A Note on Categories about Rough Sets

Using the concepts of category and functor, we provide some insights and prove an intrinsic property of the category ${\bf AprS}$ of approximation spaces and relation-preserving functions, the category ${\bf RCls}$ of rough closure spaces and continuous functions, and the category ${\bf RInt}$ of rough interior spaces and continuous functions. Furthermore, we define the category ${\bf IS}$ of information systems and O-A-D homomorphisms, and establish the relationship between the category ${\bf IS}$ and the category ${\bf AprS}$ by considering a subcategory ${\bf NeIS}$ of ${\bf IS}$ whose objects are information systems and whose arrows are non-expensive O-A-D homomorphisms with surjective attribute mappings.


Introduction
Pawlak's rough set theory (RST) [14] is intrinsically based on an equivalence relation on a domain of interests. RST starts from the notion of approximation space which is defined as a pair U, t , where U is a set and t is an equivalence relation on U .
In an approximation space U, t , the classical upper and lower approximations as mappings between subsets of U are indeed a pair of closure and interior operators on the set U . Moreover, the set of fixed points of the lower approximation is a topology in which every open set is also a closed set, namely a clopen topology.
On the other hand, categories are algebraic structures with many complementary natures. Almost every known example of a mathematical structure with the appropriate structure-preserving map yields a category. Category theory has provided the foundations for many of the twentieth century's greatest advances in pure mathematics. However, to the best of our knowledge, there are not much work done in incorporating both category theory and rough set theory. Recently, Borzooei et al. [5] considered the class of approximation spaces and discussed some categories of lower, upper and natural approximations without using much topological structures of spaces. It is not fruitful in obtaining more results by using limited tools in some categories.
In this paper, we define the category AprS of approximation spaces and relation-preserving function. We characterize clopen topologies on a set by introducing the concept of rough closure operator and rough interior operator on the set. We prove the equivalence of three categories, namely, the category of all rough interior spaces and continuous functions, the category of rough closure spaces and continuous functions, the category of approximation spaces and relation-preserving functions. We organize our paper as follows. In Section 2, we provide some preliminary backgrounds, followed by some definitions of several categories and main results in Section 3. We survey and compare some categories about rough sets in Section 4 and finally conclude with some remarks in Section 5.

Preliminaries
A (binary) relation R on a set U is a subset of the Cartesian product U × U . We write (u, v) ∈ R and uRv interchangeably.
Let f : U → V be a function from a set U into a set V . The inverse image of a set Y ⊆ V under f , designated by f ← (Y ), is the set Similarly, the direct image of a set X ⊆ U under f , designated by f [X], is the set

The notion of topological spaces
A topological space is a set U together with a set T of subsets of U that includes ∅ and the whole set U , and is closed under finite intersections and arbitrary unions. The set T is called a topology on U , and its members are called open sets. The complement of an open set is called a closed set.
The interior X 0 of a subset X of a topological space U is the union of all open sets contained in X. Accordingly, X 0 is open and is the largest open set contained in X. Dually, the closure X of X is the intersection of all closed sets containing X. Hence X is closed and is the smallest closed set containing X.
A set N in a topological space is a neighborhood of a point u in the topological space if and only if (sometimes shortened to iff) N contains an open set to which u belongs [9]. The set N u of all neighborhoods of u is called the neighborhood system of u. A neighborhood of u is called a minimal neighborhood of u if it contains no member of N u as proper subsets. In general, such a minimal member of N u may or may not exist [18].
An Alexandroff space [1] is a topological space such that every point has a minimal neighborhood, or equivalently, arbitrary intersection of open sets is open.
Recall [9] that, by definition, a function f from a topological space U into a topological space V is continuous iff the inverse image of each open set is open.
Proposition 2.1. [9] Assume that U and V are topological spaces, and that f is a function from U into V . The following statements are equivalent.
1. The function f is continuous.
2. The inverse image of each closed set is closed.
3. For each u ∈ U the inverse image of every neighborhood of f (u) is a neighborhood of u.

