Advances in Fuzzy Systems
Volume 2018 (2018), Article ID 3871282, 15 pages
https://doi.org/10.1155/2018/3871282

Research Article

On -Double Fuzzy Filter Spaces

A. A. Abd El-Latif,1,2 H. Aygün,³ and V. Çetkin³

¹Department of Mathematics, Faculty of Science and Arts at Balgarn, P.O. Box 60, University of Bisha, Sabt Al-Alaya 61985, Saudi Arabia
²High Institute of Computer King Marriott, P.O. Box 3135, Alexandria, Egypt
³Department of Mathematics, Kocaeli University, 41380 Kocaeli, Turkey

Correspondence should be addressed to V. Çetkin; vcetkin@gmail.com

Received 1 November 2017; Accepted 26 February 2018; Published 15 May 2018

Academic Editor: Kemal Kilic

Copyright © 2018 A. A. Abd El-Latif et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We give in this paper the definitions of -double fuzzy filter base and -double fuzzy filter structures where and are strictly two-sided commutative quantales, and we also investigate the relations between them. Moreover, we propose second-order image and preimage operators of -double fuzzy filter base and study some of its fundamental properties. Finally, we handle the given structures in the categorical aspect. For instance, we show that the category -DFIL of -double fuzzy filter spaces and filter maps between these spaces is a topological category over the category SET.

1. Introduction

Kubiak [1] and Šostak [2] introduced the notion of -fuzzy topological space as a generalization of -topological spaces introduced by Chang [3]. At the bottom of it lies the degree of openness of an -fuzzy set. A general approach to the study of topological-type structures on fuzzy powersets was developed in [4–14].

On the other hand, Atanassov [15] introduced the idea of intuitionistic (double graded) fuzzy set. Çoker and his coworker(s) [16, 17] introduced the idea of topology of intuitionistic fuzzy sets. Recently, Mondal and Samanta [18] introduced the notion of intuitionistic gradation of openness which is a generalization of both fuzzy topological spaces [2] and the topology of intuitionistic fuzzy sets [16].

Working under the name “intuitionistic” did not continue because doubts were thrown about the suitability of this term, especially when working in the case of complete lattice These doubts were quickly ended in 2005 by Gutiérrez García and Rodabaugh [19]. They argued that this term is unsuitable in mathematics and applications. They concluded that they work under the name “double.”

The notion of -filter was introduced by Höhle and Šostak [7] as an expansion of fuzzy filter [20–25]. In recent years, -filters were used to introduce many kinds of lattice-valued convergence spaces [26–28]. -filter is an important tool to study -fuzzy topology [29, 30] and -fuzzy uniform space [26]. The structure of this paper is as follows. In Section 2, we recall some fundamental definitions related to quantale lattice by giving illustrative examples and also recall some definitions necessary for the main sections. In Section 3, we define -double fuzzy filter and -double fuzzy filter base and then study relations between them. In the next two sections, we consider two types of second-order Zadeh image and preimage operators of -double fuzzy filter base and examine their characteristics by giving examples.

2. Preliminaries

Throughout this paper, let be a nonempty set. Let be a complete lattice with the least element and the greatest element . For ,   for all . The second lattice belonging to the context of our work is denoted by and and

A complete lattice is called completely distributive, if for any family in the following identity holds:

Definition 1 (see [24, 31–33]). A triple is called a strictly two-sided commutative quantale (stsc-quantale, for short) iff it satisfies the following properties:

(L1) is a commutative semigroup.(L2), for all (L3) is distributive over arbitrary joins:

An stsc-quantale is an -distributive quantale (or stsc-biquantale [34]) if is distributive over nonempty meets:

Remark 2 (see [24, 25, 31–33, 35]). (1) A complete lattice satisfying the infinite distributive law is an stsc-quantale. In particular, the unit interval is an -distributive quantale.
(2) Every left-continuous t-norm on is an stsc-quantale.
(3) Every continuous t-norm on is an –distributive quantale.
(4) Every GL-monoid is an stsc-quantale.
(5) Let be an stsc-quantale. For each , we define

Then, it satisfies Galois correspondence; that is,

Definition 3 (see [1, 7, 24, 29, 31, 33, 35–38]). Let be an stsc-quantale. A mapping is called an order-reversing involution if it satisfies the following conditions:
(1) , for each .
(2) If , then , for each .
An stsc-quantale is called a Girard monoid [37] if
Hence, in case is a Girard monoid, residuation induces an order-reversing involution In this paper, we always assume that (resp., ) is a Girard monoid with an order-reversing involution , and the operation is defined by

unless otherwise specified, where denote the quantale operations on .

