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Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces

Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy... axioms Article Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces Jirí ˇ Mocko ˇ r ˇ Institute for Research and Applications of Fuzzy Modelling, NSC IT4Innovations, University of Ostrava, 70200 Ostrava, Czech Republic; mockor@osu.cz Received: 5 May 2020; Accepted: 27 May 2020; Published: 2 June 2020 Abstract: Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Cech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined. Keywords: categories with relational morphisms; topological and pre-topological structures; functors 1. Introduction In fuzzy set theory many structures are used, which are based on various modifications of topological operators. These structures include variants of fuzzy topological spaces, fuzzy rough sets, fuzzy approximation spaces, fuzzy closure operators, fuzzy pretopological operators and their dual terms, such as fuzzy interior operators. For examples of these structures see, e.g., [1–8]. Although these structures are generally based on the common basis of different topological spaces and their modifications, the tools and language they use are often very different and it is difficult to identify deeper relationships between the different types of these structures. One way to effectively identify and describe the relationships among these objects is to use category theory methods. Description of transformation processes between objects of one type using the category theory language ensure morphisms and functors. The significance of morphisms and functors in categories is that morphisms between objects and functors between categories actually represent the processes of transforming objects of one type into objects of another type, whereby these transformations define not only how to create an object of one type from another type, but also how to transform relationships between objects of one type into relationships between objects of the other type. In the paper [9], the relationships among some categories of fuzzy structures related to topological operators were discussed. The common feature of these categories was that the morphisms in these categories were based on mappings between the underlying sets of corresponding objects. Recently, however, a number of results have emerged in the theory of fuzzy sets, which are based on the application of fuzzy relations as morphisms in suitable categories. A typical example of that use of fuzzy relations is the category of sets as objects and L-valued fuzzy relations between sets as Axioms 2020, 9, 63; doi:10.3390/axioms9020063 www.mdpi.com/journal/axioms Axioms 2020, 9, 63 2 of 17 morphisms. This category is frequently used in approximation functors, which represent various approximations of fuzzy sets. The approximation defined by a fuzzy relation, which can be described as a functor between appropriate categories, was defined for the first time by Goguen [10], when he introduced the notion of the image of a fuzzy set under a fuzzy relation. Many examples using explicitly or implicitly approximation functors defined by various types of fuzzy relations can be found in rough fuzzy sets theory and many others (see, e.g., [5,11–13]). In this paper we want to significantly expand the information on the relationships between categories of topological L-operators expressing the concept of “proximity” in different ways, while in accordance with the current trend of using L-fuzzy relations, the morphisms in these categories are L-fuzzy relations with various properties. In detail, we consider 15 categories and some of their subcategories of Cech closure or interior L-valued operators, categories of L-fuzzy pretopological and L-fuzzy co-pretopological operators, the category of L-valued fuzzy relations, categories of upper and lower F-transforms and the category of spaces with fuzzy partitions, where L is a complete residuated lattice or MV-algebra, and where morphisms in all of these categories are L-valued fuzzy relations. The main results of this paper are Theorems 1 and 2, which prove the existence of 23 functors among these categories, including how these functors combine with each other. It follows from these theorems that the key category between the above categories is the category of spaces with fuzzy partitions with special fuzzy relations as morphisms. As follows from the commutative diagrams in both Theorems, any space with a fuzzy partition can be transformed into an L-fuzzy relation, Cech closure or interior operator, L-fuzzy pretopological or co-pretopological operators or strong-Alexandroff variants of these operators. The structure of the paper is as follows. In Section 1 we repeat basic properties of residuated lattices and we recall definitions of principal structures representing various concepts of proximity. In Section 2, which represents the principal content of the paper, we introduce 15 new categories, whose objects are various types of proximity structures and whose morphisms are various types of relations between these structures. The main result are then two theorems identifying functors among these categories, which in fact represent the transformation processes among the individual structures. 2. Preliminaries In this section we repeat basic properties of residuated lattices and recall definitions of principal structures representing various concepts of proximity, which are frequently used in L-valued fuzzy set theory, including interior and closure operators, pretopologies and co-pretopologies, and fuzzy partitions, sometimes with additional special properties. We refer to [14,15] for additional details regarding residuated lattices. Definition 1. A residuated lattice L is an algebra L = (L,^,_, ,!, 0, 1) such that: 1. (L,^,_, 0, 1) is a bounded lattice with the least element 0 and the greatest element 1; 2. (L, , 1) is a commutative monoid, and 3. 8a, b, c 2 L, a b  c () a  b ! c. A residuated lattice (L,^,_, ,!, 0, 1) is complete if it is complete as a lattice. The following is the derived unary operation of negation :: :a = a ! 0, A residuated lattice L is called an MV-algebra if it satisfies (a ! b) ! b = a_ b. In a MV-algebra the following identities hold: ::a = a, :_ a = ^:a , a b = :(a ! :b). i i Axioms 2020, 9, 63 3 of 17 Unless otherwise stated, throughout this paper, a complete residuated lattice L = (L,^,_, ,! , 0, 1) will be fixed and for simplicity, instead of L we use only L. Let X be a nonempty set and L a set of all L-fuzzy sets (=L-valued functions) of X. For all a 2 L, a(x) = a is a constant L-fuzzy set on X. For all u 2 L , the core(u) is a set of all elements x 2 X, X X X such that u(x) = 1. An L-fuzzy set u 2 L is called normal, if core(u) 6= Æ. An L-fuzzy set c 2 L fyg is a singleton of y 2 X, if it has the following form: 1, if x = y, 8x 2 X, c (x) = fyg 0, otherwise. In the next definitions we repeat basic definitions of the L-valued operators that were mentioned above. These operators are very useful tools in several areas of mathematical structures with direct applications, both mathematical (e.g., topology, logic) and outside of mathematics (e.g., data mining, knowledge representation). In fuzzy set theory, several particular cases as well as general theory of interior and closure operators, which operate with fuzzy sets (so-called fuzzy interior or closure operators), are studied. The original notions of a Kuratowski closure and interior operators were introduced in several papers, see [1–3,16,17]. In this paper we use a more general form of these operators, called Cech operators or preclosure operators, where the idempotence of operators is not required. X X X Definition 2. The map i : L ! L is called a Cech (L-fuzzy) interior operator, if for every a, u, v 2 L , it fulfils: 1. i(a) = a, 2. i(u)  u, 3. i(u^ v) = i(u)^ i(v). X X ˇ ˇ A Cech interior operator i : L ! L is said to be a strong Cech–Alexandroff interior operator, if: ^ ^ i(a ! u) = a ! i(u) and i( u ) = i(u ), j j j2 J j2 J and is said to be a Kuratowski interior operator if ii(u) = i(u). X X X Definition 3. The map c : L ! L is called a Cech (L-fuzzy) closure operator, if for every a, u, v 2 L , it fulfils: 1. c(a) = a, 2. c(u)  u, 3. c(u_ v) = c(u)_ c(v). X X ˇ ˇ A Cech closure operator c : L ! L is said to be a strong Cech–Alexandroff closure operator, if _ _ c(a u) = a c(u) and c( u ) = c(u ), j j j2 J j2 J and is said to be a Kuratowski closure operator if cc(u) = c(u). We recall the notion of an L-fuzzy pretopological space and L-fuzzy co-pretopological space as it has been introduced in [8]. Definition 4. An L-fuzzy pretopology on X is a set of functions t = f p 2 L : x 2 Xg, such that for all u, v 2 L , a 2 L and x 2 X, Axioms 2020, 9, 63 4 of 17 1. p (a) = a, 2. p (u)  u(x), 3. p (u^ v) = p (u)^ p (v). x x x An