa,y upb x atoms F X,y atoms F Y:x[f]y f R,X upa,y upb:x[f]y f R:a[f]b aδb. So, by the theorem 60, δ can be continued till [p] for some funcoid p. For every funcoid q such that f R:q f we have x[q]y f R:x[f]y xδ y x[p]y, so q p. Consequently p= FCD R. From this x [ FCD R ] y f R:x[f]y. 1. From the former y atoms F FCD R x y F FCD R x f R: y F f x y atoms F { f x f R} y atoms F F { f x f R} for every y atoms F. From this follows FCD R x= F { f x f R}. 3.9 Direct product of filter objects A generalization of direct (Cartesian) product of two sets is direct product of two filter objects as defined in the theory of funcoids: Definition 62. Direct product of filter objects A and B is such a funcoid A FCD B that X [ A FCD B ] Y X F A Y F B. Proposition 63. A FCD B is really a funcoid and A FCD B { B ifx X = F A ; ifx F A=. Obvious. Obvious 64. A B=A FCD B for sets A and B. Proposition 65. f A FCD B domf A imf B for every f FCD and A,B F. If f A FCD B then dom f dom(a FCD B) A, im f im(a FCD B) B. If domf A imf B then consequently f A FCD B. X,Y F:(X[f]Y X F A Y F B );

15 Funcoids 15 The following theorem gives a formula for calculating an important particular case of intersection on the lattice of funcoids: Theorem 66. f FCD (A FCD B)=I B FCD f I A FCD for every f FCD and A,B F. h= def I B FCD f I A FCD. For every X F h X = I B FCD f I A FCD X =B f (A X). From this, as easy to show, h f and h A B. If g f g A FCD B for a funcoid g then domg A, img B, g X =B F g (A F X) B F f (A F X)= I B FCD f I A FCD X = h X, g h. So h=f FCD (A FCD B). Corollary 67. f A FCD =f (A FCD ) for every f FCD and A F. f FCD (A FCD )=I FCD f I A FCD =f I A FCD =f A FCD. Corollary 68. f FCD (A FCD B) A[f]B for every f FCD, A,B F. f FCD (A FCD B) f FCD (A FCD B) I B FCD f I A FCD IB FCD f I A FCD B FCD f (A F ) B F f A A[f]B. Corollary 69. The filtrator of funcoids is star-separable. The set of direct products of sets is a separation subset of the lattice of funcoids. Theorem 70. If S PF 2 then FCD { A FCD B (A;B) S } = F doms FCD F ims. If x atoms F then by the theorem 61 FCD { A FCD B (A;B) S } x= F { A FCD B x (A;B) S }. If x F F doms then if x F F doms= then (A;B) S:(x F A A FCD B x=b); { A FCD B x (A;B) S } = ims; (A;B) S:(x F A= A FCD B x= ); { A FCD B x (A;B) S }. So FCD { A FCD B (A;B) S } { F ims if x x= F F doms ; if x F F doms=. From this follows the statement of the theorem. Corollary 71. (A 0 FCD B 0 ) FCD (A 1 FCD B 1 )=(A 0 F A 1 ) FCD (B 0 F B 1 ) for every A 0,A 1, B 0,B 1 F. (A 0 FCD B 0 ) FCD (A 1 FCD B 1 ) = { } FCD A 0 FCD B 0, A 1 FCD B 1 what is by the last theorem equal to (A 0 F A 1 ) FCD (B 0 F B 1 ). Theorem 72. If A FthenA FCD is a complete homomorphism of the lattice F to a complete sublattice of the lattice FCD, if also A then it is an isomorphism.