Remark 4 (see [39]). When the underlying lattice is the unit interval of the real numbers, the notion of a Girard monoid coincides with the notion of a left-continuous t-norm with strong induced negation .

Lemma 5 (see [34]). Let be a Girard monoid. For each , one has the following properties:
(1) If , then , , and
(2)
Let be a complete lattice and be a function. The Zadeh image and preimage operators and are defined by

Lemma 6 (see [40]). Let be an stsc-quantale and be a function. For each and , one has the following properties:

(1) with equality if is injective.(2).

Definition 7 (see [40]). Basic scheme for second-order image operators: let be a function.

Case 1. Consider

This is the Zadeh image operator of the Zadeh image operator. We denote it by ; that is, for all and ,

Case 2. Consider

This is the Zadeh preimage operator of the Zadeh preimage operator. We denote it by ; that is, for all and ,

Basic scheme for second-order preimage operators: let be a function.

Case 1. Consider

This is the Zadeh image operator of the Zadeh preimage operator. We denote it by ; that is, for all and ,

Case 2. Consider

This is the Zadeh preimage operator of the Zadeh image operator. We denote it by ; that is, for all and ,

In this paper, we consider additional operators as follows.
Define the operator as , for all and .
Define the operator as , for all and .
All algebraic operations on can be extended pointwise to the sets and as follows: for all , and ;

(1) iff (2)(3) iff

Definition 8 (see [41]). The pair of maps is called an -double fuzzy topology on if it satisfies the following conditions:
(LO1) , for each ,
(LO2) ,  ,
(LO3) and , for each ,
(LO4) and , for each .
The triplet is called an -double fuzzy topological space (-dfts, for short). and may be interpreted as gradation of openness and gradation of nonopenness, respectively.
Let and be -double fuzzy topologies on . We say that is finer than ( is coarser than ) if and for all .
Let and be -dfts’s. A function is called LF-continuous iff and , for all .

Thus, we have the category -DFTOP where the objects are -dfts’s and the morphisms are -continuous maps between these spaces.

Example 9. Let be a set, and Then, is a left-continuous t-norm (Lukasiewicz t-norm) with strong induced negation . Let be defined as follows: Define as follows:

Then, the pair is a -dft on .

Remark 10. (1) If with an order-reversing involution -dfts is the concept of Mondal and Samanta [18].
(2) If and are frames with 0 and 1, -dfts is the concept of Gutiérrez García and Rodabaugh [19].
(3) If , -dfts is the concept of Abd El-latif [42].

Definition 11 (see [29, 30]). A map is called an -filter if it fulfills the following conditions:
(LF1) and .
(LF2) , for each .
(LF3) If , then .
The pair, , is called an -filter space.

3. -Double Fuzzy Filters and -Double Fuzzy Filter Bases

Definition 12. The pair of maps is called an -double fuzzy filter (briefly, -dff) on if it fulfills the following axioms:
(DFF1) , for each .
(DFF2) ,   and ,  .
(DFF3) and , for each .
(DFF4) If , then and .
The triplet is called an -double fuzzy filter space (briefly, -dffs).
If and are two -dffs on , we say that is finer than (or is coarser than ), denoted by if and only if and , for each .

Definition 13. Let and be two -dffs’s. Then, a map is said to be
(i) a filter map if and only if and ;
(ii) a filter preserving map if and only if and .
Normally, the composition of filter maps (resp., filter preserving maps) is a filter map (resp., filter preserving map).
Hence, we get to the category -DFIL with objects of all the -dffs’s and the morphisms are filter maps between these spaces.

Remark 14. (i) Let be an -filter on and defined by . Then, the pair is an -dff on . Therefore, -dff is a generalization of -filter due to Höhle and Šostak [7, 29].
(ii) If , the definition of -dff coincides with the definition of a proper -intuitionistic fuzzy filter due to Abd El-latif [42].

