Funcoids and Reloids

Transcription

1 Funcoids and Reloids by Victor Porton March 29, 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 Used concepts, notation and statements Filters Earlier works 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 T 1 - and T 2 -separable funcoids Filter objects closed regarding a funcoid This document has been written using the GNU TEX MAC S text editor (see 1

2 2 Section 1 4 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 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 [5]. Theorem number clashes may appear due editing both of these manuscripts. 1.2 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 [5] where are considered filters on arbitrary posets). Note that I do not require filters to be proper.

3 Partially ordered dagger categories 3 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 [5] for formal definition of filter objects in the framework of ZF. Filters (and filter objects) are studied in the work [5]. Prior reading of [5] is needed to 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 B. Proposition 2. Let S is a generalized filter base. If A 1, C S:C A 1 A n.,a n S (n N), then 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. 1.3 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. Some references to predecessors: In [1] and [2] are studied semi-uniformities and proximities. [3] and [4] contains recent progress in quasi-uniform spaces. 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.

4 4 Section 2 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. 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 Monovalued and entirely defined 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.

5 Funcoids 5 Remark 15. Easy to show that this is a generalization of monovalued and entirely defined binary relations as morphisms of the category Rel. Definition 16. For a given partially ordered dagger category C the category of monovalued (entirely defined) 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) 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. 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. 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.

6 6 Section 3 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). 3.2 Basic definitions Definition 17. 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 18. (α;β) = def α for a funcoid (α;β). Definition 19. (α;β) 1 =(β;α) for a funcoid (α;β). Proposition 20. If f is a funcoid then f 1 is also a funcoid. Follows from symmetry in the definition of funcoid. Obvious 21. (f 1 ) 1 =f for a funcoid f. Definition 22. 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 23. X[f]Y Y F f X X F f 1 Y for every filter objects X, Y and funcoid f. Obvious 24. [ 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 [5], thus f = g.

7 Funcoids 7 Proposition 26. 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 [5]) { 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 [5]) { 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 27. Composition of funcoids is defined by the formula (α 2 ;β 2 ) (α 1 ;β 1 )=(α 2 α 1 ;β 1 β 2 ). Proposition 28. If f, g are funcoids then g f is funcoid. Let f =(α 1 ;β 1 ), g=(α 2 ;β 2 ). 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 29. g f = g f for every funcoids f and g. Proposition 30. (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 31. (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 32. 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 equalities we get X[f]Y X upx,y upy:x[f]y. 1. Y F f X 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.

8 8 Section 3 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; for every filter objects X, Y. X[f]Y X upx,y upy:x δy (3) 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 α(i J) K F (αi F αj) (K F αi) F (K F αj) K F αi K F αj KδI KδJ.

9 Funcoids 9 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 :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 proximity structures. Definition 34. 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 35. 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 [5]. 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 [5]) but we simply will (informally) assume that discrete funcoids can be equated with binary relations. I will denote FCD the set of funcoids or the category of funcoids (see below) dependently on context. 3.4 Lattice of funcoids Definition 36. f g= def [f] [g] for f,g FCD. Thus FCD is a poset. Definition 37. I will call the filtrator of funcoids (see [5] for the definition of filtrators) the filtrator (FCD; P 2 ). Conjecture 38. The filtrator of funcoids is: 1. with separable core; 2. with co-separable core. Theorem 39. 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}. 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)

10 10 Section 3 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 52 in [5]). 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 40. For every funcoids f and g and a filter object X 1. f FCD g X = f X F g X ; 2. [ f F 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. 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 41. [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 42. 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 43. 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.

11 Funcoids 11 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 44. Let A F. The identity funcoid I A =(A F ;A F ). Proposition 45. The identity funcoid is a funcoid. We need to prove that (A F X) F Y (A F Y) F X what is obvious. Obvious 46. (I A ) 1 =I A. Obvious 47. X[I A ]Y A F X F Y for anyx,y F. Definition 48. I will define restricting of a funcoid f to a filter object A by the formula f A = def f I A. Obviously the last definition does not contradict to the previous. Definition 49. 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 50. 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 51. X F domf f X for every f FCD, X F. X F domf X F f 1 F f X f X. Corollary 52. domf = F { a a atoms F, f a }. This follows from 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 ;A;A).

12 12 Section 3 To prove that it is really a category is trivial. 3.8 Specifying funcoids by functions or relations on atomic filter objects Theorem 53. 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. Theorem 54. x atoms F X,y atoms F Y:x[f]y. 1. 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.

13 Funcoids 13 Let s prove the reverse of (5): F F α atoms F upa = F F atoms F α 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; analogously Xδ (I J) Xδ I Xδ J. Let s continue δ till a funcoid f (by the theorem 25): The reverse of (7) implication is trivial, so X[f]Y X upx,y upy:x δ Y 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 proof of the following theorem: Theorem 55. 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 54, 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}.