L-fuzzy pretopological space (X, t) is said to be a strong Cech–Alexandroff L-fuzzy pretopological space, if: ^ ^ p (a ! u) = a ! p (u) and p ( u ) = p (u ). x x x x j j j2 J j2 J x L Definition 5. An L-fuzzy co-pretopology on X is a set of functions h = f p 2 L : x 2 Xg, such that for all u, v 2 L , a 2 L and x 2 X, 1. p (a) = a, 2. p (u)  u(x), x x x 3. p (u_ v) = p (u)_ p (v). An L-fuzzy co-pretopological space (X, t) is said to be a strong Cech–Alexandroff L-fuzzy co-pretopological space, if: _ _ x x x x p (a u) = a p (u) and p ( u ) = p (u ). j j j2 J j2 J Finally, we recall the notion of an L-fuzzy partition (see [18,19]). Definition 6. A set A of normal fuzzy sets f A : a 2 Lg in X is an L-fuzzy partition of X, if: 1. The corresponding set of ordinary subsets fcore( A ) : a 2 Lg is a partition of X, and 2. Core( A ) = core( A ) implies A = A . a b a b Instead of the index set L from A we use jAj. We use the notion of powerset maps defined by a fuzzy relation, which was first defined in [10]. ! X Y Y X If R : X Y ! L is an L-fuzzy relation, then the powerset maps R : L ! L and R : L ! L are defined by: X ! t 2 L , y 2 Y, R (t)(y) = t(x) R(x, y), x2X s 2 L , x 2 X, R (s)(x) = R(x, y) ! s(y). y2Y 3. Relational Categories of L-Valued Topological Objects and Functors among Them As we mentioned in the Introduction, in the paper [9], several functors among categories based on fuzzy topological structures were discussed, whose common feature was that morphisms in these categories were mappings between corresponding underlying sets. In this section we want to significantly expand the information on the relationships between categories of topological L-operators expressing the concept of “proximity”, while in accordance with the current trend of using L-fuzzy relations, the morphisms in these categories are L-fuzzy relations with various properties. The categories we will deal with have the objects defined in Section 2. Instead of classical maps between sets we use special fuzzy relations as morphisms. Axioms 2020, 9, 63 5 of 17 Definition 7. In what follows, we denote sets by X, Y, and by the composition of morphisms from the standard category Set. 1. The category RCInt is defined by: X X (a) Objects are pairs (X, i), where i : L ! L is a Cech L-fuzzy interior operator (Definition 2), (b) R : (X, i) ! (Y, j) is a morphism, if R : X Y ! L is an L-fuzzy relation and i.R  R .j. 2. The category RCClo is defined by: X X (a) Objects are pairs (X, c), where c : L ! L is a Cech L-fuzzy closure operator (Definition 3), (b) R : (X, c) ! (Y, d) is a morphism, if R : X Y ! L is an L-fuzzy relation, and ! ! R .c  d.R . 3. The category RFPreTop is defined by: (a) Objects are L-fuzzy pretopological spaces (X, t) (Definition 4), X Y L L (b) R : (X, t) ! (Y, s) is a morphism, where t = f p 2 L : x 2 Xg, s = fq 2 L : y 2 Yg, x y if R : X Y ! L is an L-fuzzy relation, and for all x 2 X, (R(x, z) ! q )  p .R . z x z2Y 4. The category RFcoPreTop is defined by: (a) Objects are L-fuzzy co-pretopological spaces (X, t) (Definition 5), X Y x L y L (b) R : (X, t) ! (Y, s) is a morphism, where t = f p 2 L : x 2 Xg, s = fq 2 L : y 2 Yg, if R : X Y ! L is an L-fuzzy relation, and for all x 2 X, y 2 Y, y ! x q .R  p R(x, y). 5. The category RFRel is defined by: (a) Objects are pairs (X, r), where r is a reflexive L-fuzzy relation on X, (b) R : (X, r) ! (Y, s) is a morphism, if R : X Y ! L is an L-fuzzy relation, and s R  R r, where  is the composition of L-fuzzy relations. 6. The category RSFP is defined by: (a) Objects are sets with an L-fuzzy partition (X,A), (Definition 6), (b) (R,S) : (X,A) ! (Y,B) is a morphism if R : X Y ! L and S : jAjjBj ! L are L-fuzzy relations such that i. For each a 2 jAj, b 2 jBj, t 2 core( A ), z 2 core(B ), a b S(a, b) = R(t, z), Axioms 2020, 9, 63 6 of 17 ii. For each a 2 jAj, b 2 jBj, x 2 X, y 2 Y, A (x) S(a, b)  B (y) R(x, y), (1) a b A (x) R(x, y)  B (y) S(a, b). (2) 7. The category RFTrans is defined by: X jAj (a) Objects are upper F-transforms F : L ! L , where (X,A) are sets with L-fuzzy partitions X,A and F (u)(a) = u(x) A (x), where u 2 L , a 2 jAj, x2X a X,A " " (b) (R,S) : F ! F is a morphism if R : X  Y ! L and S : jAjjBj ! L are L-fuzzy X,A Y,B relations and for each a 2 jAj, b 2 jBj, t 2 core( A ), z 2 core(B ), S(a, b) = R(t, z), " " ! ! S .F  F .R . X,A Y,B hold. 8. The category RFTrans is defined by: X jAj (a) Objects are lower F-transforms F : L ! L , where (X,A) are sets with L-fuzzy partitions, X,A where F (u)(a) = A (x) ! u(x), for u 2 L , a 2 jAj, x2X X,A # # (b) (R,S) : F ! F is a morphism if R : X  Y ! L and S : jAjjBj ! L are L-fuzzy X,A Y,B relations and and for each a 2 jAj, b 2 jBj, t 2 core( A ), z 2 core(B ), a b S(a, b) = R(t, z), # # S .F  F .R . Y,B X,A hold. In the next theorems we will use the following subcategories of categories from Definition 7. Definition 8. The following full subcategories of categories from Definiton 7 will be used: 1. The full subcategory RTFRel of RFRel with transitive L-fuzzy relations as objects, 2. The full subcategory RsACClo of RCClo with strong Cech–Alexandroff L-fuzzy closure operators as objects, 3. The full subcategory RKsACClo of RCClo with Kuratowski strong Cech–Alexandroff L-fuzzy closure operators, 4. The full subcategory RsACInt of RCInt with strong Cech–Alexandroff L-fuzzy interior operators as objects, 5. The full subcategory RKsACInt of RCInt with Kuratowski strong Cech–Alexandroff L-fuzzy interior operators, 6. The full subcategory RsAFPreTop of RFPreTop with strong Cech– Alexandroff L-fuzzy pretopological spaces as objects, 7. The full subcategory RsAFcoPreTop of RFcoPreTop with strong Cech–Alexandroff L-fuzzy co-pretopological spaces. The main results of the paper are the following two theorems, in which we will define 23 functors to describe relationships between pairs of categories and some of their subcategories from Definitions 7 and 8. < Axioms 2020, 9, 63 7 of 17 Theorem 1. Let L be a complete residuated lattice. There exist functors such that the following diagram of these functors commutes, RFcoPreTop RCClo [ [ RFTrans > RsAFcoPreTop RsACClo < RKsACClo " ^ ^ U ^ 1 1 W M M K K _ _ RSFP > RFRel < RTFRel # 1 # > RFTrans > RsAFPreTop RsACInt _ _ RFPreTop RCInt 1 1 1 1 where (F, F ), (F , F ), ( M, M ) and (K, K ) are pairs of inverse functors. x L Proof. (1) Let R : (X, t) ! (Y, s) be a morphisms in RFcoPrTop, where t = f p 2 L : x 2 Xg, y L s = fq 2 L : y 2 Yg. The functor F : RFcoPrTop ! RCClo is defined by: X x F(X, t) = (X, c), 8u 2 L , x 2 X, c(u)(x) = p (u), Y y F(Y, s) = (Y, d), 8v 2 L , y 2 Y, d(v)(y) = q (v), F(R) = R. According to [5], (X, c) is an object of RCClo. For arbitrary u 2 L , y 2 Y, we have: _ _ _ ! x y ! R .c(u)(y) = c(u)(x) R(x, y) = p (u) R(x, y)  q (R (u)) = x2X x2X x2X y ! ! q (R (u)) = d.R (u)(y). Therefore, F is defined correctly and R is a morphism (X, c) ! (Y, d) in RCClo. (2) Let R : (X, c) ! (Y, d) be a morphisms in RCClo. According to the same reference [5], the object function F : RCClo ! RFcoPreTop such that: 1 x x F (X, c) = (X, t), t = f p : x 2 Xg, p (u) = c(u)(x), X 1 for u 2 L , x 2 X, is defined correctly. We set F (R) = R. Then we obtain: y ! ! ! q .R (u) = d.R (u)(y)  R .c(u)(y) = c(u)(z) R(z, y) = z2X z x p (u) R(z, y)  p (u) R(x, y). z2X 1 1 Hence, R is a morphism in RFcoPreTop and F is a functor. It is clear that F and F are inverse functors. < Axioms 2020, 9, 63 8 of 17 (3) Let R : (X, r) ! (Y, s) be a morphism in the category RFRel and let the object function M : RFRel ! RsACClo be defined by: X ! M(X, r) = (X, c),8u 2 L , x 2 X, c(u)(x) = R (u)(x), Y " M(Y, s) = (Y, d),8v 2 L , y 2 Y, d(v)(y) = s (v)(y). According to results from [6] and many others, (X, c) 2 sACClo and the object function is defined correctly. For M(R) = R we obtain: _ _ _ R .c(u)(y) = c(u)(x) R(x, y) = u(z) r(z, x) R(x, y) = x2X x2X z2X _ _ (R r)(z, y) u(z)  (s R)(z, y) u(z) = z2X z2X _ _ _ ! " ! R(z, t) s(t, y) u(z) = R (u)(t) s(t, y) = s .R (u)(y) = z2X t2Y t2Y d.R (u)(y). Hence, R is a morphism in RsACClo and M is a functor. (4) In the proof we use the following identity, which was proven in [20], and which allows to express a general L-fuzzy set u 2 L in the following form: u = u(x) c , (3) fxg x2X where a 2 L is the constant function with the value a. Let R : (X, c) ! (Y, d) be a morphism in RsACClo and let the object function M : RsACClo ! RFRel be defined by: 1 0 0 X 0 M (X, c) = (X, r),8x, x 2 X, r(x, x ) = c(c )(x ), fxg 1 0 0 Y 0 M (Y, d) = (Y, s),8y, y 2 Y, s(y, y ) = d(c )(y ). fyg It is clear that r and s are reflective fuzzy relations. We set M (R) = R. Then, for x 2 X, y 2 Y we obtain: _ _ 0 0 X 0 0 R r(x, y) = r(x, x ) R(x , y) = c(c )(x ) R(x , y) = fxg 0 0 x 2X x 2X ! X ! X ! X 0 Y R .c(c )(y)  d.R (c )(y) = d( R (c )(y ) c )(y) = fxg fxg fxg fy g y 2Y ! X 0 Y R (c )(y ) d(c )(y) = fxg fy g y 2Y _ _ X 0 0 Y c (x ) R(x , y) d(c )(y) = fxg fy g 0 0 y 2Y x 2X _ _ 0 Y 0 0 R(x, y ) d(c )(y) = R(x, y ) s(y , y) = s R(x, y), fy g y2Y y 2Y which follows from the fact that d is a strong Cech–Alexandroff closure. Hence, M (R) is a morphism 1 1 in the category RFRel and M is a functor. We prove that M and M are inverse functors. 