Theorem 15. Each -dff produces an -dfts .

Proof. When contrasting the axioms of -dff and -dft, we find (DFF4) implying (DFT4).
Let . Then, for all ; due to (DFF4), we have that and for all . So,

Then, we can get an -dft defined by

Theorem 16. Let and be -dffs’s. If is a filter map, then is an LF-continuous map.

Proof. Let . If or , then the proof is easy. Let and . Then, from the definition of double filter map and Theorem 15, we have

Corollary 17. The function defined by and is a functor.

Notation 18. Let be two maps and . Then, and are defined as follows:

Definition 19. The pair of maps is called an -double fuzzy filter base (briefly, -dffb) on if it fulfills the following axioms:
(DFFB1) , for each .
(DFFB2) ,   and ,  .
(DFFB3) and , for each .
If and are two -dffb’s on , we say is finer than (or is coarser than ) denoted by if and only if and , for each .

Remark 20. (i) An -dffb is a generalization of -filter base due to Kim and Ko [40].
(ii) If is an -dff, then is an -dffb with and .
(iii) If is an -dffb, then, by (DFFB3), implies and .

Theorem 21. If is an -dffb, then is the coarsest -dff which satisfies and .

Proof. (DFF1) For each ,

(DFF2) and (DFF4) are easily checked.
(DFF3) Suppose that there exist such that

By the definition of and (), there exist with and such that

Since is an -dffb,

Since , we have

It is a contradiction. Thus, , for each . Similarly, , for each .
Let be another -dff which is finer than , that is, and . Then, we have

Theorem 22. are maps fulfilling the following conditions:
(C1) , for each ,
(C2) and and for each finite index set , if , then and .
We define the maps as

where and are taken for every finite index set such that , respectively. Then, the following properties are satisfied:
(i) is an -dffb on .
(ii) If , and is an -dffb on , then and .

Proof. (i) (DFFB1) For each , the following is valid:

(DFFB2) It is clear by condition (C2).
(DFFB3) For each and for any two finite index sets with and , since , by the definition of and , we get

If supremum and infimum are taken over finite index set , respectively, then by (2) and (3),

Thus, is an -dffb on .
(ii) For any finite family , the following are true:

Then, and .

Theorem 23. Let and be two -dffb’s on and , respectively, and be a function. Then, one has the following properties:
(i) is a filter map if and only if and .
(ii) is a filter preserving map if and only if and .
(iii) If and , then is a filter map.
(iv) If and , then is a filter preserving map.

Proof. Proving condition (i) is enough since the other conditions are similarly proved.
(i) (:) Since and , for each , it is trivial.
(:) Let and , for each . We will show that is a filter map. For arbitrary , we have

Thus, is a filter map.

Example 24. Let be a set, be the stsc-quantale with Lukasiewicz t-norm, and be defined by . Define the maps as follows:

It can be seen by easy computation that

(1) and are not -double fuzzy filters but they are -double fuzzy filter bases, so they generate -double fuzzy filters and .

(2) is not an -double fuzzy filter base and it does not satisfy condition (C2) of Theorem 22.

(3) Since , is a filter map and is a filter preserving map though and

We also note that if is considered as a frame, then is an -double fuzzy filter base.

4. The Types , of Preimages and Images of -Double Fuzzy Filter Bases

Theorem 25. Let be a function and be an -dffb on . Then, the following properties are satisfied.
(i) If implies and , then is an -dffb on and is the coarsest -dff on for which is a filter map.
(ii) If is surjective, then is an -dffb.
(iii) If implies and , is injective, and is an -dff on , then is an -dff on .

Proof. (i) (DFFB1) For each , we have

(DFFB2) Since , then and . By assumption, and .
(DFFB3) Suppose that there exist such that

By the definition of and , there exist with and such that

Since is an -dffb, the following is valid:

Then,

By the definition of , there exists with such that

On the other hand, since

then
This contradicts the assumption. Thus,

Similarly,
Hence, is an -dffb on .
Let be another -dff on such that is a filter map. Then, for each , the following inequalities are valid:

(ii) Since is surjective, implies . So, and Then, by (i), is an -dffb on .
(iii) (DFF1)–(DFF3) are obvious.
(DFF4) Let , for . Since is injective, there exists with and . It implies

If supremum and infimum are taken over , respectively, then it is clear that and .