16 16 Section 3 Let S PF, X P, x atoms F. FCD A FCD S X = F { A FCD B X B S } = = { F S ifx F A ifx F A= A FCD F S X; FCD A FCD S x = F { A FCD B x B S } = = { F S ifx F A ifx F A= A FCD F S x. If A then obviously the function A FCD is injective. The following proposition states that cutting a rectangle of atomic width from a funcoid always produces a rectangular (representable as a direct product of filter objects) funcoid (of atomic width). Proposition 73. If a is an atomic filter object, f FCD then f a FCD =a FCD f a. Let X F. X F a f a FCD X = f a, X F a= f a FCD X = Atomic funcoids Theorem 74. A funcoid is an atom of the lattice of funcoids iff it is direct product of two atomic filter objects.. Let f is an atomic funcoid. Let s get elements a atoms F dom f and b atoms F f a. Then for every X F X F a= a FCD b X = f X, X F a a FCD b X =b f X. So a FCD b f; because f is atomic we have f =a FCD b.. Let a,b atoms F, f FCD. If b F f a= then (a[f]b), f FCD (a FCD b)= ; if b f a then X F: (X F a f X b), f a FCD b. Consequently f FCD (a FCD b)= f a FCD b; that is a FCD b is an atomic filter object. Theorem 75. The lattice of funcoids is atomic. Let f is a non-empty funcoid. Then domf, thus by the theorem 46 in [14] exists a atoms F dom f. So f a thus exists b atoms f a. Finally the atomic funcoid a FCD b f. Theorem 76. The lattice of funcoids is separable. Let f, g FCD, f g. Then exists a atoms F such that f a g a. So because the lattice F is atomically separable then exists b atoms F such that f a F b = and b g a. For every x atoms F f a F a FCD b a= f a F b=, x a f x F a FCD b x= f x F =

17 Funcoids 17 Thus f x F a b x= and consequently f FCD (a FCD b)=. a FCD b a=b g a, x a a FCD b x= g a. Thus a FCD b x g x and consequently a FCD b g. So the lattice of funcoids is separable by the theorem 19 in [14]. Corollary 77. The lattice of funcoids is: 1. separable; 2. atomically separable; 3. conforming to Wallman s disjunction property. By the theorem 22 in [14]. Remark 78. For more ways to characterize (atomic) separability of the lattice of funcoids see [14], subsections Separation subsets and full stars and Atomically separable lattices. Corollary 79. The lattice of funcoids is an atomistic lattice. Let f is a funcoid. Suppose contrary to the statement to be proved that FCD atoms FCD f f. Then exists a atoms FCD f such that a FCD FCD atoms FCD f = what is impossible. Proposition 80. atoms FCD (f FCD g)=atoms FCD f atoms FCD g for every funcoids f and g. (a FCD b) FCD (f FCD g) a [ f FCD g ] b a[f]b a[g]b (a FCD b) FCD f (a FCD b) FCD g for every atomic filter objects a and b. Corollary 81. For every f,g,h FCD, R PFCD 1. f FCD (g FCD h)=(f FCD g) FCD (f FCD h); 2. f FCD FCD R= FCD f FCD R. We will take in account that the lattice of funcoids is an atomistic lattice. To be concise I will write atoms instead of atoms FCD and and instead of FCD and FCD. 1. atoms(f (g h))=atomsf atoms(g h)=atoms f (atoms g atomsh)=(atoms f atomsg) (atomsf atomsh)=atoms(f g) atoms(f h)=atoms((f g) (f h)). 2. atoms ( f FCD R ) = atoms f atoms FCD R = atoms f FCD atoms R = FCD (atoms f) atoms R= FCD atoms f R=atoms FCD f R. (Used the following equality.) (atomsf) atoms R = {(atomsf) A A atoms R} = {(atomsf) A C R:A=atomsC} = {(atomsf) (atomsc) C R} = {atoms(f C) C R} = {atomsb C R:B=f C} = {atomsb B f R} = atoms f. Note that distributivity of the lattice of funcoids is proved through using atoms of this lattice. I have never seen such method of proving distributivity.

