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

: 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 ˇCech 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.


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][2][3][4][5][6][7][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 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][12][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 ofČech 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,Čech 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.

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.
The following is the derived unary operation of negation ¬: In a MV-algebra the following identities hold: 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 X a set of all L-fuzzy sets (=L-valued functions) of X. For all α ∈ L, α(x) = α is a constant L-fuzzy set on X. For all u ∈ L X , the core(u) is a set of all elements x ∈ X, such that u(x) = 1. An L-fuzzy set u ∈ L X is called normal, if core(u) = ∅. An L-fuzzy set χ X {y} ∈ L X is a singleton of y ∈ X, if it has the following form: 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][2][3]16,17]. In this paper we use a more general form of these operators, calledČech operators or preclosure operators, where the idempotence of operators is not required. Definition 2. The map i : L X → L X is called aČech (L-fuzzy) interior operator, if for every α, u, v ∈ L X , it fulfils: AČech interior operator i : L X → L X is said to be a strongČech-Alexandroff interior operator, if: and is said to be a Kuratowski interior operator if ii(u) = i(u).

Definition 3.
The map c : L X → L X is called aČech (L-fuzzy) closure operator, if for every α, u, v ∈ L X , it fulfils: AČech closure operator c : L X → L X is said to be a strongČech-Alexandroff closure operator, if 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 5. An L-fuzzy co-pretopology on X is a set of functions η = {p x ∈ L L X : x ∈ X}, such that for all u, v ∈ L X , α ∈ L and x ∈ X, Finally, we recall the notion of an L-fuzzy partition (see [18,19]).

Definition 6.
A set A of normal fuzzy sets {A α : α ∈ Λ} in X is an L-fuzzy partition of X, if: 1.
The corresponding set of ordinary subsets {core(A α ) : α ∈ Λ} is a partition of X, and 2. Core Instead of the index set Λ from A we use |A|.
We use the notion of powerset maps defined by a fuzzy relation, which was first defined in [10]. If R : X × Y → L is an L-fuzzy relation, then the powerset maps R → : L X → L Y and R ← : L Y → L X are defined by:

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.

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:

2.
The category RCClo is defined by:

4.
The category RFcoPreTop is defined by: (a) Objects are L-fuzzy co-pretopological spaces (X, τ) (Definition 5), if R : X × Y → L is an L-fuzzy relation, and for all x ∈ X, y ∈ Y, q y .R → ≥ p x ⊗ 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, where • is the composition of L-fuzzy relations.
The category RFTrans ↑ is defined by: 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 strongČech-Alexandroff L-fuzzy closure operators as objects, 3.
The full subcategory RsACInt of RCInt with strongČech-Alexandroff L-fuzzy interior operators as objects, 5.
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. Theorem 1. Let L be a complete residuated lattice. There exist functors such that the following diagram of these functors commutes, According to [5], (X, c) is an object of RCClo. For arbitrary u ∈ L X , y ∈ Y, we have: 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 −1 : RCClo → RFcoPreTop such that: for u ∈ L X , x ∈ X, is defined correctly. We set F −1 (R) = R. Then we obtain: Hence, R is a morphism in RFcoPreTop and F −1 is a functor. It is clear that F and F −1 are inverse functors.
(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: According to results from [6] and many others, (X, c) ∈ sACClo and the object function is defined correctly. For M(R) = R we obtain: 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 ∈ L X in the following form: where α ∈ L X is the constant function with the value α. Let R : (X, c) → (Y, d) be a morphism in RsACClo and let the object function M −1 : RsACClo → RFRel be defined by: It is clear that r and s are reflective fuzzy relations. We set M −1 (R) = R. Then, for x ∈ X, y ∈ Y we obtain: which follows from the fact that d is a strongČech-Alexandroff closure. Hence, M −1 (R) is a morphism in the category RFRel and M −1 is a functor. We prove that M and M −1 are inverse functors. Let (X, r) ∈ RFRel. Then M −1 .M(X, r) = (X, r ), where On the other hand, M.M −1 (X, c) = (X, c ), where M −1 (X, c) = (X, r), r(z, x) = c(χ X {z} )(x), and we obtain: Therefore, M and M −1 are inverse functors. 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 ∈ L X , x ∈ X we have: and the definition of K is correct.
(6) Let R : (X, c) → (Y, d) be a morphism in RKsACClo and let the functor K −1 : RKsACClo → RTFRel be defined by: We prove that this definition is correct. In fact, for arbitrary u ∈ L X , according to (3), we obtain: Hence, for arbitrary y ∈ X and u = χ X {y} , we obtain: Because cc = c, it follows that r(y, x) ≥ r(y, t) ⊗ r(t, x). Therefore, r is a transitive relation.
where w A (x) ∈ |A| is the unique index such that x ∈ core(A w A (x) ). It is easy to see that the object function W is defined correctly. We set W(R, Σ) = R. For arbitrary x ∈ X, y ∈ Y and u ∈ L X , from the inequalitiy (2) we obtain: as follows from the identity Σ(w A (x), w B (y)) = R(x, y).
Hence, R is a morphisms in RsAFcoPreTop and W is a functor.
. By a simple computation it can be proven that the definition of U ↑ is correct and U ↑ is a functor.
(9) Let (R, Σ) : (X, A) → (Y, B) be a morphism in the category RSFP. We define T : RSFP → RFRel by: We prove that R is a morphism in the category RFRel. In fact, let x ∈ X, y ∈ Y. Then, according to Inequality (2), Hence, T is a functor. (10) The functor Q ↑ : RFTrans ↑ → RsAFcoPreTop can be simply defined by: (11) The functors F 1 and F −1 1 are restrictions of the functors F and F −1 . It can be proven simply that these functors are mutually inverse.
(12) Let (R, Σ) : (X, A) → (Y, B) be a morphism in RSFP. We define V : RSFP → RsAFPreTop by: 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 ∈ L Y , using Inequality (2), we obtain for x ∈ X and arbitrary y ∈ Y: Therefore, R is a morphisms and V is a functor. (13) Let (R, Σ) : (X, A) → (Y, B) be a morphism in RSFP. We define U ↓ : RSFP → RFTRans ↓ by: It is clear that the object function U ↓ is defined correctly. We show that R is a morphism, i.e., Σ ← .F ↓ Y,B ≤ F ↓ X,A .R ← holds. In fact, for v ∈ L Y , α ∈ |A|, we have: Therefore, U ↓ is a functor. (14) The functor Q ↓ : RFTrans ↓ → RsAFPreTop can be defined simply by: By a simple verification we can see that H(X, τ) is an object of RCInt. We prove that R is a morphism in RCInt. Using the properties of morphisms from RFPreTop, for arbitrary v ∈ L Y , x ∈ X, y ∈ Y, we obtain: Hence, R is a morphism and H is a functor. The inverse functor H −1 is defined symmetrically, i.e., H −1 (X, i) = (X, τ), τ = {p x : x ∈ X}, p x (u) = i(u)(x). It is clear that H −1 is a functor and H, H −1 are mutually inverse. (16) The functors H 1 and H −1 1 are restrictions of functors H and H −1 . 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]. Lemma 1. Let L be a complete MV-algebra and f ∈ L X . For α ∈ L, let α ∈ L X be a constant function with the constant value α. Then for each f ∈ L X we have: Theorem 2. Let L be a complete MV-algebra. Then the following diagram of functors from Theorem 1 and new functors commutes, where (H, H −1 ), (N, N −1 ) and (P, P −1 ) are inverse pairs of functors.
(1) Let (R : (X, i) → (Y, j) be a morphism in RsACInt. We define N by: The definition of objects is correct, because r is a reflexive relation. In fact, if r(x, x) = ¬i(¬χ X {x} )(x) = α, then ¬α = i(χ X {x} )(x) ≤ ¬χ X {x} (x) = 0 and it follows that α = 1. Let x ∈ X, y ∈ Y, then we have: On the other hand, using Lemma 1, for f = ¬R ← (¬χ Y {y} ) ∈ L X we obtain: Because i and j are strongČech-Alexandroff operators, we obtain: Therefore, we obtain: 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 −1 : RFRel → RsACInt by: It is clear that object function N −1 is defined correctly. We show that R is a morphism in RsACInt, i.e., we need to prove that for arbitrary v ∈ L Y , i.R ← (v) ≥ R ← .j(v) holds. Because both operators i.R ← and R ← .j have properties of strongČech-Alexandroff operators, according to Lemma 1, for arbitrary v ∈ L Y we obtain: Therefore, we obtain the following equivalence: for arbitrary t ∈ Y. Let t ∈ Y, x ∈ X. We have:

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 ofČech 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.