Theorem 26. Let be a family of functions and be a family of -dffb’s on satisfying the following condition:
(C) For each finite subset of , if , then and .
We define the maps as

where and are taken for every finite index subset of such that .
Let and be given. Then, the following properties are satisfied.
(i) is an -dffb on and is the coarsest -dff for which is a filter map for each .
(ii) A function is a filter map if and only if, for each ,   is a filter map.
(iii) and .

Proof. (i) (DFFB1) Let with . Then, the following inequality is valid:

(DFFB2) By condition (C), and . Since, , then and .
(DFFB3) Suppose that there exist such that
By the definition of , and , there exists a finite subset of with such that
Again, by the definition of and , there exists a finite subset of with such that

Put such that

Since, for each , we have

From the definition of , there exists with such that

On the other hand, since

and since is finite, we have

This contradicts the assumption. Then, , for each . Similarly, , for each, Hence, is an -dffb on .
Since and , for each , by Theorem 23(iii), is a filter map.
Let be an -dff on such that, for each , the map is a filter map. Then,

For any finite subset of with , since and , for each , we have

Hence, by the definition of and , it is obvious that
(ii) Necessity of the composition condition is obvious.
Conversely, for every finite subset of with , since, for each ,   is a filter map, that is, and . Since , we have

By the definition of and , we have and .
(iii) Put and ; by applying (i) to both and , the desired equality is obtained.

The following corollaries are the direct results of Theorem 26.

Corollary 27. Let be a family of -dffb’s on satisfying the following condition:
(C) For any finite subset of , if , then and .
We define the maps as

where and are taken for every finite index subset of such that Then, is an -dffb on and is the coarsest -dff which is finer than for each .

Example 28. Let be a set and be an stsc-quantale with (Lukasiewicz t-norm). We define maps as follows :

Each for is an -double fuzzy filter base but is not.

Corollary 29. Let be projection maps, for all , where is the product set. Let be a family of -dffb’s on satisfying the following condition:
(C) For any finite subset of , if , then and .
We define the maps as

where and are taken for every finite subset of such that .
Let and be given. Then, the following properties are satisfied:
(i) is an -dffb on and is the coarsest -dff on for which is a filter map.
(ii) A map is a filter map if and only if, for each , is a filter map.

In Corollary 29, the structure is called a product of -dffs’s on .

Theorem 30. Let be an injective function and be an -dffb on . Then, the following properties are satisfied:
(i) is an -dffb on , and is the coarsest -dff, for which the function is a filter preserving map.
(ii) is an -dffb on with and .

Proof. (i) (DFFB1) For each , we have

(DFFB2) It is straightforward from the definition.
(DFFB3) Suppose that there exist such that

By the definition of , we have

Since is an -dffb, the following is obtained:

Thus,

By the definition of , there exists with such that

Since and is injective,

This contradicts the assumption. Then,

Similarly, it can be verified that

Hence, is an -dffb on .
For each , we have

Hence, by Theorem 23(ii), is a filter preserving map.
Let be another -dff such that is a filter preserving map. So, for each , the following is valid:

Hence, . Similarly, it can be proved that .
(ii) If , then , and By Theorem 25(i), is an -dffb on . For each , following equalities are obtained:

Example 31. Let be sets and be the stsc-quantale with Lukasiewicz t-norm Let be a function defined by and be defined by We define maps as follows:

Then, is an -double fuzzy filter base but is not an -double fuzzy filter base.

Theorem 32. Let be a family of injective functions and be a family of -dffb’s on satisfying the following condition:
(C) For any finite subset of , if , then , and .
We define the maps as

where and are taken for every finite index subset of . Then, the following properties are satisfied:
(i) is an -dffb on and is the coarsest -dff for which is a filter preserving map.
(ii) A map is a filter preserving map if and only if, for each , is a filter preserving map.