18 18 Section 3 Corollary 82. The lattice of funcoids is co-brouwerian. The next proposition is one more (among the theorem 48) generalization for funcoids of composition of relations. Proposition 83. For every f,g FCD atoms FCD (g f)= { x FCD z x,z atoms F, y atoms F :(x FCD y atoms FCD f y FCD z atoms FCD g) }. (x FCD z) FCD (g f) x[g f]z y atoms F :(x[f]y y[g]z) y atoms F : ((x FCD y) FCD f (y FCD z) FCD g ) (were used the theorem 48). Conjecture 84. The set of discrete funcoids is the center of the lattice of funcoids Complete funcoids Definition 85. I will call co-complete such a funcoid f that X P : f X P. Remark 86. I will call generalized closure such a function α P P that 1. α = ; 2. I,J P :α(i J)=αI αj. Obvious 87. A funcoid f is co-complete iff f P is a generalized closure. Remark 88. Thus funcoids can be considered as a generalization of generalized closures. A topological space in Kuratowski sense is the same as reflexive and transitive generalized closure. So topological spaces can be considered as a special case of funcoids. Definition 89. I will call a complete funcoid a funcoid whose reverse is co-complete. Theorem 90. The following conditions are equivalent for every funcoid f: 1. funcoid f is complete; 2. S PF,J P :( F S[f]J I S:I[f]J); 3. S PP,J P :( S[f]J I S:I[f]J); 4. S PF: f F S= F f S; 5. S PP : f S= F f S; 6. A P : f A= F { f {a} a A}. (3) (1). For every S PP, J P S F f 1 J I S:I F f 1 J, (9) consequently by the theorem 52 in [14] we have f 1 J P. (1) (2). For every S PF, J P we have f 1 J P, consequently the formula (9) is true. From this follows (2). (6) (5). f S = F { f {a} a S} = F { F { f {a} a A} A S } = F { f A A S}= F f S. (2) (4). J F f F S 0 F S[f]J I S: I[f]J I S: J F f I J F F f S (used the theorem 53 in [14]).

19 Funcoids 19 (2) (3), (4) (5), (5) (3), (5) (6). Obvious. The following proposition shows that complete funcoids are a direct generalization of pre-topological spaces. Proposition 91. To specify a complete funcoid f it is enough to specify f on one-element sets, values of f on one element sets can be specified arbitrarily. From the above theorem is clear that knowing f on one-element sets f can be found on every set and then its value can be inferred for every filter objects. Choosing arbitrarily the values of f on one-element sets we can define a complete funcoid the following way: f X= def F { f {α} α X} for every X P. Obviously it is really a complete funcoid. Theorem 92. A funcoid is discrete iff it is both complete and co-complete.. Obvious.. Let f is both a complete and co-complete funcoid. Consider the relation g defined by that g {α}= f {α} (g is correctly defined because f is a generalized closure). Because f is a complete funcoid f =g. Theorem 93. If R is a set of (co-)complete funcoids then FCD R is a (co-)complete funcoid. It is enough to prove only for co-complete funcoids. Let R is a set of co-complete funcoids. Then for every X P FCD R X = { f X f R} P (used the theorem 45). Corollary 94. If R is a set of binary relations then FCD R= R. From two last theorems. Theorem 95. The filtrator of funcoids is filtered. It s enough to prove that every funcoid is representable as (infinite) meet (on the lattice of funcoids) of some set of discrete funcoids. Let f FCD, A P, B up f A, g(a;b)= def A FCD B FCD A FCD. For every X P g(a;b) X= A FCD B X A FCD ifx= X= B if X A f X; ifx A so g(a;b) f. For every A P F { g(a;b) A B up f A}= F {B B up f A}= f A; consequently FCD {g(a;b) A P,B up f A}=f. Conjecture 96. If f is a complete funcoid and R is a set of funcoids then f FCD R = FCD f R. This conjecture can be weakened:

20 20 Section 3 Conjecture 97. If f is a discrete funcoid and R is a set of funcoids then f FCD R = FCD f R. I will denote ComplFCD and CoComplFCD the sets of complete and co-complete funcoids correspondingly. Obvious 98. ComplFCD and CoComplFCD are closed regarding composition of funcoids. Proposition 99. ComplFCD and CoComplFCD (with induced order) are complete lattices. Follows from the theorem Completion of funcoids Theorem 100. Cor f = Cor f for an element f of the filtrator of funcoids. (Core part is taken for the filtrator of funcoids.) From the theorem 26 in [14] and the corollary 94 and theorem 95. Definition 101. Completion of a funcoid f is the complete funcoid Compl f defined by the formula Complf {α}= f {α} for α. Definition 102. Co-completion of a funcoid f is defined by the formula CoComplf =(Complf 1 ) 1. Obvious 103. Complf f and CoComplf f for every funcoid f. Proposition 104. The filtrator (FCD; ComplFCD) is filtered. Because the filtrator (FCD; P 2 ) is filtered. Theorem 105. Complf = Cor (FCD;ComplFCD) f = Cor (FCD;ComplFCD) f. Cor (FCD;ComplFCD) f = Cor (FCD;ComplFCD) f since (the theorem 26 in [14]) the filtrator (FCD; ComplFCD) is filtered and with join closed core (the theorem 93). Let g up (FCD;ComplFCD) f. Then g ComplFCD and g f. Thus g= Complg Complf. Thus g up (FCD;ComplFCD) f:g Complf. Let g up (FCD;ComplFCD) f:h g for some h ComplFCD. Then h FCD up (FCD;ComplFCD) f =f and consequently h=complh Complf. Thus Complf = ComplFCD up (FCD;ComplFCD) f = Cor (FCD;ComplFCD) f. Theorem 106. Atoms of the lattice ComplFCD are exactly direct products of the form {α} FCD b where α and b is an atomic f.o. First, easy to see that {α} FCD b are elements of ComplFCD. Also is an element of ComplFCD. {α} FCD b are atoms of ComplFCD because these are atoms of FCD. Remain to prove that if f is an atom of ComplFCD then f ={α} FCD b for some α and an atomic f.o. b. Suppose f is a non-empty complete funcoid. Then exists α such that f {α}. Thus {α} FCD b f for some atomic f.o. b. If f is an atom then f ={α} FCD b. Theorem 107. CoComplf X = Cor f X for every funcoid f and set X. CoCompl f f thus CoCompl f X f X, but CoCompl f X P thus CoComplf X Cor f X.