1 0 Let (X, r) 2 RFRel. Then M .M(X, r) = (X, r ), where 0 0 X 0 ! X 0 X 0 r (x, x ) = c(c )(x ) = R (c )(x ) = c (z) r(z, x) = r(x, x ). fxg fxg fxg z2X Axioms 2020, 9, 63 9 of 17 1 0 1 X On the other hand, M.M (X, c) = (X, c ), where M (X, c) = (X, r), r(z, x) = c(c )(x), fzg and we obtain: _ _ 0 ! X c (u)(x) = R (u)(x) = u(z) r(z, x) = u(z) c(c )(x) = fzg z2X z2X c( u(z) c )(x) = c(u)(x). fxg z2X Therefore, M and M are inverse functors. (5) Let R : (X, r) ! (Y, s) be a morphism in the category RTFRel. The functor K : RTFRel ! RKsACClo is defined by: K(X, r) = M(X, r), K(R) = M(R). We prove that this definition is correct, i.e., M(X, r) = (X, c) and c is a Kuratowski closure operator. In fact, for u 2 L , x 2 X we have: ! ! cc(u)(x) = R (R (u))(x) = u(z) r(z, y) r(y, x) y,z2X u(z) r(z, x) = R (u)(x) = c(u)(x), y,z2X and the definition of K is correct. (6) Let R : (X, c) ! (Y, d) be a morphism in RKsACClo and let the functor K : RKsACClo ! RTFRel be defined by: 1 1 1 1 K (X, c) = M (X, c) = (X, r), K (R) = M (R). We prove that this definition is correct. In fact, for arbitrary u 2 L , according to (3), we obtain: _ _ X X c(u)(x) = c( u(z) c )(x) = u(z) c(c )(x) = fzg fzg z2X z2X u(z) r(z, x) = R (u)(x). z2x Hence, for arbitrary y 2 X and u = c , we obtain: fyg _ _ X ! ! X X cc(c )(x) = R (R (c ))(x) = c (z) r(z, t) r(t, y) = fyg fyg fyg z2X t2X r(y, t) r(t, x), t2X X ! X c(c )(x) = R (c )(x) = r(y, x). fyg fyg Because cc = c, it follows that r(y, x)  r(y, t) r(t, x). Therefore, r is a transitive relation. (7) Let (R,S) : (X,A) ! (Y,B) be a morphism in RSFP and let the object function W : RSFP ! RsAFcoPreTop be defined by: x L x W(X,A) = (X,f p 2 L : x 2 Xg), p (u) = u(t) A (x), w (t) t2X y L y W(Y,B) = (Y,fq 2 L : y 2 Yg), q (v) = v(s) B (y), w (s) s2Y Axioms 2020, 9, 63 10 of 17 where w (x) 2 jAj is the unique index such that x 2 core( A ). It is easy to see that the object w (x) function W is defined correctly. We set W(R,S) = R. For arbitrary x 2 X, y 2 Y and u 2 L , from the inequalitiy (2) we obtain: _ _ _ y ! ! q .R (u) = R (u)(z) B (y) = u(t) R(t, z) B (y) = w (z) w (z) B B z2Y z2Y t2X _ _ _ u(t) S(w (t), w (z)) B (y)  u(t) A (x) R(x, y) A B w (z) w (t) B A t2X t2X z2Y p (u) R(x, y) as follows from the identity S(w (x), w (y)) = R(x, y). A B Hence, R is a morphisms in RsAFcoPreTop and W is a functor. " " " (8) Let U : RSFP ! RFTrans be defined by U (X,A) = F , U (R,S) = (R,S). By a simple X,A " " computation it can be proven that the definition of U is correct and U is a functor. (9) Let (R,S) : (X,A) ! (Y,B) be a morphism in the category RSFP. We define T : RSFP ! RFRel by: 0 0 T(X,A) = (X, r), r(x, x ) = A (x ), w (x) 0 0 T(Y,B) = (Y, s), s(y, y ) = B (y ), w (y) T(R,S) = R. We prove that R is a morphism in the category RFRel. In fact, let x 2 X, y 2 Y. Then, according to Inequality (2), _ _ (R r)(x, y) = r(x, z) R(z, y) = A (z) R(z, y) w (x) z2X z2X _ _ S(w (x), w (t)) B (y) = R(x, t) s(t, y) = s R(x, y). A B w (t) t2Y t2Y Hence, T is a functor. " " (10) The functor Q : RFTrans ! RsAFcoPreTop can be simply defined by: " " Q (F ) = W(X,A), Q (R,S) = R. X,A (11) The functors F and F are restrictions of the functors F and F . It can be proven simply that these functors are mutually inverse. (12) Let (R,S) : (X,A) ! (Y,B) be a morphism in RSFP. We define V : RSFP ! RsAFPreTop by: V(X,A) = (X,f p 2 L : x 2 Xg), p (u) = F (u)(w (x)), x x A X,A V(Y,B) = (Y,fq 2 L : y 2 Yg), q (v) = F (v)(w (y)), y x Y,B V(R,S) = R. Axioms 2020, 9, 63 11 of 17 It can be proven by a simple computation that the object function of V is defined correctly. We show that R is a morphism. In fact, for arbitrary v 2 L , using Inequality (2), we obtain for x 2 X and arbitrary y 2 Y: ^ ^ ^ p .R (v) = A (z) ! R (v)(z) = A (z) ! ( R(z, t) ! v(t)) = w (x) w (x) A A z2X z2X t2Y ^ ^ ^ ^ ( A (z) ! (R(z, t) ! v(t))) = A (z) R(z, t) ! v(t) w (x) w (x) A A z2X z2X t2Y t2Y ^ ^ B (t) S(w (x), w (y)) ! v(t) = B (t) R(x, y) ! v(t) = A B w (y) w (y) B B t2Y t2Y ^ ^ R(x, y) ! (B (t) ! v(t)) = R(x, y) ! ( B (t) ! v(t)) = w (y) w (y) B B t2Y t2Y R(x, y) ! q (v)  R(x, y) ! q (v). y y y2Y Therefore, R is a morphisms and V is a functor. # # (13) Let (R,S) : (X,A) ! (Y,B) be a morphism in RSFP. We define U : RSFP ! RFTRans by: # # U (X,A) = F , U (R,S) = R. X,A It is clear that the object function U is defined correctly. We show that R is a morphism, # # i.e., S .F  F .R holds. In fact, for v 2 L , a 2 jAj, we have: Y,B X,A ^ ^ F .R (v)(a) = A (x) ! ( R(x, t) ! v(t)) = X,A x2X t2Y ^ ^ A (x) R(x, t) ! v(t) x2X t2Y ^ ^ ^ ^ B (t) S(a, b) ! v(t) = S(a, b) ! ( B (t) ! v(t)) = b b t2Y t2Y b2jBj b2jBj S .F (v)(a). Y,B Therefore, U is a functor. # # (14) The functor Q : RFTrans ! RsAFPreTop can be defined simply by: # # Q (F ) = V(X,A), Q (R,S) = R. X,A (15) Let R : (X, t) ! (Y, s) be a morphism in RFPreTop, where t = f p : x 2 Xg, s = fq : y 2 Yg. We define H : RFPreTop ! RCInt by: H(X, t) = (X, i), i(u)(x) = p (u), H(Y, s) = (Y, j), j(v)(y) = q (v), H(R) = R. By a simple verification we can see that H(X, t) is an object of RCInt. We prove that R is a morphism in RCInt. Using the properties of morphisms from RFPreTop, for arbitrary v 2 L , x 2 X, y 2 Y, we obtain: i.R (v)(x) = p (R (v))  R(x, t) ! q (v) = R (j(v))(x). x t t2Y < Axioms 2020, 9, 63 12 of 17 Hence, R is a morphism and H is a functor. The inverse functor H is defined symmetrically, 1 1 1 i.e., H (X, i) = (X, t), t = f p : x 2 Xg, p (u) = i(u)(x). It is clear that H is a functor and H, H x x are mutually inverse. 1 1 (16) The functors H and H are restrictions of functors H and H . It can be verified easily that the diagram of functors commutes. For the proof of the next theorem we use the following lemma, which was proven in [21]. X X Lemma 1. Let L be a complete MV-algebra and f 2 L . For a 2 L, let a 2 L be a constant function with the constant value a. Then for each f 2 L we have: : f = f (x) ! :c . fxg x2X Theorem 2. Let L be a complete MV-algebra. Then the following diagram of functors from Theorem 1 and 1 1 1 new functors commutes, where ( H, H ), (N, N ) and (P, P ) are inverse pairs of functors. RFcoPreTop RCClo [ [ Q > RFTrans > RsAFcoPreTop RsACClo < RKsACClo " ^ ^ 1 1 W M M K K _ _ RSFP > RFRel < RTFRel # ^ ^ 1 1 V N N P P _ _ _ RFTrans > RsAFPreTop RsACInt < RKsACInt < \ _ H _ RFPreTop RCInt Proof. (1) Let (R : (X, i) ! (Y, j) be a morphism in RsACInt. We define N by: 0 X N(X, i) = (X, r), r(x, x ) = :i(:c )(x), fx g 0 Y N(Y, j) = (Y, s), s(y, y ) = :j(:c )(y), N(R) = R. fy g The definition of objects is correct, because r is a reflexive relation. In fact, if r(x, x) = X X X :i(:c )(x) = a, then :a = i(c )(x)  :c (x) = 0 and it follows that a = 1. Let x 2 X, y 2 Y, fxg fxg fxg then we have: Y Y R (j(:c ))(x) = R(x, t) ! j(:c )(t) = fyg fyg t2Y _ _ Y Y : :(R(x, t) ! j(:c )(t)) = : R(x, t) :j(:c )(t) = :s R(x, y). fyg fyg t2Y t2Y Y X On the other hand, using Lemma 1, for f = :R (:c ) 2 L we obtain: fyg Y Y X R (:c ) = :R (:c )(z) ! :c . fyg fyg fzg z2X < Axioms 2020, 9, 63 13 of 17 Because i and j are strong Cech–Alexandroff operators, we obtain: Y Y X i(R (:c ))(x) = :R (:c )(z) ! i(:c )(x) = fyg fyg fzg z2X ^ ^ Y X :( R(z, t) ! :c (t)) ! i(:c )(x) = fyg fzg z2X t2Y Y X :(R(z, y) ! :c (y)) ! i(:c )(x) = fyg fzg z2X ^ ^ X X ::R(z, y) ! i(:c )(x) = R(z, y) ! ::i(:c )(x) = fzg fzg z2X z2X ^ ^ R(z, y) ! :r(x, z) = :(R(z, y) r(x, z)) = z2X z2X : r(x, z) R(z, y) = :(r R)(x, y) z2X Therefore, we obtain: Y Y :(s R)(x, y) = R (j(:c ))(x)  i(R (:c ))(x) = :(R r)(x, y), fyg fyg and it follows s R(x, y)  R r(x, y). Therefore, R is a morphism in RFRel. (2) Let R : (X, r) ! (Y, s) be a morphism in RFRel. We define N : RFRel ! RsACInt by: N (X, r) = (X, i), i(u)(x) = R (u)(x), 1 # 1 N (Y, s) = (Y, j), j(v)(y) = s (v)(y), N (R) = R. It is clear that object function N is defined correctly. We show that R is a morphism in RsACInt, i.e., we need to prove that for arbitrary v 2 L , i.R (v)  R .j(v) holds. Because both operators i.R and R .j have properties of strong Cech–Alexandroff operators, according to Lemma 1, for arbitrary v 2 L we obtain: ^ ^ i.R (v) = i.R ( :v(t) ! :c ) = :v(t) ! i.R (:c ), ftg ftg t2Y t2Y ^ ^ R .j(v) = R .j( :v(t) ! :c ) = :v(t) ! R .j(:c ). ftg ftg t2Y t2Y Therefore, we obtain the following equivalence: Y Y i.R (v)  R .j(v) , i.R (:c )  R .j(:c ) ftg ftg for arbitrary t 2 Y. Let t 2 Y, x 2 X. We have: Y Y i(R (:c ))(x) = r(x, z) ! R (:c )(z) = ftg ftg z2X ^ ^ ^ r(x, z) ! ( R(z, u) ! :c (u)) = r(z, x) ! :R(z, t) = ftg z2X u2Y z2X ^ _ :(r(x, z) R(z, t)) = : r(x, z) R(z, t) = :(R r)(x, t). z2X z2X Axioms 2020, 9, 63 14 of 17 On the other hand, we have: Y Y R .j(:c )(x) = R(x, u) ! j(:c )(u) = ftg ftg u2Y ^ ^ ^ R(x, u) ! ( s(u, p) ! :c ( p)) = R(x, u) ! :s(u, t) = ftg u2Y p2Y u2Y ^ _ :(R(x, u) s(u, t)) = : R(x, u) s(u, t) = :(s R)(x, y). u2Y u2Y Therefore, we obtain: Y Y R .j(:c ))(x) = :(s R)(x, t)  :(R r)(x, t) = i.R (:c )(x), ftg ftg and R is a morphism in RsACInt. (3) We prove that N, N are inverse functors. In fact, let (X, i) 2 RsACInt. Then we have: 1 1 N .N(X, i) = N (X, r) = (X, j), r(x, t) = :i(:c )(x), ^ ^ j(u)(x) = R (u)(x) = r(x, t) ! u(t) = :i(:c (x) ! u(t) = ftg t2X t2X ^ ^ X X :u(t) ! i(:c ) (x) = i( :u(t) ! :c )(x) = i(u)(x), ftg ftg t2X t2X as follows from Lemma 1. Hence, N .N(X, i) = (X, i). Conversely, for (X, r) 2 RFRel, we obtain: N.N (X, r) = N(X, i) = (X, s), i(u)(x) = R (u)(x), X X s(x, t) = :i(:c )(x) = :R (:c )(x) = ftg ftg :( r(x, t) ! :c (z)) = ::r(x, t) = r(x, t). ftg t2X Hence r = s and N, N are inverse functors. (4) Let R : (X, i) ! (Y, j) be a morphism in RKsACInt. We define P by: P(X, i) = N(X, i) P(R) = N(R). Hence, if P(X, i) = (X, r), then r(x, t) = i(:c )(x). We need to prove only that r is a transitive ftg L-fuzzy relation. For arbitrary u 2 L we have: ^ ^ R (u)(x) = r(x, z) ! u(z) = :i(:c )(x) ! u(z) = fzg z2X z2X ^ ^ X X :u(z) ! i(:c )(x) = i( :u(z) ! :c )(x) = i(u)(x), fzg fzg z2X z2X as follows from Lemma 1. Hence, for u = :c we obtain: ftg X X X R (:c )(x) = r(x, z) ! :c (z) = :r(x, t) = i(:c )(x). ftg ftg ftg z2X Axioms 2020, 9, 63 15 of 17 Analogously we obtain: X X ii(:c )(x) = R (R (:c ))(x) = r(x, z) ! R (:c )(z) = ftg ftg ftg z2X ^ ^ r(x, z) ! ( r(z, y) ! :c (y)) = ftg z2X y2X ^ ^ ^ r(x, z) r(z, y) ! :c (y) = :(r(x, z) r(z, t)). ftg z2X y2X z2X Therefore, from ii = i we obtain r(x, t)  r(x, z) r(z, t) and r is a transitive relation. (5) Let R : (X, r) ! (Y, s) be a morphism in RTFRel. We define P : RTFRel ! RKsACInt by: 1 1 1 P (X, r) = N (X, r), P (R) = R. Hence, if P (X, r) = (X, i), then i(u)(x) = R (u)(x). We need to prove only that i is a Kuratowski interior operator. For arbitrary t 2 X we have: i(:c )(x) = r(x, z) ! :c (z) = :r(x, t). ftg ftg z2X Analogously, as in previous part (5), we prove that: X X ii(:c )(x) = :(r(x, z) r(z, t)) = :r(x, t) = i(:c )(x). ftg ftg z2X Using Lemma 1, for arbitrary u 2 L we obtain: ^ ^ X X ii(u)(x) = ii( :u(z) ! :c )(x) = :u(z) ! ii(:c )(x) = fzg fzg z2X z2X ^ ^ :u(z) ! i(:c )(x) = i( :u(z) ! :c )(x) = i(u)(x). fzg fzg z2X z2X For the illustration of previous functors we show how from one topological type structure another structure can be defined. Example 1. Let L be a complete residuated lattice. Using Theorem 1 we show how a strong Cech–Alexandroff L-fuzzy closure operator c in a set X can be constructed from an equivalence relation s in X. In fact, using s we can define a fuzzy partitionA = f A : a 2 X/sg, where X/s is the set of equivalence classes defined by s and A (x) = 1 iff x 2 a; otherwise the value is 0. Using functors from the Theorem 1, the L-fuzzy closure operator c in X can be defined by (X, c) = M.T(X,A), i.e., for arbitrary u 2 L , x 2 X, _ _ c(u)(x) = u(t) A (t) = u(t). w (x) t2X t2X,(t,x)2s Example 2. Let L be a complete MV-algebra. Using the Theorem 2 we show how from a strong ˇ ˇ Cech–Alexandroff L-fuzzy pretopological space (X, t), a strong Cech–Alexandroff L-fuzzy co-pretopological space (X, r) can be defined. In fact, we can put (X, r) = G .M.N.H(X, t). If t = f p : x 2 Xg then r = f p : x 2 Xg is defined by: x X p (u) = u(t) : p (:c )(t), fxg t2X as can be verified by a simple calculation. Axioms 2020, 9, 63 16 of 17 4. Conclusions The article follows the paper [9], in which the issue of relationships between categories motivated by topological structures was investigated. These structures include variants of fuzzy topological spaces, fuzzy rough sets, fuzzy approximation spaces, fuzzy closure operators and fuzzy pretopological operators, and their dual terms, such as fuzzy interior operators, are frequently used in fuzzy set theory and applications. The common feature of these categories was that the morphisms in these categories with topological objects were mappings among the supports of these structures. Recently, however, a number of results have emerged in the theory of fuzzy sets, which are based on the application of fuzzy relations as morphisms in suitable categories. In this sequel, we looked at a more general situation where morphisms in these categories of topological structures are not mappings, but L-fuzzy relations. In detail, we considered categories and some of their subcategories of Cech closure or interior L-valued operators, categories of L-fuzzy pretopological and L-fuzzy co-pretopological operators, the category of L-valued fuzzy relation, categories of upper and lower F-transforms and the category of spaces with fuzzy partitions, where morphisms between objects are based on L-valued relations. We investigated functors between these categories with relational morphisms, which can thus represent transformation processes between different types of topological structures. As an interesting consequence of these relationships among relational categories, it follows that the category of spaces with fuzzy partitions and relational morphisms plays the key role. From the corresponding functor diagrams it follows that a space with a fuzzy partition can be used to create an object of any of the above categories of topological structures. Moreover, from a relational variant of a morphism between two spaces with fuzzy partitions we can derive a relational morphism between corresponding transformed objects from these categories of topological structures. This paper presents the first systematic contribution to the study of relationships between categories of different variants of fuzzy topological structures, where the morphisms between structures are L-valued fuzzy relations. Although there are isomorphic functors between some pairs of these categories, it is not yet known which of the functors between the remaining pairs of categories are also isomorphisms. It is also not known in all cases whether the special subcategories presented in this paper constitute reflective subcategories, which would make it possible to extend the general forms of a given topological structure to these special structures from these subcategories. These issues will be the subject of further research. Funding: This research was partially supported by the Grand Agency of the Czech Republic, grand number 18-06915S. Conflicts of Interest: The author declares no conflict of interest. References 1. Belohlávek ˇ , R. Fuzzy closure operators. J. Math. Anal. Appl. 2001, 262, 473–489. [CrossRef] 2. Belohlávek ˇ , R. Fuzzy closure operators II. Soft Comput. 2002, 7, 53–64. [CrossRef] 3. Biacino, V.; Gerla, G. Closure systems and L-subalgebras. Inf. Sci. 1984, 33, 181–195. [CrossRef] 4. Elkins, A.; Šostak, A.; Uljane, I. On a category of extensional fuzzy rough approximation operators. In Communication in Computer and Information Science; Springer: Berlin, Germany, 2016. 5. Perfilieva, I.; Singh, A.P.; Tiwari, S.P. On the relationship among F-transform, fuzzy rough set and fuzzy topology. Soft Comput. 2017, 21, 3513–3523. [CrossRef] 6. Ramadan, A.A.; Li, L. Categories of lattice-valued closure (interior) operators and Alexandroff L-fuzzy topologies. Iran. J. Fuzzy Syst. 2018. [CrossRef] 7. Perfilieva, I., Singh, A.P., Tiwari, S.P. On F-transform, L-fuzzy partitions and l-fuzzy pretopological spaces. In Proceedings of the 2017 IEEE Symposium Series on Computational Intelligence (SSCI), Honolulu, HI, USA, 27 November–1 December 2017; pp. 1–8. 8. Zhang, D. Fuzzy pretopological spaces, an extensional topological extension of FTS. Chin. Ann. Math. 1999, 3, 309–316. [CrossRef] Axioms 2020, 9, 63 17 of 17 9. Mocko ˇ r, ˇ J. Perfiljeva, I. Functors among categories of L-fuzzy partitions, L-fuzzy pretopological spaces and L-fuzzy closure spaces. In IFSA World Congress and NAFIPS Annual Conference: Fuzzy Techniques: Theory and Applications. IFSA/NAFIPS Louisiana, USA; Springer: Cham, Switzerland, 2019; pp. 382–393. 10. Goguen, J.A. L-fuzzy sets. J. Math. Anal. Appl. 1967, 18, 145–174. [CrossRef] 11. Dikranjan, D.; Šostak, A. On two categories of many-valued fuzzy relations. In IFSA-EUSFLAT Proceedings; Atlantis Press: Paris, France, 2015; pp. 933–938. 12. Wang, C.Y. Fuzzy rough sets based on generalized residuated lattices. Inf. Sci. 2013, 248, 31–49. [CrossRef] 13. Wu, W.Z.; Leung, Y.; Mi, J.S. On characterization of (J ,T )-fuzzy rough approximation operators. Fuzzy Sets Syst. 2005, 154, 76–102. [CrossRef] 14. Höhle, U. Fuzzy sets and sheaves. Part I, Basic concepts. Fuzzy Sets Syst. 2007, 158, 1143–1174. [CrossRef] 15. Novák, V.; Perfilijeva, I.; Mocko ˇ r, ˇ J. Mathematical Principles of Fuzzy Logic; Kluwer Academic Publishers: Boston, MA, USA, 1999. 16. Bandler, W.; Kohout, L. Special properties, closures and interiors of crisp and fuzzy relations. Fuzzy Sets Syst. 1988, 26, 317–331. [CrossRef] 17. Bodenhofer, U.; De Cock, M.; Kerre, E.E. Openings and closures of fuzzy preorderings: Theoretical basics and applications to fuzzy rule-based systems. Int. J. Gen. Syst. 2003, 32, 343–360. [CrossRef] 18. Mocko ˇ r, ˇ J. Spaces with fuzzy partitions and fuzzy transform. Soft Comput. 2017, 13, 3479–3492. [CrossRef] 19. Perfilieva, I. Fuzzy transforms: Theory and applications. Fuzzy Sets Syst. 2006, 157, 993–1023. [CrossRef] 20. Rodabaugh, S.E. Powerset operator foundation for poslat fuzzy sst theories and topologies. In Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbook of Fuzzy Sets Series; Höhle, U., Rodabaugh, S.E., Eds.; Kluwer Academic Publishers: Boston, MA, USA, 1999; Volume 3, pp. 91–116. 21. Mocko ˇ r, ˇ J. Axiomatic of lattice-valued F-transform. Fuzzy Sets Syst. 2018, 342, 53–66. [CrossRef] © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Axioms Multidisciplinary Digital Publishing Institute

Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces

Močkoř, Jiří
Axioms , Volume 9 (2) – Jun 2, 2020
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axioms Article Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces Jirí ˇ Mocko ˇ r ˇ Institute for Research and Applications of Fuzzy Modelling, NSC IT4Innovations, University of Ostrava, 70200 Ostrava, Czech Republic; mockor@osu.cz Received: 5 May 2020; Accepted: 27 May 2020; Published: 2 June 2020 Abstract: Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Cech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined. Keywords: categories with relational morphisms; topological and pre-topological structures; functors 1. Introduction In fuzzy set theory many structures are used, which are based on various modifications of topological operators. These structures include variants of fuzzy topological spaces, fuzzy rough sets, fuzzy approximation spaces, fuzzy closure operators, fuzzy pretopological operators and their dual terms, such as fuzzy interior operators. For examples of these structures see, e.g., [1–8]. Although these structures are generally based on the common basis of different topological spaces and their modifications, the tools and language they use are often very different and it is difficult to identify deeper relationships between the different types of these structures. One way to effectively identify and describe the relationships among these objects is to use category theory methods. Description of transformation processes between objects of one type using the category theory language ensure morphisms and functors. The significance of morphisms and functors in categories is that morphisms between objects and functors between categories actually represent the processes of transforming objects of one type into objects of another type, whereby these transformations define not only how to create an object of one type from another type, but also how to transform relationships between objects of one type into relationships between objects of the other type. In the paper [9], the relationships among some categories of fuzzy structures related to topological operators were discussed. The common feature of these categories was that the morphisms in these categories were based on mappings between the underlying sets of corresponding objects. Recently, however, a number of results have emerged in the theory of fuzzy sets, which are based on the application of fuzzy relations as morphisms in suitable categories. A typical example of that use of fuzzy relations is the category of sets as objects and L-valued fuzzy relations between sets as Axioms 2020, 9, 63; doi:10.3390/axioms9020063 www.mdpi.com/journal/axioms Axioms 2020, 9, 63 2 of 17 morphisms. This category is frequently used in approximation functors, which represent various approximations of fuzzy sets. The approximation defined by a fuzzy relation, which can be described as a functor between appropriate categories, was defined for the first time by Goguen [10], when he introduced the notion of the image of a fuzzy set under a fuzzy relation. Many examples using explicitly or implicitly approximation functors defined by various types of fuzzy relations can be found in rough fuzzy sets theory and many others (see, e.g., [5,11–13]). In this paper we want to significantly expand the information on the relationships between categories of topological L-operators expressing the concept of “proximity” in different ways, while in accordance with the current trend of using L-fuzzy relations, the morphisms in these categories are L-fuzzy relations with various properties. In detail, we consider 15 categories and some of their subcategories of Cech closure or interior L-valued operators, categories of L-fuzzy pretopological and L-fuzzy co-pretopological operators, the category of L-valued fuzzy relations, categories of upper and lower F-transforms and the category of spaces with fuzzy partitions, where L is a complete residuated lattice or MV-algebra, and where morphisms in all of these categories are L-valued fuzzy relations. The main results of this paper are Theorems 1 and 2, which prove the existence of 23 functors among these categories, including how these functors combine with each other. It follows from these theorems that the key category between the above categories is the category of spaces with fuzzy partitions with special fuzzy relations as morphisms. As follows from the commutative diagrams in both Theorems, any space with a fuzzy partition can be transformed into an L-fuzzy relation, Cech closure or interior operator, L-fuzzy pretopological or co-pretopological operators or strong-Alexandroff variants of these operators. The structure of the paper is as follows. In Section 1 we repeat basic properties of residuated lattices and we recall definitions of principal structures representing various concepts of proximity. In Section 2, which represents the principal content of the paper, we introduce 15 new categories, whose objects are various types of proximity structures and whose morphisms are various types of relations between these structures. The main result are then two theorems identifying functors among these categories, which in fact represent the transformation processes among the individual structures. 2. Preliminaries In this section we repeat basic properties of residuated lattices and recall definitions of principal structures representing various concepts of proximity, which are frequently used in L-valued fuzzy set theory, including interior and closure operators, pretopologies and co-pretopologies, and fuzzy partitions, sometimes with additional special properties. We refer to [14,15] for additional details regarding residuated lattices. Definition 1. A residuated lattice L is an algebra L = (L,^,_, ,!, 0, 1) such that: 1. (L,^,_, 0, 1) is a bounded lattice with the least element 0 and the greatest element 1; 2. (L, , 1) is a commutative monoid, and 3. 8a, b, c 2 L, a b  c () a  b ! c. A residuated lattice (L,^,_, ,!, 0, 1) is complete if it is complete as a lattice. The following is the derived unary operation of negation :: :a = a ! 0, A residuated lattice L is called an MV-algebra if it satisfies (a ! b) ! b = a_ b. In a MV-algebra the following identities hold: ::a = a, :_ a = ^:a , a b = :(a ! :b). i i Axioms 2020, 9, 63 3 of 17 Unless otherwise stated, throughout this paper, a complete residuated lattice L = (L,^,_, ,! , 0, 1) will be fixed and for simplicity, instead of L we use only L. Let X be a nonempty set and L a set of all L-fuzzy sets (=L-valued functions) of X. For all a 2 L, a(x) = a is a constant L-fuzzy set on X. For all u 2 L , the core(u) is a set of all elements x 2 X, X X X such that u(x) = 1. An L-fuzzy set u 2 L is called normal, if core(u) 6= Æ. An L-fuzzy set c 2 L fyg is a singleton of y 2 X, if it has the following form: 1, if x = y, 8x 2 X, c (x) = fyg 0, otherwise. In the next definitions we repeat basic definitions of the L-valued operators that were mentioned above. These operators are very useful tools in several areas of mathematical structures with direct applications, both mathematical (e.g., topology, logic) and outside of mathematics (e.g., data mining, knowledge representation). In fuzzy set theory, several particular cases as well as general theory of interior and closure operators, which operate with fuzzy sets (so-called fuzzy interior or closure operators), are studied. The original notions of a Kuratowski closure and interior operators were introduced in several papers, see [1–3,16,17]. In this paper we use a more general form of these operators, called Cech operators or preclosure operators, where the idempotence of operators is not required. X X X Definition 2. The map i : L ! L is called a Cech (L-fuzzy) interior operator, if for every a, u, v 2 L , it fulfils: 1. i(a) = a, 2. i(u)  u, 3. i(u^ v) = i(u)^ i(v). X X ˇ ˇ A Cech interior