Proof. (i) By Corollary 27 and Theorem 30, is an -dffb on .
Since is injective, for each ,

Hence, is a filter preserving map, for each .
According to Theorem 26(i), other cases are similarly proved.
(ii) It is proved in the same way as Theorem 26(ii).

Definition 33 (see [43]). (a) Let be a concrete category over . is said to be amnestic provided that its fibres are partially ordered classes; that is, no two different -objects are equivalent.
(b) Let and be categories. A functor is called topological provided that every -structured source has a unique -initial lift .

Proposition 34 (see [43]). If is a functor such that every -structured source has a -initial lift, then the following conditions are equivalent:

(1) is topological.(2) is uniquely transportable.(3) is amnestic.

Theorem 35. The forgetful functor - defined by and is topological.

Proof. The proof follows from Definition 33, Proposition 34, and Theorem 26.

5. The Types , of Images and Preimages of -Double Fuzzy Filter Bases

Theorem 36. Let be a surjective function and be an -dffb on . Then, the following properties are satisfied:
(i) is an -dffb on .
(ii) is the coarsest -dff on for which is a filter preserving map.
(iii) If is an -dff, then and .

Proof. (i) and (ii) are proved in the same manner as Theorem 25(i).
(iii) Let be an -dff. Since is surjective, is equivalent to . Then, for each , it is clear that

Hence, . Similarly, is obtained.

Remark 37. Let be a bijective function, be an -dffb on , and be an -dffb on . Then, the following equalities are clear.

(i) and .(ii) and .

Remark 38. Let be a bijective function and be an -dffb on . Then, it follows from Remark 37(ii) and Theorem 25 that is an -dffb on and is the coarsest -dff on for which is a filter map.

Theorem 39. Let be a function and be a family of -dffb’s on satisfying the following condition:
(C) For every finite subset of , if , then and . Then, the following properties are satisfied:
(i) If is bijective, then and .
(ii) If is injective, then and

Proof. (i) Let us consider the following condition:
For every finite subset of , if , then and .
For the proof, it is enough to show that .
: For any finite subset of with , since is injective, by Lemma 6(!),

By (C1), we have

and thus and is satisfied.
: Suppose that, for every finite subset of with ,  . Then, for each , there exists with such that

By (C), . By Lemma 6(!),

This contradicts the assumption. Thus, . Similarly, for every finite subset of , if , then .
Since is surjective, by Theorem 36, exists for each . By Corollary 27 and (C1), exists.
For each finite subset of such that with , the following inequalities are satisfied:

This implies that

So, the following are clear:

For any finite subset of with , there exist with . Thus,

This implies that

From the above inequalities, we have

(ii) It is proved by the same method as in (i) and Theorem 36(ii).

Theorem 40. Let be a family of functions and be a family of -dffbs’ on satisfying the following condition:
(C) For any finite subset of , if , then and .
We define the maps as

where and are taken for every finite subset of . Let and . Then, the following properties are satisfied:
(i) If is surjective for some , then is an -dffb on and is the coarsest -dff for which the map is a filter preserving map.
(ii) A function is a filter preserving map if and only if, for each , is a filter preserving map.
(iii) If are surjective for all , then

Proof. (i) (DFFB2) Since is surjective for some and (C), ,   and , .
According to Theorems 26(i) and 32(i), other cases are similarly proved.
(ii) The proof is similar to Theorem 26(ii).
(iii) Let us consider the following condition:
(C1) For any finite subset of , if , then and .
For the proof, it is enough to show that .
: For any finite subset of , if , by (C1), we have

: Suppose that, for any finite subset of with , we have . Then, for each , there exists with such that

By (C),

This is a contradiction. Thus, . Similarly, .
For any finite index set with , by the definition of
and , the following inequalities are obtained:

Hence,

For any finite index set with , since is surjective, for each , there exists with such that
Thus,

Hence,

6. Conclusion

In this study, we introduced the notions of -double fuzzy filter space and -double fuzzy filter base where and are stsc-quantales as an extension of frames. We showed the existence of initial and also final -double fuzzy filter structures. We also proved that the category -DFIL is a topological category over SET. By giving illustrative examples, we considered two types of second-order Zadeh image and preimage operators of -double fuzzy filter.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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