21 Funcoids 21 Let αx= Cor f X. Then α = and α(x Y)=Cor f (X Y)=Cor( f X f Y)=Cor f X Cor f Y =αx αy. (used the theorem 64 from [14]). Thus α can be continued till g for some funcoid g. This funcoid is co-complete. Evidently g is the greatest co-complete funcoid which is lower than f. Thus g= CoComplf and so Cor f X=αX = g X = CoComplf X. Theorem 108. ComplFCD is an atomistic lattice. Let f ComplFCD. f X = F { f {x} x X} = { F { } F FCD f {x} X x X, thus f = { FCD f FCD {x} x X union of atoms of ComplFCD. } FCD f {x} {x} x X }. It is trivial that every f {x} = FCD is a Theorem 109. A funcoid is complete iff it is a join (on the lattice FCD) of atomic complete funcoids. Follows from the theorem 93 and the previous theorem. Corollary 110. ComplFCD is join-closed. Theorem 111. Compl( FCD R)= FCD Compl R for every set R of funcoids. Compl( FCD R) X = { F FCD R {α} α X } = F { F { f {α} f R} α X } = F { F { f {α} α X} f R } = F { Compl f X f R}= FCD Compl R X for every set X. Corollary 112. Compl is a lower adjoint. Conjecture 113. Compl is not an upper adjoint (in general). Conjecture 114. Complf =f \ FCD (Ω FCD ) for every funcoid f. This conjecture may be proved by considerations similar to these in the section Fréchet filter in [14]. Lemma 115. Co-completion of a complete funcoid is complete. Let f is a complete funcoid. CoCompl f X = Cor f X = Cor F { f {x} x X} = {Cor f {x} x X} = { CoComplf {x} x X} for every set X. Thus CoComplf is complete. Theorem 116. Compl CoComplf = CoCompl Complf = Corf for every funcoid f. Compl CoCompl f is co-complete since (used the lemma) CoCompl f is co-complete. Thus Compl CoCompl f is a discrete funcoid. CoCompl f is the the greatest co-complete funcoid under f and Compl CoCompl f is the greatest complete funcoid under CoCompl f. So Compl CoCompl f is greater than any discrete funcoid under CoCompl f which is greater than any discrete funcoid under f. Thus Compl CoCompl f it is the greatest discrete funcoid under f. Thus Compl CoComplf = Corf. Similarly CoCompl Complf = Corf. Question 117. Is ComplFCD a co-brouwerian lattice? 3.13 Monovalued funcoids Following the idea of definition of monovalued morphism let s call monovalued such a funcoid f that f f 1 I FCD im f.