operator i : L ! L is said to be a strong Cech–Alexandroff interior operator, if: ^ ^ i(a ! u) = a ! i(u) and i( u ) = i(u ), j j j2 J j2 J and is said to be a Kuratowski interior operator if ii(u) = i(u). X X X Definition 3. The map c : L ! L is called a Cech (L-fuzzy) closure operator, if for every a, u, v 2 L , it fulfils: 1. c(a) = a, 2. c(u)  u, 3. c(u_ v) = c(u)_ c(v). X X ˇ ˇ A Cech closure operator c : L ! L is said to be a strong Cech–Alexandroff closure operator, if _ _ c(a u) = a c(u) and c( u ) = c(u ), j j j2 J j2 J and is said to be a Kuratowski closure operator if cc(u) = c(u). We recall the notion of an L-fuzzy pretopological space and L-fuzzy co-pretopological space as it has been introduced in [8]. Definition 4. An L-fuzzy pretopology on X is a set of functions t = f p 2 L : x 2 Xg, such that for all u, v 2 L , a 2 L and x 2 X, Axioms 2020, 9, 63 4 of 17 1. p (a) = a, 2. p (u)  u(x), 3. p (u^ v) = p (u)^ p (v). x x x An L-fuzzy pretopological space (X, t) is said to be a strong Cech–Alexandroff L-fuzzy pretopological space, if: ^ ^ p (a ! u) = a ! p (u) and p ( u ) = p (u ). x x x x j j j2 J j2 J x L Definition 5. An L-fuzzy co-pretopology on X is a set of functions h = f p 2 L : x 2 Xg, such that for all u, v 2 L , a 2 L and x 2 X, 1. p (a) = a, 2. p (u)  u(x), x x x 3. p (u_ v) = p (u)_ p (v). An L-fuzzy co-pretopological space (X, t) is said to be a strong Cech–Alexandroff L-fuzzy co-pretopological space, if: _ _ x x x x p (a u) = a p (u) and p ( u ) = p (u ). j j j2 J j2 J Finally, we recall the notion of an L-fuzzy partition (see [18,19]). Definition 6. A set A of normal fuzzy sets f A : a 2 Lg in X is an L-fuzzy partition of X, if: 1. The corresponding set of ordinary subsets fcore( A ) : a 2 Lg is a partition of X, and 2. Core( A ) = core( A ) implies A = A . a b a b Instead of the index set L from A we use jAj. We use the notion of powerset maps defined by a fuzzy relation, which was first defined in [10]. ! X Y Y X If R : X Y ! L is an L-fuzzy relation, then the powerset maps R : L ! L and R : L ! L are defined by: X ! t 2 L , y 2 Y, R (t)(y) = t(x) R(x, y), x2X s 2 L , x 2 X, R (s)(x) = R(x, y) ! s(y). y2Y 3. Relational Categories of L-Valued Topological Objects and Functors among Them As we mentioned in the Introduction, in the paper [9], several functors among categories based on fuzzy topological structures were discussed, whose common feature was that morphisms in these categories were mappings between corresponding underlying sets. In this section we want to significantly expand the information on the relationships between categories of topological L-operators expressing the concept of “proximity”, while in accordance with the current trend of using L-fuzzy relations, the morphisms in these categories are L-fuzzy relations with various properties. The categories we will deal with have the objects defined in Section 2. Instead of classical maps between sets we use special fuzzy relations as morphisms. Axioms 2020, 9, 63 5 of 17 Definition 7. In what follows, we denote sets by X, Y, and by the composition of morphisms from the standard category Set. 1. The category RCInt is defined by: X X (a) Objects are pairs (X, i), where i : L ! L is a Cech L-fuzzy interior operator (Definition 2), (b) R : (X, i) ! (Y, j) is a morphism, if R : X Y ! L is an L-fuzzy relation and i.R  R .j. 2. The category RCClo is defined by: X X (a) Objects are pairs (X, c), where c : L ! L is a Cech L-fuzzy closure operator (Definition 3), (b) R : (X, c) ! (Y, d) is a morphism, if R : X Y ! L is an L-fuzzy relation, and ! ! R .c  d.R . 3. The category RFPreTop is defined by: (a) Objects are L-fuzzy pretopological spaces (X, t) (Definition 4), X Y L L (b) R : (X, t) ! (Y, s) is a morphism, where t = f p 2 L : x 2 Xg, s = fq 2 L : y 2 Yg, x y if R : X Y ! L is an L-fuzzy relation, and for all x 2 X, (R(x, z) ! q )  p .R . z x z2Y 4. The category RFcoPreTop is defined by: (a) Objects are L-fuzzy co-pretopological spaces (X, t) (Definition 5), X Y x L y L (b) R : (X, t) ! (Y, s) is a morphism, where t = f p 2 L : x 2 Xg, s = fq 2 L : y 2 Yg, if R : X Y ! L is an L-fuzzy relation, and for all x 2 X, y 2 Y, y ! x q .R  p R(x, y). 5. The category RFRel is defined by: (a) Objects are pairs (X, r), where r is a reflexive L-fuzzy relation on X, (b) R : (X, r) ! (Y, s) is a morphism, if R : X Y ! L is an L-fuzzy relation, and s R  R r, where  is the composition of L-fuzzy relations. 6. The category RSFP is defined by: (a) Objects are sets with an L-fuzzy partition (X,A), (Definition 6), (b) (R,S) : (X,A) ! (Y,B) is a morphism if R : X Y ! L and S : jAjjBj ! L are L-fuzzy relations such that i. For each a 2 jAj, b 2 jBj, t 2 core( A ), z 2 core(B ), a b S(a, b) = R(t, z), Axioms 2020, 9, 63 6 of 17 ii. For each a 2 jAj, b 2 jBj, x 2 X, y 2 Y, A (x) S(a, b)  B (y) R(x, y), (1) a b A (x) R(x, y)  B (y) S(a, b). (2) 7. The category RFTrans is defined by: X jAj (a) Objects are upper F-transforms F : L ! L , where (X,A) are sets with L-fuzzy partitions X,A and F (u)(a) = u(x) A (x), where u 2 L , a 2 jAj, x2X a X,A " " (b) (R,S) : F ! F is a morphism if R : X  Y ! L and S : jAjjBj ! L are L-fuzzy X,A Y,B relations and for each a 2 jAj, b 2 jBj, t 2 core( A ), z 2 core(B ), S(a, b) = R(t, z), " " ! ! S .F  F .R . X,A Y,B hold. 8. The category RFTrans is defined by: X jAj (a) Objects are lower F-transforms F : L ! L , where (X,A) are sets with L-fuzzy partitions, X,A where F (u)(a) = A (x) ! u(x), for u 2 L , a 2 jAj, x2X X,A # # (b) (R,S) : F ! F is a morphism if R : X  Y ! L and S : jAjjBj ! L are L-fuzzy X,A Y,B relations and and for each a 2 jAj, b 2 jBj, t 2 core( A ), z 2 core(B ), a b S(a, b) = R(t, z), # # S .F  F .R . Y,B X,A hold. In the next theorems we will use the following subcategories of categories from Definition 7. Definition 8. The following full subcategories of categories from Definiton 7 will be used: 1. The full subcategory RTFRel of RFRel with transitive L-fuzzy relations as objects, 2. The full subcategory RsACClo of RCClo with strong Cech–Alexandroff L-fuzzy closure operators as objects, 3. The full subcategory RKsACClo of RCClo with Kuratowski strong Cech–Alexandroff L-fuzzy closure operators, 4. The full subcategory RsACInt of RCInt with strong Cech–Alexandroff L-fuzzy interior operators as objects, 5. The full subcategory RKsACInt of RCInt with Kuratowski strong Cech–Alexandroff L-fuzzy interior operators, 6. The full subcategory RsAFPreTop of RFPreTop with strong Cech– Alexandroff L-fuzzy pretopological spaces as objects, 7. The full subcategory RsAFcoPreTop of RFcoPreTop with strong Cech–Alexandroff L-fuzzy co-pretopological spaces. The main results of the paper are the following two theorems, in which we will define 23 functors to describe relationships between pairs of categories and some of their subcategories from Definitions 7 and 8. < Axioms 2020, 9, 63 7 of 17 Theorem 1. Let L be a complete residuated lattice. There exist functors such that the following diagram of these functors commutes, RFcoPreTop RCClo [ [ RFTrans > RsAFcoPreTop RsACClo < RKsACClo " ^ ^ U ^ 1 1 W M M K K _ _ RSFP > RFRel < RTFRel # 1 # > RFTrans > RsAFPreTop RsACInt _ _ RFPreTop RCInt 1 1 1 1 where (F, F ), (F , F ), ( M, M ) and (K, K ) are pairs of inverse functors. x L Proof. (1) Let R : (X, t) ! (Y, s) be a morphisms in RFcoPrTop, where t = f p 2 L : x 2 Xg, y L s = fq 2 L : y 2 Yg. The functor F : RFcoPrTop ! RCClo is defined by: X x F(X, t) = (X, c), 8u 2 L , x 2 X, c(u)(x) = p (u), Y y F(Y, s) = (Y, d), 8v 2 L , y 2 Y, d(v)(y) = q (v), F(R) = R. According to [5], (X, c) is an object of RCClo. For arbitrary u 2 L , y 2 Y, we have: _ _ _ ! x y ! R .c(u)(y) = c(u)(x) R(x, y) = p (u) R(x, y)  q (R (u)) = x2X x2X x2X y ! ! q (R (u)) = d.R (u)(y). Therefore, F is defined correctly and R is a morphism (X, c) ! (Y, d) in RCClo. (2) Let R : (X, c) ! (Y, d) be a morphisms in RCClo. According to the same reference [5], the object function F : RCClo ! RFcoPreTop such that: 1 x x F (X, c) = (X, t), t = f p : x 2 Xg, p (u) = c(u)(x), X 1 for u 2 L , x 2 X, is defined correctly. We set F (R) = R. Then we obtain: y ! ! ! q .R (u) = d.R (u)(y)  R .c(u)(y) = c(u)(z) R(z, y) = z2X z x p (u) R(z, y)  p (u) R(x, y). z2X 1 1 Hence, R is a morphism in RFcoPreTop and F is a functor. It is clear that F and F are inverse functors. < Axioms 2020, 9, 63 8 of 17 (3) Let R : (X, r) ! (Y, s) be a morphism in the category RFRel and let the object function M : RFRel ! RsACClo be defined by: X ! M(X, r) = (X, c),8u 2 L , x 2 X, c(u)(x) = R (u)(x), Y " M(Y, s) = (Y, d),8v 2 L , y 2 Y, d(v)(y) = s (v)(y). According to results from [6] and many others, (X, c) 2 sACClo and the object function is defined correctly. For M(R) = R we obtain: _ _ _ R .c(u)(y) = c(u)(x) R(x, y) = u(z) r(z, x) R(x, y) = x2X x2X z2X _ _ (R r)(z, y) u(z)  (s R)(z, y) u(z) = z2X z2X _ _ _ ! " ! R(z, t) s(t, y) u(z) = R (u)(t) s(t, y) = s .R (u)(y) = z2X t2Y t2Y d.R (u)(y). Hence, R is a morphism in RsACClo and M is a functor. (4) In the proof we use the following identity, which was proven in [20], and which allows to express a general L-fuzzy set u 2 L in the following form: u = u(x) c , (3) fxg x2X where a 2 L is the constant function with the value a. Let R : (X, c) ! (Y, d) be a morphism in RsACClo and let the object function M : RsACClo ! RFRel be defined by: 1 0 0 X 0 M (X, c) = (X, r),8x, x 2 X, r(x, x ) = c(c )(x ), fxg 1 0 0 Y 0 M (Y, d) = (Y, s),8y, y 2 Y, s(y, y ) = d(c )(y ). fyg It is clear that r and s are reflective fuzzy relations. We set M (R) = R. Then, for x 2 X, y 2 Y we obtain: _ _ 0 0 X 0 0 R r(x, y) = r(x, x ) R(x , y) = c(c )(x ) R(x , y) = fxg 0 0 x 2X x 2X ! X ! X ! X 0 Y R .c(c )(y)  d.R (c )(y) = d( R (c )(y ) c )(y) = fxg fxg fxg fy g y 2Y ! X 0 Y R (c )(y ) d(c )(y) = fxg fy g y 2Y _ _ X 0 0 Y c (x ) R(x , y) d(c )(y) = fxg fy g 0 0 y 2Y x 2X _ _ 0 Y 0 0 R(x, y ) d(c )(y) = R(x, y ) s(y , y) = s R(x, y), fy g y2Y y 2Y which follows from the fact that d is a strong Cech–Alexandroff closure. Hence, M (R) is a morphism 1 1 in the category RFRel and M is a functor. We prove that M and M are inverse functors. 