Q-Convergence in graded ditopological texture spaces

Convergence of graded difilters have been presented and investigated by the authors in [9]. In this paper, using graded Q-dinhd systems defined in [8] the authors define a different convergence type of graded difilters called Q-convergence which has some advantages and some disadvantages in comparison with the convergence defined in [9].


Introduction
In 1968, C. Chang defined the concept of fuzzy topological space as ordinary subset of the family of all fuzzy subsets of a given set [7].As a more suitable approach to the idea of fuzzyness, in 1985, Šostak and Kubiak independently redefined fuzzy topology where a fuzzy subset has a degree of openness rather than being open or not [12,14] (for historical developments and basic ideas of the theory of fuzzy topology see [15]).
Ditopological texture spaces were introduced by L. M. Brown as a natural extention of the work on the representation of latticevalued topologies by bitopologies in [10].The concept of ditopology is more general than general topology, bitopology and fuzzy topology in Chang's sense.An adequate introduction to the theory of texture spaces and ditopological texture spaces may be obtained from [1-5, 17, 18].
Recently, L. M. Brown and A. Šostak have presented the concept "graded ditopology" on textures as an extention of the concept of ditopology to the case where openness and closedness are given in terms of a priori unrelated grading functions [6].The concept of graded ditopology is more general than ditopology and fuzzy topology in Šostak's sense.Two sorts of neighborhood structure (graded dineighborhoods and graded Q-dineighborhoods) on graded ditopological texture spaces are presented and investigated in [8].In [9] graded difilters are introduced and their convergence is studied by means of graded dineighborhoods.
The aim of this paper is to introduce an alternative definition of convergence of graded difilters using graded Q-dineighborhoods.There is no relation between graded dineighborhoods and graded Q-dineighborhoods, i.e. they are completely different concepts.So, convergence and Q-convergence of graded difilters are also completely different.Convergence of graded difilters is not defined using graded dineighborhoods but a version of it.Contrary to the concept of convergence, Q-convergence is defined directly by means of Q-dineighborhoods.However, Q-convergence is defined on complemented grounded textures.

Preliminaries
We recall some basic concepts and properties from [3][4][5]: Let S be a set.A texturing S on S is a subset of P(S) which is a point separating, complete, completely distributive lattice with respect to inclusion which contains S, ∅ and for which meet coincides with intersection and finite joins with unions .The pair (S, S) is then called a texture or a texture space.
In general, a texturing of S need not be closed under set complementation, but there may exist a mapping σ : S → S satisfying σ(σ(A)) = A and A ⊆ B ⇒ σ(B) ⊆ σ(A) for all A, B ∈ S. In this case σ is called a complementation on (S, S) and (S, S, σ) is said to be a complemented texture.
For a texture (S, S), most properties are conveniently defined in terms of the p − sets A texture (S, S) is called a plain texture if it satisfies any of the following equivalent conditions: (1) For a set A ∈ S, the core of A (denoted by A ) is defined by ).In any texture space (S, S), the following statements hold: (3) For A j ∈ S, j ∈ J we have (4) A is the smallest element of S containing A for all A ∈ S.
(5) For A, B ∈ S, if A B then there exists s ∈ S with A Q s and (4) S = {∅, {a, b}, {b}, {b, c}, S} is a simple texturing of S = {a, b, c}.P a = {a, b}, P b = {b}, P c = {b, c}.It is not possible to define a complementation on (S, S).
(5) If (S, S), (V, V) are textures, the product texturing S ⊗ V of S × V consists of arbitrary intersections of sets of the form A dichotomous topology, or ditopology for short, on a texture (S, S) is a pair (τ, κ) of subsets of S, where the set of open sets τ satisfies Hence a ditopology is essentially a "topology" for which there is no priori relation between the open and closed sets.
A complementation σ on a texture (S, S) is called grounded [13] if there is an involution s → s on S such that σ(P s ) = Q s and σ(Q s ) = P s (s will be denoted by σ(s)) for all s ∈ S and in this case the complemented texture space (S, S, σ) is called "complemented grounded texture space".In [16], it is shown that a complemented plain texture is grounded.
It is well known that, in the classical set theory, "A ∩ B = ∅ if and only if A ⊆ X \ B for any subsets, A and B of X" however, in the fuzzy set theory, "A ∩ B = 0 implies A ⊂ 1 − B but in general, A ⊂ 1 − B doesn't imply A ∩ B = 0 for any fuzzy subsets, A and B of X".So it is defined an alternative binary implication in the fuzzy set theory such as; "A is quasicoincident with B (denoted by AqB) if and only if there exists an x ∈ X such that A(x) + B(x) > 1 for any fuzzy sets, A and B of X" and "A is not quasi-coincident with B (denoted by AqB) if and only if A(x) + B(x) ≤ 1 for all x ∈ X".These notions are generalized to the complemented texture spaces as follows: Let (S, S, σ) be a complemented texture space and A, B ∈ S. It is called that "A is quasi-coincident with B" (denoted by AqB) if A σ(B) and "A is not quasi-coincident with B" (denoted by AqB) if A ⊆ σ(B) [11].
For v ∈ V it is defined that Then (T v , K v ) is a ditopology on (S, S) for each v ∈ V .That is, if (S, S, T , K, V, V) is any graded ditopological texture space, then there exists a ditopology (T v , K v ) on the texture space (S, S) for each v ∈ V If (S, S, σ) is a complemented texture and (T , K) a (V, V)-graded ditopology on (S, S), then Example 2.2.Let (S, S, τ, κ) be a ditopological texture space and (V, V) the discrete texture on a singleton.Take (V, V) = (1, P( 1)) (The notation 1 denotes the set {0}) and define τ g : S → P(1) by τ g (A) = 1 ⇔ A ∈ τ .Then τ g is a (V, V)-graded topology on (S, S).Likewise, κ g defined by κ g (A) = 1 ⇔ A ∈ κ is a (V, V)-graded cotopology on (S, S). (τ g , κ g ) is called then the graded ditopology on (S, S) corresponding to ditopology (τ, κ).
Therefore graded ditopological texture spaces are more general than ditopological texture spaces.
The graded Q-dineighborhood (Q-dinhd) systems of the graded ditopological texture spaces were defined in [8].To avoid a long preliminaries we will give the following equivalent proposition instead of the definition.