22 22 Section 3 Obvious 118. A morphism (f;a;b) of the category of funcoids is monovalued iff the funcoid f is monovalued. Theorem 119. The following statements are equivalent for a funcoid f: 1. f is monovalued. 2. a atoms F A: f a atoms F { }. 3. I,J F: f 1 (I F J)= f 1 I F f 1 J. 4. I,J P : f 1 (I J)= f 1 I F f 1 J. (2) (3). Let a atoms F, f a=b. Then because b atoms F { } (I F J) F b I F b J F b ; a[f](i F J) a[f]i a[f]j; (I F J) [ f 1] a I [ f 1] a J [ f 1] a; a F f 1 (I F J) a F f 1 I a F f 1 J ; f 1 (I F J)= f 1 I F f 1 J. (3) (1). f 1 a F f 1 b = f 1 (a F b) = f 1 = for every two distinct atomic filter objects a and b. This is equivalent to ( f 1 a[f]b); b F f f 1 a= ; b F f f 1 a= ; (a [ f f 1] b). So a [ f f 1] b a=b for every atomic filter objects a and b. FCD. This is possible only when f f 1 I im f (4) (3). f 1 (I F J) = F f 1 up(i F J) = F f 1 {I J I up I, J upj}= F { f 1 (I J) I upi,j upj } = F { f 1 I F f 1 J I upi,j upj } = F { f 1 I I upi } F F { f 1 J J upj } = f 1 I F f 1 J. (3) (4). Obvious. (2) (1). Suppose f a atoms F B { } for some a atoms F A. Then there exist two atomic filter objects p q such that f a p f a q. Consequently p F f a ; a F f 1 p ; a f 1 p; f f 1 p= f f 1 p f a q; f f 1 p p and f f 1 p. So it cannot be f f 1 I FCD im f. Corollary 120. A binary relation is a monovalued funcoid iff it is a function. Because I,J P : f 1 (I J)= f 1 I F f 1 J is true for a binary relation f if and only if it is a function. Remark 121. This corollary can be reformulated as follows: For binary relations the classic concept of monovaluedness and monovaluedness in the above defined sense of monovaluedness of a funcoid are the same T 0 -, T 1 - and T 2 -separable funcoids For funcoids can be generalized T 0 -, T 1 - and T 2 - separability. Worthwhile note that T 0 and T 2 separability is defined through T 1 separability. Definition 122. Let call T 1 -separable such funcoid f that for every α,β is true α β ({α}[f]{β}). Definition 123. Let call T 0 -separable such funcoid f that f FCD f 1 is T 1 -separable.

23 Reloids 23 Definition 124. Let call T 2 -separable such funcoid f that the funcoid f 1 f is T 1 -separable. For symmetric transitive funcoids T 1 - and T 2 -separability are the same (see theorem 12). Obvious 125. A funcoid f is T 2 -separable iff α β f {α} F f {β}= for every α,β Filter objects closed regarding a funcoid Definition 126. Let s call closed regarding a funcoid f such filter object A that f A A. This is a generalization of closedness of a set regarding an unary operation. Proposition 127. If I and J are closed (regarding some funcoid), S is a set of closed filter objects, then 1. I F J is a closed filter object; 2. F S is a closed filter object. Let denote the given funcoid as f. f (I F J) = f I F f J I F J, f F S F f S F S. Consequently the filter objects I F J and F S are closed. Proposition 128. If S is a set of filter objects closed regarding a complete funcoid, then the filter object F S is also closed regarding our funcoid. f F S= F f S F S where f is the given funcoid. 4 Reloids Definition 129. I will call a reloid a filter object on the set of binary relations. Reloids are a generalization of uniform spaces. Also reloids are generalization of binary relations (the set of binary relations is a subset of the set of reloids, I will call discrete these reloids which are binary relations). Definition 130. The reverse reloid of a reloid f is defined by the formula upf 1 = { F 1 F upf 1}. Reverse reloid is a generalization of conjugate quasi-uniformity. I will denote RLD either the set of reloids or the category of reloids (defined below), dependently on context. 4.1 Composition of reloids Definition 131. Composition of reloids is defined by the formula g f = RLD {G F F upf,g upg}. Composition of reloids is a reloid. Theorem 132. (h g) f =h (g f) for every reloids f, g, h. For two nonempty collections A and B of sets I will denote A B ( K A L B:L K) ( K B L A:L K).