Closure and interior spaces
A closure operator on a set U is a function c : 2 U −→ 2 U from the power set 2 U of U into itself which satisfies the Kuratowski closure axioms. Namely, which satisfies corresponding axioms: The pair (U, c) is called a closure space. As it is known [9], c : 2 U −→ 2 U defines the topology on U , called the topology associated with the closure operator c. It follows from (3), (KC3), and (KC5) that for any subset X of U , c(X) is the smallest closed set containing X.
It is easily seen [13] that u / ∈ c(X) iff there exists an open set including u that does not intersect X. Equivalently, u ∈ c(X) ⇐⇒ every neighborhood of x intersects X.
Furthermore, for all u, v ∈ U u ∈ c({v}) ⇐⇒ u is contained in all closed sets that include v ⇐⇒ v is contained in all open sets that include u.
According to Proposition 2.1 and (4), a function f between closure spaces (U,

Approximation spaces
The basic construct of rough set theory is the approximation space, that is, a pair U, t , where U is the universe and t ⊆ U × U is an equivalence relation on U . In an approximation space U, t , the upper and lower approximations, apr t (X) and apr t (X), of a set X ⊆ U are defined respectively as follows [14]: where [u] t is the t-equivalence class of u ∈ U . They are indeed a pair of closure and interior operators on the set U . Moreover, for every X ⊆ U [14]: apr t (X) = apr t (apr t (X)) and apr t (X) = apr t (apr t (X)).
Consequently, the topology associated with the upper approximation is clopen and therefore Alexandroff. This, together with (4) and (9), leads to the following: For an approximation space U, t , the topology associated with apr t : 2 U −→ 2 U is clopen and therefore Alexandroff. Moreover, the set of all t-equivalence classes is unique minimal base for τ t .

Information systems
As rough set theory is applied to data analysis, an approximation space is usually induced from information systems. According to [14], an information system 1 is defined as a tuple T = (U, A, {D a | a ∈ A}, {f a | a ∈ A}), where U is a nonempty finite set, called the universe; A is a nonempty finite set of primitive attributes; for each a ∈ A, D a is the domain of values for a; and for each a ∈ A, f a : U → D a is the information function for the attribute a. To simplify the presentation, we can identify each attribute a with its information function and set D = a∈A D a . Then, an information system is simply a triplet (U, A, D), where A is a set of functions from U to D. Given an information system (U, A, D) and a subset of attributes B ⊆ A, the indiscernibility relation with respect to B is defined as is an approximation space. In particular, we can associate with each information system On the other hand, given an approximation space U, t , we can (somewhat trivially) associate with it a single-attribute information system (U, {a t }, D), where D = U/t is the set of equivalence classes (i.e. the quotient set) of U with respect to t and a t : In [7], homomorphisms between information systems are introduced. Given two information sys- for any a ∈ A 1 and x ∈ U 1 .

Categories and functors
A category C is a class of objects A, B, C, . . . together with a family of disjoint sets hom C (A, B), one for each ordered pair A, B of objects. Write f : , and call f an arrow of C with source A and target B. Given each pair f : A → B, g : B → C of arrows, there is unique arrow gf = g • f : A → C, called the composite of f and g. The composition is associative, and each object has an identity arrow that serves as a unit under composition [11].
A subcategory of C is a category D whose objects are objects of C and whose arrows are arrows of C with the same identities and composition of arrows.
An arrow f : A → B in a category C is called an isomorphism, and A and B are said to be isomorphic in C, if and only if it has an inverse arrow, i.e., if there is and only if an arrow g : and an epimorphism (or epic) if for any g, h : In what follows, a full subcategory D of C is a subcategory of C such that A small category is a category whose objects, and hence all of those arrows, form a set.
Following are some examples of small categories: • The category Set of sets and functions has objects including all sets A, B, · · · , and arrows of all (total) functions from A into B with the usual composition.
• The category Top of topological spaces and continuous functions.
• The category Alex of Alexandroff topological spaces and continuous functions.
• The category Clop of clopen topological spaces and continuous functions.
• The category Cls of closure spaces and continuous functions [16].
• The category Int of interior spaces and continuous functions [16].
The category Clop is a full subcategory of Alex. The categories Alex and Clop are full subcategories of the category Top.
Definition 2.3. A functor F : C → D from a category C to a category D consists of an object function F and an arrow function, also written F , in such a way that Under this composition, a functor can be viewed as an arrow of categories. In particular, the identity function on objects and arrows in a category C is the identity functor 1 C . Thus, we can define the category Cat of all small categories, whose arrows are the functors between categories. This allows us to define isomorphic categories, i.e., two categories are isomorphic if they are isomorphic as objects of Cat.

