Serial relation and textural rough set

The generalized rough set theory is based on the lower and upper approximation operators defined on the binary relation. The rough sets obtained from serial relations take an important place in topological applications. In this paper, we consider serial relation for texture spaces. A texturing U of a set U is a complete and completely distributive lattice of subset of the power set P(U) which satisfies some certain conditions. Serial relation is defined by using textural sections and presections under a direlation on a texturing. We give some properties of serial direlation and a discussion on rough set theory from the textural point of view under serial direlation. Further, the concept of serial direlation has been characterized in terms of lower and upper textural approximation operators.


Introduction
The rough set was introduced by Pawlak in 1982 as a tool for dealing with the incomplete knowledge in information and decision systems. The main concept of the theory is the lower and upper approximation operators formed by an equivalence relation on a finite universe [14]. However, it is clear that the equivalence relation has some limitations in rough set theory applications. In order to expand the application areas of the theory, the equivalence relation has been replaced by an arbitrary binary relation. In this way, it has been the generalization of concepts of Pawlak rough set approximation operators [17][18][19].
A texturing U of a set U is a complete and completely distributive lattice of subset of the power set P(U ) which satisfies some certain conditions. It was shown that [1][2][3] texture spaces provide a a unified setting for the study of fuzzy lattices and their topologies and bitopologies. Further direlations which are compatible the lattice structure of texture spaces defined as a suitable morphisms in categories of texture spaces. In recent years, generalizations of some concepts have been studied in texture space theory [10][11][12][13].
In [6], textural rough set algebra was introduced to approach for generalized rough set, and it is obtained effective results for classical rough sets. Note that (U, U, σ, r ← , R ← ) is called a textural rough set algebra where (r, R) is a complemented direlation on complemented texture space (U, U, σ) and (R ← , r ← ) is inverse direlation of (r, R). The pair (r ← A, R ← A) is approximations of a set A ∈ U where r ← A and R ← A are presections of A. It has been studied rough sets through algebraic approach in texture theory [7][8][9]. The aim of this study is introduced the notion of serial direlation and analyze the corresponding textural rough sets.

Texture spaces
Let U be a set. A texturing U of U is a subset of P(U ) which is a pointseparating, complete, completely distributive lattice containing U and ∅, and for which meet coincides with intersection and finite joins with union. The pair (U, U) is then called a texture space, or shortly texture.
For u ∈ U , the p-sets and, as a dually, the q-sets are defined by A mapping σ U : U → U is called a complementation on (U, U) if it satisfies the conditions σ U (σ U (A)) = A for all A ∈ U and A ⊆ B =⇒ σ U (B) ⊆ σ U (A) for all A, B ∈ U. In this case (U, U, σ) is said to be complemented texture.    1]. Again (I, I, ι) is a complemented texture, which is called unit interval texture. Here P t = [0, t] and Q t = [0, t) for all t ∈ I.

Direlations
Let (U, U), (V, V) be textures. Consider the product texture P(U ) ⊗ V of the textures (U, P(U )) and (V, V) (see Example 1 (4)). We denote the p-sets and the q-sets by P (u,v) and Q (u,v) , respectively. From the product texturing, it is obtained that .
(2) Let (r, R) be a direlation from the discrete texture (U, P(U )) to the discrete texture (V, P(V )). Since P(U × V ) = P(U ) ⊗ P(V ), r and R are point relation from U to V .
Inverses of a direlation: The inverses of r and R are defined by The complement of a direlation: Let (r, R) be a direlation between the complemented textures (U, U, σ U ) and (V, V, σ V ).
(1) The complement r of the relation r is the co-relation Serial relation and textural rough set (2) The complement R of the co-relation R is the relation Order between Direlations: Let (r 1 , R 1 ) and (r 2 , R 2 ) be direlations from (U, U) to (V, V). The inclusion between direlations is defined by

Sections and presections
Let us recall that [4] some properties of sections and presections are given in this subsection.
Let (r, R) be a direlation on (U, U) and A, B ∈ U. The A-sections under (r, R) are given as: Likewise, the B-presections under (r, R) are given as Note that section and presection are related as in the next result: Lemma 3. Let (r, R) be a direlation (U, U) and A ∈ U. Then: Some basic properties of sections and presections such as inclusion and meet and join are given below.

Serial direlation
As is known, in the rough set theory, the characterizations of reflexive and transitive binary relations are given by the lower and upper approximation operators [18]. This leads to the following serialness concepts for a direlation on texture spaces. Proof. (i) Let (r, R) be a reflexive direlation on (U, U). Since r ← (A) ⊆ A ⊆ R ← (A) for all A ∈ U, (r, R) is a serial direlation.
(ii) Let (r, R) be a serial and symmetric and transitive direlation on (U, U).
Lemma 9. Let (r, R) be a direlation on (U, U). We have: Proof. Suppose that (r, R) is a direlation on (U, U).
(i) From Lemma 4, R ← ∅ = ∅ and r ← U = U . Then Proof. We suppose that (r, R) is a symmetric direlation on (U, U). Since (r, R) = (r, R) ← , we have r ← A = R ← A and R ← A = r ← A for all A ∈ U. Then, for A ∈ U: Corollary 11. Let (r, R) ← be inverse direlation of (r, R). Then (r, R) is (inverse-) serial if and only if (r, R) ← is (inverse-) serial.
Let (U, U, σ) be a complemented texture and L, H : U → U be two unary operators. Recall that [6] L and H are called dual operators on (U, U)if L(A) = H(σ(A)) and H(A) = L(σ(A)) for all A ∈ U. Now, we consider the following property: Then we have (L 1 ) ⇐⇒ (H 1 ) and (L 2 ) ⇐⇒ (H 2 ), for the dual operators L, H : U → U on the texture (U, U, σ).
Recall that [6] if the dual operators L and H satisfy conditions (L 1 ) and (L 2 ) (or equivalently,(H 1 ) and (H 2 )), Note that if (r, R) is a complemented direlation on (U, U, σ), then the system (U, U, σ, r ← , R ← ) is a textural rough set algebra where r ← , R ← : U → U are approximation operators.
that (r ← A, R ← A) = (app r A, app r A) for all A ∈ U. Then we have: r is serial ⇐⇒ app r A ⊆ app r A, A ⊆ U ⇐⇒ r ← A ⊆ R ← A, A ∈ P(U ) ⇐⇒ (r, R) is serial.

Conclusion
Rough sets are defined on the basis of lower and upper approximation operators obtained from the equivalence relations. In addition, generalized rough sets obtained from reflexive, transitive or serial relations are used in solving various problems in many areas as well as in topology.
Textural rough set algebra was introduced to approach for generalized rough set, and it is obtained effective results for classical rough sets. On the other hand, ditopologies (dichotomous topologies) on textures unify the fuzzy topologies, topologies and bitopologies in a non-complemented setting by means of duality in the textural concepts [5].
In this study, serial direlations are defined and their algebraic properties are investigated. In further studies, the conditions for obtaining a ditopology from the serial direlation and the algebraic properties of serial textural rough sets can be examined.