24 24 Section 4 It is easy to see that is a transitive relation. I will denote B A={L K K A,L B}. Let first prove that for every nonempty collections of relations A, B, C A B A C B C. Suppose A B and P A C that is K A and M C such that P =K M. K B:K K because A B. We have P =K M B C. Obviously P P. So for every P A C exist P B C such that P P ; vice versa is analogous. So A C B C. up((h g) f) up(h g) upf, up(h g) (uph) (upg). By proven above up((h g) f) (uph) (upg) (upf). Analogously up(h (g f)) (uph) (upg) (upf). So up((h g) f) up(h (g f)) what is possible only if up((h g) f)=up(h (g f)). Theorem f f = RLD {F F F upf}; 2. f 1 f = RLD { F 1 F F upf } ; 3. f f 1 = RLD { F F 1 F upf }. I will prove only (1) and (2) because (3) is analogous to (2). 1. Enough to show that F,G upf H upf:h H G F. To prove it take H =F G. 2. Enough to show that F,G up f H up f:h 1 H G 1 F. To prove it take H = F G. Then H 1 H =(F G) 1 (F G) G 1 F. Conjecture 134. If f, g, h are reloids then 1. f (g RLD h)=f g RLD f h; 2. (g RLD h) f =g f RLD h f. Conjecture 135. If f and g are reloids, then g f = RLD { G F F atoms RLD f,g atoms RLD g }. 4.2 Direct product of filter objects In theory of reloids direct product of filter objects A and B is defined by the formula A RLD B= def F {A B A upa,b upb}. Obvious 136. A RLD B=A B for every sets A and B. Theorem 137. A RLD B= F { a RLD b a atoms F A, b atoms F B } for every A,B F. Obviously A RLD B F { a RLD b a atoms F A, b atoms F B } Reversely, let K up { F a RLD b a atoms F A, b atoms F B }. Then K up(a RLD b) for every a atoms F A, b atoms F B; K X a RLD Y b for some X a up a, Y b upb; K { X a Y b a atoms F A, b atoms F B } = { X a a atoms F A } { Y b b atoms F A } A B where A upa, B upb; K up(a RLD B). Theorem 138. (A 0 RLD B 0 ) RLD (A 1 RLD B 1 )=(A 0 RLD A 1 ) RLD (B 0 RLD B 1 ) for every A 0, A 1,B 0,B 1 F.

25 Reloids 25 (A 0 RLD B 0 ) RLD (A 1 RLD B 1 ) = RLD { P Q P up(a 0 RLD B 0 ),Q up(a 1 RLD B 1 ) } = RLD {(A 0 B 0 ) (A 1 B 1 ) A 0 upa 0, B 0 upb 0, A 1 upa 1,B 1 upb 1 } = RLD {(A 0 A 1 ) (B 0 B 1 ) A 0 upa 0, B 0 upb 0, A 1 upa 1,B 1 upb 1 } = RLD {K L K up(a 0 A 1 ),L up(b 0 B 1 )} = (A 0 RLD A 1 ) RLD (B 0 RLD B 1 ). Theorem 139. If S PF 2 then RLD { A RLD B (A;B) S } = F doms RLD F ims. Let P = F doms, Q= F ims; l= RLD { A RLD B (A;B) S }. P RLD Q l is obvious. Let F up(p RLD Q). Then exist P upp and Q upq such that F P Q. P =P 1 P n where P i up doms and Q=Q 1 Q m where Q i up ims. P Q= i,j (P i Q i ). P i Q i A RLD B for some (A;B) S. P Q= i,j (P i Q i ) l. So F upl. Conjecture 140. If A F then A RLD is a complete homomorphism of the lattice F to a complete sublattice of the lattice RLD, if also A then it is an isomorphism. Definition 141. I will call a reloid convex iff it is a union of direct products. Example 142. Non-convex reloids exist. Let a is a non-trivial atomic f.o. Then (=) a is non-convex. This follows from the fact that only direct products which are below (=) are direct products of atomic f.o. and (=) a is not their join. I will call two filter objects isomorphic when the corresponding filters are isomorphic (in the sense defined in [14]). Theorem 143. The reloid {a} RLD F is isomorphic to the filter object F for every a. Consider B={a} and f ={(x;(a;x)) x }. Then f is a bijection from to B. If X upf then f X B and f X={a} X up({a} RLD F). For every Y up({a} RLD F) PB we have Y ={a} X for some X upf and thus Y = f X. So f up F P = f up F is a bijection from upf P to up({a} RLD F) PB. We have upf P and up({a} RLD F) PB directly isomorphic and thus upf is isomorphic to up({a} RLD F). 4.3 Restricting reloid to a filter object. Domain and image Definition 144. I call restricting a reloid f to a filter object A as f RLD A =f RLD (A RLD ). Definition 145. Domain and image of a reloid f are defined as follows: domf = F dom upf; imf = F im upf. Proposition 146. f A RLD B domf A imf B.