Proposition 2.1 ([8]
).Let (T , K) be a (V, V)-graded ditopology on (S, S, σ) and N , M : S → V S mappings where N (s) = N s : S → V, M (s) = M s : S → V for each s ∈ S. Then ( N , M ) is a graded Q-dinhd system of the graded ditopological texture space (S, S, T , K, V, V) iff for each s ∈ S and A ∈ S.
For s ∈ S, the graded difilter (F, G) is called Q-diconvergent if F −→ q s and G −→ q σ(s).In this case, the couple (s, σ(s)) or s for short is called Q-limit of (F, G).Proposition 3.1.For a graded difilter (F, G) on (S, S, T , K, V, V) the following hold: (a) F −→ q s ⇔ "P s qA ⇒ T (A) ⊆ F(A)" (b) G −→ q s ⇔ "P s A ⇒ K(A) ⊆ G(A)".
Proof.(a) (⇒) : Let F −→ q s and P s qA.Since F −→ q s we have N s ⊆ F and so N s (A) ⊆ F(A).On the other hand, P s qA implies T (A) ⊆ N s (A).
Hence T (A) ⊆ F(A) is obtained.
(⇐) : Considering the definition of N s , we get N s (A) ⊆ F(A) because of the implication P s qA ⇒ T (A) ⊆ F(A).Thus we have F −→ q s.
The proof of (b) is similar to (a).
Proposition 3.2.For a regular graded difilter (F, G) on (S, S, T , K, V, V) the following hold: Proof.(a) Let F −→ q s and suppose that there exists v ∈ G(A) such that P s q]A[ v for a set A ∈ S. Since P s q]A[ v we have P s σ( {B ∈ S | B ⊆ A, P v ⊆ T (B)}) = {σ(B) | B ⊆ A, P v ⊆ T (B)} and so, there exists B ∈ S such that P s σ(B), B ⊆ A, P v ⊆ T (B).Because F −→ q s and P s qB, using Proposition 3.1 we have T (B) ⊆ F(B).So we get P v ⊆ T (B) ⊆ F(B) ⊆ F(A) and remembering v ∈ G(A) we obtain F(A) ∩ G(A) = ∅ which contradicts with the regularity of (F, G).The proof of (b) is similar to (a).Definition 3.2.Let (F, G) be a regular (V, V)-graded difilter on a graded ditopological texture space (S, S, T , K, V, V). ( all A ∈ S. (3) For s ∈ S if s is a Q-cluster point of F and σ(s) is a Q-cluster point of G then (F, G) is called Q-diclustering in (S, S, T , K, V, V).
(2) F ∨ G = V (i.e.∀A ∈ S, F(A) ∨ G(A) = F(A) ∪ G(A) = V ).some disadvantages in comparison with the convergence defined in [9].Convergence of graded difilters is not defined using graded dineighborhoods but a version of it because in general, a graded nhd is not a graded difilter.Contrary to the concept of convergence, Q-convergence is defined directly by means of Q-dinhds since ( N s , M σ(s) ) is a regular graded difilter (see Example 3.1).However, there is a handicap of this new structure: Q-convergence is defined on complemented grounded textures.