14 14 Section 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 56. 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 57. A FCD B is really a funcoid and A FCD B { B ifx X = F A ; ifx F A=. Obvious. Obvious 58. A B=A FCD B for sets A and B. Proposition 59. 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 X,Y F:(X[f]Y X F A Y F B ); consequently f A FCD B. The following theorem gives a formula for calculating an important particular case of intersection on the lattice of funcoids: Theorem 60. f FCD (A FCD B)=I B f I A for every f FCD and A,B F. h= def I B f I A. For every X F h X = I B f I A 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 f I A X = h X, g h. So h=f FCD (A FCD B). Corollary 61. f A =f (A FCD ) for every f FCD and A F. f FCD (A FCD )=I f I A =f I A =f A. Corollary 62. 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 f I A I B f I A B FCD f (A F ) B F f A A[f]B. Corollary 63. The filtrator of funcoids is star-separable. The set of direct products of sets is a separation subset of the lattice of funcoids. Theorem 64. If S PF 2 then FCD { A FCD B (A;B) S } = F doms FCD F ims.

15 Funcoids 15 If x atoms F then by the theorem 55 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 65. (A 0 FCD B 0 ) FCD (A 1 FCD B 1 )=(A 0 FCD A 1 ) FCD (B 0 FCD B 1 ) for every A 0, A 1,B 0,B 1 F. (A 0 FCD B 0 ) FCD (A 1 FCD B 1 )= { } F A 0 FCD B 0,A 1 FCD B 1 what is by the last theorem equal to (A 0 FCD A 1 ) FCD (B 0 FCD B 1 ). Theorem 66. 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. 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 67. If a is an atomic filter object, f FCD then f a =a FCD f a. Let X F. X F a f a X = f a, X F a= f a X = Atomic funcoids Theorem 68. A funcoid is an atom of the lattice of funcoids iff it is direct product of two atomic filter objects.

16 16 Section 3. 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 an atomic funcoid f =a FCD b.. Let a,b atoms F, f FCD. If b F f a= then (a[f]b), f F (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 69. The lattice of funcoids is atomic. Let f is a non-empty funcoid. Then dom f, thus by the theorem 46 in [5] exists a atoms F dom f. So f a thus exists b atoms f a. Finally the atomic funcoid a FCD b f. Theorem 70. 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 = 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 [5]. Corollary 71. The lattice of funcoids is: 1. separable; 2. atomically separable; 3. conforming to Wallman s disjunction property. By the theorem 22 in [5]. Remark 72. For more ways to characterize (atomic) separability of the lattice of funcoids see [5], subsections Separation subsets and full stars and Atomically separable lattices. Corollary 73. The lattice of funcoids is an atomistic lattice. Let f is a funcoid. Suppose contrary to the statement to be proved that F atoms FCD f f. Then exists a atoms FCD f such that a F F atoms FCD f = what is impossible. Proposition 74. atoms FCD (f F 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 75. For every f,g,h FCD, R PFCD 1. f FCD (g FCD h)=(f FCD g) FCD (f FCD h);

17 Funcoids 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. Corollary 76. The lattice of funcoids is co-brouwerian. The next proposition is one more (among the theorem 42) generalization for funcoids of composition of relations. Proposition 77. 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 42). Conjecture 78. The set of discrete funcoids is the center of the lattice of funcoids Complete funcoids Definition 79. I will call co-complete such a funcoid f that X P : f X P. Remark 80. I will call generalized closure such a function α P P that 1. α = ; 2. I,J P :α(i J)=αI αj. Obvious 81. A funcoid f is co-complete iff f P is a generalized closure. Remark 82. 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 83. I will call a complete funcoid a funcoid whose reverse is co-complete.

18 18 Section 3 Theorem 84. 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 [5] 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 52 in [5]). (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 85. 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 86. 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 87. 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 FCDR X = { f X f R} P (used the theorem 39). Corollary 88. If R is a set of binary relations then FCD R= R. From two last theorems.

19 Funcoids 19 Theorem 89. 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 X= ifx= B if X A 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; f X; consequently FCD {g(a;b) A P,B up f A}=f. Conjecture 90. If f, g are (co-)complete funcoids then f FCD g is a (co-)complete funcoid. Its consequence: Conjecture 91. If f, g are discrete funcoids then f FCD g is a discrete funcoid. An equivalent conjecture: Conjecture 92. If f,gare discrete funcoids thenf FCD g=f g. Conjecture 93. 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: Conjecture 94. 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 95. ComplFCD and CoComplFCD are closed regarding composition of funcoids. Proposition 96. ComplFCD and CoComplFCD (with induced order) are complete lattices. Follows from the corollary Completion of funcoids Theorem 97. 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 [5] and the corollary 88 and theorem 89. Definition 98. Completion of a funcoid f is the complete funcoid Compl f defined by the formula Complf {α}= f {α} for α. Definition 99. Co-completion of a funcoid f is defined by the formula CoComplf =(Complf 1 ) 1. Obvious 100. Complf f and CoComplf f for every funcoid f. Proposition 101. The filtrator (FCD; ComplFCD) is filtered.