Categories of Clopen Topologies
In this section, we define the category AprS of approximation spaces and relation-preserving functions, and show that the relation-preserving functions between approximation spaces are precisely the continuous functions between the corresponding clopen topological spaces. We characterize clopen topologies on a set by introducing the concept of rough closure operator and rough interior operator on the set, and define the category RCls of rough closure spaces and continuous functions and the category RInt of rough interior spaces and continuous functions. We prove the equivalence of AprS, RCls and RInt. Furthermore, we define the category IS of information systems and O-A-D homomorphisms. We establish the relationship between IS and AprS by considering a subcategory NeIS of information systems non-expansive O-A-D homomorphisms.

The category of approximation spaces
In [15], Rydeheard and Burstall considered the category Rel with objects ordered pairs (U, R), where U is a set and R is a binary relation on U . Arrows f : (U, R) → (V, S) are relation-preserving functions; i.e., functions f : Considering the category Rel, we can define a second category AprS of approximation spaces and relation-preserving functions, which is the subcategory of Rel with objects ordered being pairs U, t , where U is a set and t is an equivalence relation on U , and arrows f : U, t → V, s being relation-preserving functions; i.e., functions f : When thought of as a relation-preserving function from U, t to V, s , we will symbolize the function as f : U, t → V, s . Consider two approximation spaces U, t and V, s , we have the two sets of arrows Thus, AprS is a full subcategory of Rel.
Proof Assume that f : U, t → V, s is a relation-preserving function. Let u ∈ U . If s . This gives (17). Conversely, assume that (17) This gives (15). This establishes Lemma 3.1.
Since the set of all equivalence classes is unique minimal base for the clopen topology associated with apr t : 2 U −→ 2 U , it follows from Item 4 of Proposition 2.1 that a function f : U, t → V, s is continuous iff (17) holds. This observation, together with Proposition2.1, Lemma 2.2, Lemma 3.1 and (8), leads to the following: Proposition 3.2. Assume that U, t and V, s are approximation spaces, and that f is a function from U into V . The following statements are equivalent.
1. The function f between approximation spaces is relation-preserving.
2. The function f between clopen spaces associated with the upper (resp. lower) approximations is continuous.
Since homeomorphisms are the isomorphisms in Top, it follows from Proposition 3.2 that an arrow f : U, t → V, s is an isomorphism if and only if the function f : U → V is bijective and its inverse function f −1 : V → U is relation-preserving. Therefore, from Lemma 3.1, we obtain the following:

The categories of rough closure and rough interior spaces
Note that (10) and (11) may be written as equivalent conditions apr t (X) = U − apr t (U − apr t (X)), or apr t (X) = U − apr t (U − apr t (X)).
This motivates the following: The pair (U, c U ) is called a rough closure space. Dually, an interior operator i U : 2 U −→ 2 U on a set U will be called a rough interior operator on U if it satisfies The pair (U, i U ) is called a rough interior space.
Notice that (20) is equivalent to c U (X) = i c U (c U (X)). This, together with (4), leads to the following: Proposition 3.5. Let c U : 2 U −→ 2 U be a rough closure operator on a set U . Then it dual interior operator is a rough interior operator on U , and the topology associated with c U is clopen and therefore Alexandroff. Moreover, the set of c U ({u}) is a basis for U . Indeed, it is unique minimal basis for U .
Consider the category Cls of closure spaces and continuous functions. We can define a second category RCls of all rough closure spaces and continuous functions, which is the subcategory of Cls with objects ordered pairs (U, c U ), where U is a set and c U is a rough closure operator on U . Arrows f : (U, c U ) → (V, c V ) are continuous functions; i.e., functions f : This shows that RCls is a full subcategory of Cls.
Similarly, we define a category RInt of all rough interior spaces and continuous functions, which is the subcategory of Int with objects ordered pairs (U, i U ), where U is a set and i U is a rough interior operator on U . Arrows f : (U, i U ) → (V, i V ) are continuous functions; i.e., functions f : For each ordered pair (U, i U ), (V, i V ) of objects of RInt, we have This shows that RInt is a full subcategory of Int.