1 0 Let (X, r) 2 RFRel. Then M .M(X, r) = (X, r ), where 0 0 X 0 ! X 0 X 0 r (x, x ) = c(c )(x ) = R (c )(x ) = c (z) r(z, x) = r(x, x ). fxg fxg fxg z2X Axioms 2020, 9, 63 9 of 17 1 0 1 X On the other hand, M.M (X, c) = (X, c ), where M (X, c) = (X, r), r(z, x) = c(c )(x), fzg and we obtain: _ _ 0 ! X c (u)(x) = R (u)(x) = u(z) r(z, x) = u(z) c(c )(x) = fzg z2X z2X c( u(z) c )(x) = c(u)(x). fxg z2X Therefore, M and M are inverse functors. (5) Let R : (X, r) ! (Y, s) be a morphism in the category RTFRel. The functor K : RTFRel ! RKsACClo is defined by: K(X, r) = M(X, r), K(R) = M(R). We prove that this definition is correct, i.e., M(X, r) = (X, c) and c is a Kuratowski closure operator. In fact, for u 2 L , x 2 X we have: ! ! cc(u)(x) = R (R (u))(x) = u(z) r(z, y) r(y, x) y,z2X u(z) r(z, x) = R (u)(x) = c(u)(x), y,z2X and the definition of K is correct. (6) Let R : (X, c) ! (Y, d) be a morphism in RKsACClo and let the functor K : RKsACClo ! RTFRel be defined by: 1 1 1 1 K (X, c) = M (X, c) = (X, r), K (R) = M (R). We prove that this definition is correct. In fact, for arbitrary u 2 L , according to (3), we obtain: _ _ X X c(u)(x) = c( u(z) c )(x) = u(z) c(c )(x) = fzg fzg z2X z2X u(z) r(z, x) = R (u)(x). z2x Hence, for arbitrary y 2 X and u = c , we obtain: fyg _ _ X ! ! X X cc(c )(x) = R (R (c ))(x) = c (z) r(z, t) r(t, y) = fyg fyg fyg z2X t2X r(y, t) r(t, x), t2X X ! X c(c )(x) = R (c )(x) = r(y, x). fyg fyg Because cc = c, it follows that r(y, x)  r(y, t) r(t, x). Therefore, r is a transitive relation. (7) Let (R,S) : (X,A) ! (Y,B) be a morphism in RSFP and let the object function W : RSFP ! RsAFcoPreTop be defined by: x L x W(X,A) = (X,f p 2 L : x 2 Xg), p (u) = u(t) A (x), w (t) t2X y L y W(Y,B) = (Y,fq 2 L : y 2 Yg), q (v) = v(s) B (y), w (s) s2Y Axioms 2020, 9, 63 10 of 17 where w (x) 2 jAj is the unique index such that x 2 core( A ). It is easy to see that the object w (x) function W is defined correctly. We set W(R,S) = R. For arbitrary x 2 X, y 2 Y and u 2 L , from the inequalitiy (2) we obtain: _ _ _ y ! ! q .R (u) = R (u)(z) B (y) = u(t) R(t, z) B (y) = w (z) w (z) B B z2Y z2Y t2X _ _ _ u(t) S(w (t), w (z)) B (y)  u(t) A (x) R(x, y) A B w (z) w (t) B A t2X t2X z2Y p (u) R(x, y) as follows from the identity S(w (x), w (y)) = R(x, y). A B Hence, R is a morphisms in RsAFcoPreTop and W is a functor. " " " (8) Let U : RSFP ! RFTrans be defined by U (X,A) = F , U (R,S) = (R,S). By a simple X,A " " computation it can be proven that the definition of U is correct and U is a functor. (9) Let (R,S) : (X,A) ! (Y,B) be a morphism in the category RSFP. We define T : RSFP ! RFRel by: 0 0 T(X,A) = (X, r), r(x, x ) = A (x ), w (x) 0 0 T(Y,B) = (Y, s), s(y, y ) = B (y ), w (y) T(R,S) = R. We prove that R is a morphism in the category RFRel. In fact, let x 2 X, y 2 Y. Then, according to Inequality (2), _ _ (R r)(x, y) = r(x, z) R(z, y) = A (z) R(z, y) w (x) z2X z2X _ _ S(w (x), w (t)) B (y) = R(x, t) s(t, y) = s R(x, y). A B w (t) t2Y t2Y Hence, T is a functor. " " (10) The functor Q : RFTrans ! RsAFcoPreTop can be simply defined by: " " Q (F ) = W(X,A), Q (R,S) = R. X,A (11) The functors F and F are restrictions of the functors F and F . It can be proven simply that these functors are mutually inverse. (12) Let (R,S) : (X,A) ! (Y,B) be a morphism in RSFP. We define V : RSFP ! RsAFPreTop by: V(X,A) = (X,f p 2 L : x 2 Xg), p (u) = F (u)(w (x)), x x A X,A V(Y,B) = (Y,fq 2 L : y 2 Yg), q (v) = F (v)(w (y)), y x Y,B V(R,S) = R. Axioms 2020, 9, 63 11 of 17 It can be proven by a simple computation that the object function of V is defined correctly. We show that R is a morphism. In fact, for arbitrary v 2 L , using Inequality (2), we obtain for x 2 X and arbitrary y 2 Y: ^ ^ ^ p .R (v) = A (z) ! R (v)(z) = A (z) ! ( R(z, t) ! v(t)) = w (x) w (x) A A z2X z2X t2Y ^ ^ ^ ^ ( A (z) ! (R(z, t) ! v(t))) = A (z) R(z, t) ! v(t) w (x) w (x) A A z2X z2X t2Y t2Y ^ ^ B (t) S(w (x), w (y)) ! v(t) = B (t) R(x, y) ! v(t) = A B w (y) w (y) B B t2Y t2Y ^ ^ R(x, y) ! (B (t) ! v(t)) = R(x, y) ! ( B (t) ! v(t)) = w (y) w (y) B B t2Y t2Y R(x, y) ! q (v)  R(x, y) ! q (v). y y y2Y Therefore, R is a morphisms and V is a functor. # # (13) Let (R,S) : (X,A) ! (Y,B) be a morphism in RSFP. We define U : RSFP ! RFTRans by: # # U (X,A) = F , U (R,S) = R. X,A It is clear that the object function U is defined correctly. We show that R is a morphism, # # i.e., S .F  F .R holds. In fact, for v 2 L , a 2 jAj, we have: Y,B X,A ^ ^ F .R (v)(a) = A (x) ! ( R(x, t) ! v(t)) = X,A x2X t2Y ^ ^ A (x) R(x, t) ! v(t) x2X t2Y ^ ^ ^ ^ B (t) S(a, b) ! v(t) = S(a, b) ! ( B (t) ! v(t)) = b b t2Y t2Y b2jBj b2jBj S .F (v)(a). Y,B Therefore, U is a functor. # # (14) The functor Q : RFTrans ! RsAFPreTop can be defined simply by: # # Q (F ) = V(X,A), Q (R,S) = R. X,A (15) Let R : (X, t) ! (Y, s) be a morphism in RFPreTop, where t = f p : x 2 Xg, s = fq : y 2 Yg. We define H : RFPreTop ! RCInt by: H(X, t) = (X, i), i(u)(x) = p (u), H(Y, s) = (Y, j), j(v)(y) = q (v), H(R) = R. By a simple verification we can see that H(X, t) is an object of RCInt. We prove that R is a morphism in RCInt. Using the properties of morphisms from RFPreTop, for arbitrary v 2 L , x 2 X, y 2 Y, we obtain: i.R (v)(x) = p (R (v))  R(x, t) ! q (v) = R (j(v))(x). x t t2Y < Axioms 2020, 9, 63 12 of 17 Hence, R is a morphism and H is a functor. The inverse functor H is defined symmetrically, 1 1 1 i.e., H (X, i) = (X, t), t = f p : x 2 Xg, p (u) = i(u)(x). It is clear that H is a functor and H, H x x are mutually inverse. 1 1 (16) The functors H and H are restrictions of functors H and H . It can be verified easily that the diagram of functors commutes. For the proof of the next theorem we use the following lemma, which was proven in [21]. X X Lemma 1. Let L be a complete MV-algebra and f 2 L . For a 2 L, let a 2 L be a constant function with the constant value a. Then for each f 2 L we have: : f = f (x) ! :c . fxg x2X Theorem 2. Let L be a complete MV-algebra. Then the following diagram of functors from Theorem 1 and 1 1 1 new functors commutes, where ( H, H ), (N, N ) and (P, P ) are inverse pairs of functors. RFcoPreTop RCClo [ [ Q > RFTrans > RsAFcoPreTop RsACClo < RKsACClo " ^ ^ 1 1 W M M K K _ _ RSFP > RFRel < RTFRel # ^ ^ 1 1 V N N P P _ _ _ RFTrans > RsAFPreTop RsACInt < RKsACInt < \ _ H _ RFPreTop RCInt Proof. (1) Let (R : (X, i) ! (Y, j) be a morphism in RsACInt. We define N by: 0 X N(X, i) = (X, r), r(x, x ) = :i(:c )(x), fx g 0 Y N(Y, j) = (Y, s), s(y, y ) = :j(:c )(y), N(R) = R. fy g The definition of objects is correct, because r is a reflexive relation. In fact, if r(x, x) = X X X :i(:c )(x) = a, then :a = i(c )(x)  :c (x) = 0 and it follows that a = 1. Let x 2 X, y 2 Y, fxg fxg fxg then we have: Y Y R (j(:c ))(x) = R(x, t) ! j(:c )(t) = fyg fyg t2Y _ _ Y Y : :(R(x, t) ! j(:c )(t)) = : R(x, t) :j(:c )(t) = :s R(x, y). fyg fyg t2Y t2Y Y X On the other hand, using Lemma 1, for f = :R (:c ) 2 L we obtain: fyg Y Y X R (:c ) = :R (:c )(z) ! :c . fyg fyg fzg z2X < Axioms 2020, 9, 63 13 of 17 Because i and j are strong Cech–Alexandroff operators, we obtain: Y Y X i(R (:c ))(x) = :R (:c )(z) ! i(:c )(x) = fyg fyg fzg z2X ^ ^ Y X :( R(z, t) ! :c (t)) ! i(:c )(x) = fyg fzg z2X t2Y Y X :(R(z, y) ! :c (y)) ! i(:c )(x) = fyg fzg z2X ^ ^ X X ::R(z, y) ! i(:c )(x) = R(z, y) ! ::i(:c )(x) = fzg fzg z2X z2X ^ ^ R(z, y) ! :r(x, z) = :(R(z, y) r(x, z)) = z2X z2X : r(x, z) R(z, y) = :(r R)(x, y) z2X Therefore, we obtain: Y Y :(s R)(x, y) = R (j(:c ))(x)  i(R (:c ))(x) = :(R r)(x, y), fyg fyg and it follows s R(x, y)  R r(x, y). Therefore, R is a morphism in RFRel. (2) Let R : (X, r) ! (Y, s) be a morphism in RFRel. We define N : RFRel ! RsACInt by: N (X, r) = (X, i), i(u)(x) = R (u)(x), 1 # 1 N (Y, s) = (Y, j), j(v)(y) = s (v)(y), N (R) = R. It is clear that object function N is defined correctly. We show that R is a morphism in RsACInt, i.e., we need to prove that for arbitrary v 2 L , i.R (v)  R .j(v) holds. Because both operators i.R and R .j have properties of strong Cech–Alexandroff operators, according to Lemma 1, for arbitrary v 2 L we obtain: ^ ^ i.R (v) = i.R ( :v(t) ! :c ) = :v(t) ! i.R (:c ), ftg ftg t2Y t2Y ^ ^ R .j(v) = R .j( :v(t) ! :c ) = :v(t) ! R .j(:c ). ftg ftg t2Y t2Y Therefore, we obtain the following equivalence: Y Y i.R (v)  R .j(v) , i.R (:c )  R .j(:c ) ftg ftg for arbitrary t 2 Y. Let t 2 Y, x 2 X. We have: Y Y i(R (:c ))(x) = r(x, z) ! R (:c )(z) = ftg ftg z2X ^ ^ ^ r(x, z) ! ( R(z, u) ! :c (u)) = r(z, x) ! :R(z, t) = ftg z2X u2Y z2X ^ _ :(r(x, z) R(z, t)) = : r(x, z) R(z, t) = :(R r)(x, t). z2X z2X Axioms 2020, 9, 63 14 of 17 On the other hand, we have: Y Y R .j(:c )(x) = R(x, u) ! j(:c )(u) = ftg ftg u2Y ^ ^ ^ R(x, u) ! ( s(u, p) ! :c ( p)) = R(x, u) ! :s(u, t) = ftg u2Y p2Y u2Y ^ _ :(R(x, u) s(u, t)) = : R(x, u) s(u, t) = :(s R)(x, y). u2Y u2Y Therefore, we obtain: Y Y R .j(:c ))(x) = :(s R)(x, t)  :(R r)(x, t) = i.R (:c )(x), ftg ftg and R is a morphism in RsACInt. (3) We prove that N, N are inverse functors. In fact, let (X, i) 2 RsACInt. Then we have: 1 1 N .N(X, i) = N (X, r) = (X, j), r(x, t) = :i(:c )(x), ^ ^ j(u)(x) = R (u)(x) = r(x, t) ! u(t) = :i(:c (x) ! u(t) = ftg t2X t2X ^ ^ X X :u(t) ! i(:c ) (x) = i( :u(t) ! :c )(x) = i(u)(x), ftg ftg t2X t2X as follows from Lemma 1. Hence, N .N(X, i) = (X, i). Conversely, for (X, r) 2 RFRel, we obtain: N.N (X, r) = N(X, i) = (X, s), i(u)(x) = R (u)(x), X X s(x, t) = :i(:c )(x) = :R (:c )(x) = ftg ftg :( r(x, t) ! :c (z)) = ::r(x, t) = r(x, t). ftg t2X Hence r = s and N, N are inverse functors. (4) Let R : (X, i) ! (Y, j) be a morphism in RKsACInt. We define P by: P(X, i) = N(X, i) P(R) = N(R). Hence, if P(X, i) = (X, r), then r(x, t) = i(:c )(x). We need to prove only that r is a transitive ftg L-fuzzy relation. For arbitrary u 2 L we have: ^ ^ R (u)(x) = r(x, z) ! u(z) = :i(:c )(x) ! u(z) = fzg z2X z2X ^ ^ X X :u(z) ! i(:c )(x) = i( :u(z) ! :c )(x) = i(u)(x), fzg fzg z2X z2X as follows from Lemma 1. Hence, for u = :c we obtain: ftg X X X R (:c )(x) = r(x, z) ! :c (z) = :r(x, t) = i(:c )(x). ftg ftg ftg z2X Axioms 2020, 9, 63 15 of 17 Analogously we obtain: X X ii(:c )(x) = R (R (:c ))(x) = r(x, z) ! R (:c )(z) = ftg ftg ftg z2X ^ ^ r(x, z) ! ( r(z, y) ! :c (y)) = ftg z2X y2X ^ ^ ^ r(x, z) r(z, y) ! :c (y) = :(r(x, z) r(z, t)). ftg z2X y2X z2X Therefore, from ii = i we obtain r(x, t)  r(x, z) r(z, t) and r is a transitive relation. (5) Let R : (X, r) ! (Y, s) be a morphism in RTFRel. We define P : RTFRel ! RKsACInt by: 1 1 1 P (X, r) = N (X, r), P (R) = R. Hence, if P (X, r) = (X, i), then i(u)(x) = R (u)(x). We need to prove only that i is a Kuratowski interior operator. For arbitrary t 2 X we have: i(:c )(x) = r(x, z) ! :c (z) = :r(x, t). ftg ftg z2X Analogously, as in previous part (5), we prove that: X X ii(:c )(x) = :(r(x, z) r(z, t)) = :r(x, t) = i(:c )(x). ftg ftg z2X Using Lemma 1, for arbitrary u 2 L we obtain: ^ ^ X X ii(u)(x) = ii( :u(z) ! :c )(x) = :u(z) ! ii(:c )(x) = fzg fzg z2X z2X ^ ^ :u(z) ! i(:c )(x) = i( :u(z) ! :c )(x) = i(u)(x). fzg fzg z2X z2X For the illustration of previous functors we show how from one topological type structure another structure can be defined. Example 1. Let L be a complete residuated lattice. Using Theorem 1 we show how a strong Cech–Alexandroff L-fuzzy closure operator c in a set X can be constructed from an equivalence relation s in X. In fact, using s we can define a fuzzy partitionA = f A : a 2 X/sg, where X/s is the set of equivalence classes defined by s and A (x) = 1 iff x 2 a; otherwise the value is 0. Using functors from the Theorem 1, the L-fuzzy closure operator c in X can be defined by (X, c) = M.T(X,A), i.e., for arbitrary u 2 L , x 2 X, _ _ c(u)(x) = u(t) A (t) = u(t). w (x) t2X t2X,(t,x)2s Example 2. Let L be a complete MV-algebra. Using the Theorem 2 we show how from a strong ˇ ˇ Cech–Alexandroff L-fuzzy pretopological space (X, t), a strong Cech–Alexandroff L-fuzzy co-pretopological space (X, r) can be defined. In fact, we can put (X, r) = G .M.N.H(X, t). If t = f p : x 2 Xg then r = f p : x 2 Xg is defined by: x X p (u) = u(t) : p (:c )(t), fxg t2X as can be verified by a simple calculation. Axioms 2020, 9, 63 16 of 17 4. Conclusions The article follows the paper [9], in which the issue of relationships between categories motivated by topological structures was investigated. These structures include variants of fuzzy topological spaces, fuzzy rough sets, fuzzy approximation spaces, fuzzy closure operators and fuzzy pretopological operators, and their dual terms, such as fuzzy interior operators, are frequently used in fuzzy set theory and applications. The common feature of these categories was that the morphisms in these categories with topological objects were mappings among the supports of these structures. Recently, however, a number of results have emerged in the theory of fuzzy sets, which are based on the application of fuzzy relations as morphisms in suitable categories. In this sequel, we looked at a more general situation where morphisms in these categories of topological structures are not mappings, but L-fuzzy relations. In detail, we considered categories and some of their subcategories of Cech closure or interior L-valued operators, categories of L-fuzzy pretopological and L-fuzzy co-pretopological operators, the category of L-valued fuzzy relation, categories of upper and lower F-transforms and the category of spaces with fuzzy partitions, where morphisms between objects are based on L-valued relations. We investigated functors between these categories with relational morphisms, which can thus represent transformation processes between different types of topological structures. As an interesting consequence of these relationships among relational categories, it follows that the category of spaces with fuzzy partitions and relational morphisms plays the key role. From the corresponding functor diagrams it follows that a space with a fuzzy partition can be used to create an object of any of the above categories of topological structures. Moreover, from a relational variant of a morphism between two spaces with fuzzy partitions we can derive a relational morphism between corresponding transformed objects from these categories of topological structures. This paper presents the first systematic contribution to the study of relationships between categories of different variants of fuzzy topological structures, where the morphisms between structures are L-valued fuzzy relations. Although there are isomorphic functors between some pairs of these categories, it is not yet known which of the functors between the remaining pairs of categories are also isomorphisms. It is also not known in all cases whether the special subcategories presented in this paper constitute reflective subcategories, which would make it possible to extend the general forms of a given topological structure to these special structures from these subcategories. These issues will be the subject of further research. Funding: This research was partially supported by the Grand Agency of the Czech Republic, grand number 18-06915S. Conflicts of Interest: The author declares no conflict of interest. References 1. Belohlávek ˇ , R. Fuzzy closure operators. J. Math. Anal. Appl. 2001, 262, 473–489. [CrossRef] 2. Belohlávek ˇ , R. Fuzzy closure operators II. Soft Comput. 2002, 7, 53–64. [CrossRef] 3. Biacino, V.; Gerla, G. Closure systems and L-subalgebras. Inf. Sci. 1984, 33, 181–195. [CrossRef] 4. Elkins, A.; Šostak, A.; Uljane, I. On a category of extensional fuzzy rough approximation operators. In Communication in Computer and Information Science; Springer: Berlin, Germany, 2016. 5. Perfilieva, I.; Singh, A.P.; Tiwari, S.P. On the relationship among F-transform, fuzzy rough set and fuzzy topology. Soft Comput. 2017, 21, 3513–3523. [CrossRef] 6. Ramadan, A.A.; Li, L. Categories of lattice-valued closure (interior) operators and Alexandroff L-fuzzy topologies. Iran. J. Fuzzy Syst. 2018. [CrossRef] 7. 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Fuzzy transforms: Theory and applications. Fuzzy Sets Syst. 2006, 157, 993–1023. [CrossRef] 20. Rodabaugh, S.E. Powerset operator foundation for poslat fuzzy sst theories and topologies. In Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbook of Fuzzy Sets Series; Höhle, U., Rodabaugh, S.E., Eds.; Kluwer Academic Publishers: Boston, MA, USA, 1999; Volume 3, pp. 91–116. 21. Mocko ˇ r, ˇ J. Axiomatic of lattice-valued F-transform. Fuzzy Sets Syst. 2018, 342, 53–66. [CrossRef] © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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