20 20 Section 3 Because the filtrator (FCD; P 2 ) is filtered. Theorem 102. Complf = Cor (FCD;ComplFCD) f = Cor (FCD;ComplFCD) f. Cor (FCD;ComplFCD) f = Cor (FCD;ComplFCD) f since (the theorem 26 in [5]) the filtrator (FCD; ComplFCD) is filtered and with join closed core (the theorem 87). 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 103. 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 104. 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. 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 [5]). 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 105. ComplFCD is an atomistic lattice. Let f ComplFCD. f X = F { f {x} x X} = F { f {x} {x} x X } = F { f {x} X x X }, thus f = FCD { f {x} x X }. It is trivial that every f {x} is a union of atoms of ComplFCD. Theorem 106. A funcoid is complete iff it is a join (on the lattice FCD) of atomic complete funcoids. Follows from the theorem 87 and the previous theorem. Corollary 107. ComplFCD is join-closed. Theorem 108. 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. Lemma 109. Co-completion of a complete funcoid is complete. Let f is a complete funcoid.

21 Funcoids 21 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 110. 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 111. 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 im f. Obvious 112. A morphism (f;a;b) of the category of funcoids is monovalued iff the funcoid f is monovalued. Theorem 113. 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. (4) (1). f 1 a F f 1 b = 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. This is possible only when f f 1 I Dst f. (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. So it cannot be f f 1 I Dst f. Corollary 114. 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 115. 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.

22 22 Section 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 116. Let call T 1 -separable such funcoid f that for every α,β is true α β ({α}[f]{β}) Definition 117. Let call T 0 -separable such funcoid f that f FCD f 1 is T 1 -separable. Definition 118. 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) Filter objects closed regarding a funcoid Definition 119. 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 120. 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 121. If S is a set of closed regarding a complete funcoid filter objects, 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 122. 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 123. 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 124. Composition of reloids is defined by the formula g f = RLD {G F F upf,g upg}.

23 Reloids 23 Composition of reloids is a reloid. Theorem 125. (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). 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 127. 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. 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}. Theorem 128. 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 129. (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.

24 24 Section 4 (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 130. 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. F upl. Conjecture 131. 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 132. I will call a reloid convex iff it is a union of direct products. Example 133. 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 [5]). Theorem 134. 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 135. I call restricting a reloid f to a filter object A as f A =f RLD (A RLD ). Definition 136. Domain and image of a reloid f are defined as follows: domf = F dom upf; imf = F im upf. Proposition 137. f A RLD B domf A imf B.

25 Reloids 25. Follows from dom(a RLD B) A im(a RLD B) B.. domf A A upa F upf: domf A. Analogously imf B B upb G upf: img B. Let dom f A im f B, A upa, B upb. Then exist F up f, G up f such that domf A img B. Consequently F G upf, dom(f G) A, im(f G) B that is F G A B. So exists H up f such that H A B for every A upa,b upb. So f A RLD B. Definition 138. I call identity reloid for a filter object A the reloid I A = def (=) A. Theorem 139. I A = F {I A A upa} where I A is the identity relation on a set A. Let K up F {I A A up A}, then exists A upa such that K I A. Then I A = (=) A =(=) RLD (A ) (=) (A )=I A K; K upi A. Reversely let K up I A = up((=) RLD (A RLD )), then exists A upa such that K up((=) (A ))=upi A up F {I A A upa}. Proposition 140. I A 1 =I A. Follows from the previous theorem. Theorem 141. f A =f I A for every reloid f and filter object A. We need to prove that f RLD (A ) = f RLD {I A A upa}. f RLD {I A A upa}= RLD {F I A F up f,a upa}= RLD {F A F up f,a upa}= RLD {F (A ) F up f, A upa} = RLD {F F up f} RLD {A A upa} = f RLD (A RLD ). Theorem 142. (g f) A =g (f A ) for every reloids f and g and filter object A. (g f) A =(g f) I A =g (f I A )=g (f A ). Theorem 143. f RLD (A RLD B)=I B f I A for every reloid f and filter objects A and B. f RLD (A RLD B) = f RLD (A RLD ) RLD ( RLD B) = f A RLD ( B) = f I A RLD ( B) = ((f I A ) 1 RLD ( RLD B) 1 ) 1 = ((I A f 1 ) RLD (B RLD )) 1 = (I A f 1 I B ) 1 =I B f I A. 4.4 Category of reloids I will define the category RLD of reloids: 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 reloid 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 ;A;A). To prove that it is really a category is trivial Monovalued reloids Following the idea of definition of monovalued morphism let s call monovalued such a reloid f that f f 1 I im f.