Equivalence of AprS, RCls and RInt
Using (11) and Definition 3.4, we obtain Lemma 3.6. Let U, t be an approximation space. Then 1. the upper approximation apr t (X) : 2 U −→ 2 U is a rough closure operator on U .
2. the lower approximation apr t (X) : 2 U −→ 2 U is a rough interior operator on U .
Assume that c : 2 U −→ 2 U is a rough closure operator on a set U . It follows from (5) and Proposition 3.5, for each u ∈ U , c({u}) is not only the smallest closed set including u, but also the smallest open neighborhood of u. This, together with (7), gives that for any u, v ∈ U , u ∈ c({v}) if and only if v ∈ c({u}). It follows that c({u}) and c({u}) are either equal or they are disjoint. Therefore, the relation t ⊆ U × U defined by letting is an equivalence relation on U . Moreover, [u] t = c({u})) for each u ∈ U . Therefore, from (6) and (11), we have apr t (X) = c(X) for each X ⊆ U . This proves the following: Then t is an equivalence relation on U , and apr t (X) = c(X) for each X ⊆ U .
According to Proposition 3.2, Lemma 3.6, (22) and Proposition 3.7, approximation spaces on a set U are in bijective correspondence with rough closure operators on the set U ; moreover, the relationpreserving functions between approximation spaces are precisely the continuous functions between the corresponding rough closure.
According to Proposition 3.2 and Lemma 3.6, we can define a functor According to Lemma 3.6, (22) and Proposition 3.7, we can define a functor for any object (U, c U ) of RCls, and for f : It is easy to check that This proves that the two categories AprS and RCls are isomorphic as objects of Cat.
Similarly, according to (3) and Proposition 3.5, it is easy to check that the functor , in the category Cat is an isomorphism. Consequently, the three categories AprS, RCls and RInt are isomorphic as objects of Cat.

The category of information systems
The category of information systems IS has information systems as its objects and O-A-D homomorphisms as its arrows. We say that an O-A-D homomorphism is non-expansive if its attribute mapping is onto. To establish the relationship between IS and AprS, we consider a subcategory NeIS whose objects are information systems and arrows are non-expansive O-A-D homomorphisms. Then, it is is a non-expansive O-A-D homomorphism between information systems (U, A 1 , D 1 ) and (V, A 2 , D 2 ), then h O is a relation-preserving function between approximation space U, t A 1 and V, t A 2 by using (13). Hence, we can define a functor We can see that H • H ′ = 1 AprS but H ′ • H = 1 NeIS does not hold. This means that an information system is more informative than the finest approximation space derived from it. Indeed, we can usually induce the same approximation space from several different information systems by using different sets of attributes, and this is the main idea of attribute reduction in rough set theory.

Related Works
While rough set theory has been extensively studied from diverse perspectives, there were a few works on its categorical aspect. In [5], three categories of approximations, denoted by AprS, AprS, and AprS, are defined. Objects of the three categories, as well as our definition of AprS, are all approximation spaces. However, the require arrows to preserve not only the underlying equivalence relation but also lower and/or upper approximations. These arrows are called lower/upper transformations although they are simply morphisms instead of natural transformations in the sense of category theory. More precisely, let U, t and V, s be two approximation spaces. Then, a function f : U → V is called a lower and upper natural transformation if it satisfies, for any X ⊆ U , and respectively. In addition, f is simply a natural transformation if it is both a lower and upper natural transformation. It is then proved that a lower natural transformation preserves equivalence classes ( [5],Proposition 2.4). On the other hand, they also proved that a mapping is an upper natural transformation if and only if it preserves equivalence classes ( [5], Theorem 3.4). This indicates that a lower natural transformation is automatically an upper natural transformation, although it is generally not the case. Because a function preserving equivalence classes is necessarily relation-preserving (but not conversely), the categories introduced in [5] imposes a much stricter restriction on morphisms than ours. As a result, all three categories in [5] are subcategories of AprS (or equivalently, RCls or RInt) in this paper.
It is well-known that lower and upper approximations are set-theoretic counterparts of modal operators and in S5 modal logic. Hence, in the current context, we need to consider a special BAO with only one modal operator ⋄ with arity 1. In such a BAO, the normality and additivity conditions reduce to ⋄0 = 0 and ⋄(a + a ′ ) = ⋄a + ⋄a ′ respectively. In addition, extra algebraic axioms for S5 logic are T: a + ⋄a = ⋄a, 4: ⋄ ⋄ a = ⋄a, and 5: −⋄ a = ⋄(−⋄ a). It is easy to see that the rough closure space for any X ⊆ U ), then it is also a lower natural transformation. Unlike categories defined in [5] and this paper, there are also categories of rough sets whose objects are not only approximation spaces. The earliest categorical analysis of rough sets is the category ROUGH defined in [3]. Objects of ROUGH are triples R, t, X , where R, t is an approximation space and X ⊆ U . Let X t and X t denote the quotient sets of apr t (X) and apr t (X) respectively. Then, an arrow f : R, t, X → V, s, Y in ROUGH is a map f : X t → Y s such that f [X t ] ⊆ Y s . Thus, arrows of ROUGH must preserve lower approximations.
Yet another category of rough sets is based on the algebraic interpretation of rough sets [8]. A pair (U, B) is called a rough universe in [8], where U is the domain and B is a subalgebra of the power set Boolean algebra (2 U , ∪, −, ∅). Any pair (X 1 , X 2 ) such that X 1 , X 2 ∈ B and X 1 ⊆ X 2 is an I-rough set of (U, B). Then, the category RSC has all I-rough sets as its objects and an arrow f : [12]. While ROUGH and RSC looks different at first glance, they are shown to be equivalent in [12].
A subcategory of ROUGH, called ξ-ROUGH, is defined to have same objects as ROUGH, but its arrows must satisfy the additional condition of preserving the boundary region. That is, f : Analogously, ξ-RSC is the subcategory of RSC with the same collection of objects and its arrows f : (X 1 , It is then shown that ξ-ROUGH and ξ-RSC are equivalent and both are equivalent to Set 2 , whose objects are pairs of sets (X 1 , X 2 ) and arrows are pairs of functions (f, g) : (X 1 , X 2 ) → (Y 1 , Y 2 ) where f : X 1 → Y 1 and g : X 2 → Y 2 . The result shows that the role of approximation spaces has become hardly visible in the categories of algebraic rough sets. By contrast, our categories of approximation spaces, rough closure spaces, and rough interior spaces arise from a topological interpretation of rough sets. Hence, approximation spaces and continuous functions between them play the major role in such categories. In some sense, this means that categories based on I-rough sets somewhat lose structural information behind the construction of rough approximations.
On the other hand, the advantage of using I-rough sets is that more general categorical construction is possible. In [12], by changing the base from Set to an arbitrary topos, a natural generalization of RSC and ξ-RSC is obtained. Recalling that a topos, as a category-theoretic abstraction of Set, is a category C that has all finite limits, a subobject classifier, and all exponentials [2]. Then, the generalization of RSC (resp. ξ-RSC) to a parameterized class of categories RSC(C ) (resp. ξ-RSC(C )) is proposed and its topos-theoretic properties are explored in [12]. Objects of RSC(C ) and ξ-RSC(C ) are both triples (X, Y, m), where X and Y are objects of C and m : X → Y is a monic arrow in C . An arrow of RSC(C ) is a pair (f, g) : (X, Y, m) → (X ′ , Y ′ , m ′ ) where f : X → X ′ and g : Y → Y ′ are arrows in C such that g • m = m ′ • f , and it is also a ξ-RSC(C ) arrow if there exist objects (¬X, Y, ¬m) and (¬X ′ , Y ′ , ¬m ′ ) and arrow f − : ¬X → ¬X ′ in C such that ¬m ′ • f − = g • ¬m. By instantiating C to a particular topos M-Set (the category of actions of a monoid M on sets, see [6]), a possible application of RSC(C ) to monoid actions on rough sets is suggested in [12].

Conclusion
With the notions of rough closure operator and rough interior operator together with the functor, we obtain the equivalence of the category of all rough interior spaces and continuous functions, the category of rough closure spaces and continuous functions, the category of approximation spaces and relation-preserving functions. Our approach using information system to characterize the category of approximation space and its subcategory gives rise to a deeper understanding of the interplay among rough set theory, information system and category theory. It gives us not only a better understanding of the rough set theory structures in the context of category theory but also the relationships of different categories arising from some topological tools. By using the concepts of categories, one can further derive useful properties and applications of rough set theory. More precisely, category theory can be used to show how mathematical structures of a given type can be transformed into one another by functions that preserve some aspect of their structures; it provides a uniform language for speaking of various mathematical or topological structures and the mappings within those types. The theory of abstract categories, however, totally ignores the sets, operations, relations and axioms that specify the structure of objects in question. It just provides a language in which one can talk about how mappings preserve certain structure behaviors without a good understanding of the relationship of various properties or implications of what have been proved. For that reason, more technical insights and knowledge will help explore further theory and applications. We envision to work out more results along this line of research.