# On (L, M)-Double Fuzzy Filter Spaces.

1. IntroductionKubiak [1] and Sostak [2] introduced the notion of L-fuzzy topological space as a generalization of L-topological spaces introduced by Chang [3]. At the bottom of it lies the degree of openness of an L-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. Coker 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 L. These doubts were quickly ended in 2005 by Gutierrez Garcia 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 L-filter was introduced by Hohle and Sostak [7] as an expansion of fuzzy filter [20-25]. In recent years, L-filters were used to introduce many kinds of lattice-valued convergence spaces [26-28]. L-filter is an important tool to study L-fuzzy topology [29, 30] and L-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 (L, M)-double fuzzy filter and (L, M)-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 (L, M)-double fuzzy filter base and examine their characteristics by giving examples.

2. Preliminaries

Throughout this paper, let X bea nonempty set. Let L = (L, [less than or equal to], [disjunction], [conjunction]) be a complete lattice with the least element [0.sub.L] and the greatest element [1.sub.L]. For [alpha] [member of] L, [[alpha].bar](x) = [alpha] for all x [member of] X. The second lattice belonging to the context of our work is denoted by M and [M.sub.0] = M - {[0.sub.M]} and [M.sub.1] = M - {[1.sub.M]}.

A complete lattice L = (L, [less than or equal to], [disjunction], [conjunction]) is called completely distributive, if for any family {{[a.sub.i,j] : j [member of] [J.sub.i]}: i [member of] I} in L the following identity holds:

[mathematical expression not reproducible]. (1)

Definition 1 (see [24, 31-33]). A triple L = (L, [less than or equal to], [dot encircle]) is called a strictly two-sided commutative quantale (stsc-quantale, for short) iff it satisfies the following properties:

(L1) (L, [dot encircle]) is a commutative semigroup.

(L2) x [dot encircle] [i.sub.L] = X, for all x [member of] L.

(L3) [dot encircle] is distributive over arbitrary joins:

[mathematical expression not reproducible]. (2)

An stsc-quantale L = (L, [less than or equal to], [dot encircle]) is an [??]-distributive quantale (or stsc-biquantale [34]) if [dot encircle] is distributive over nonempty meets:

[mathematical expression not reproducible]. (3)

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 ([0, 1], [less than or equal to], [conjunction], 0, 1) is an [??]-distributive quantale.

(2) Every left-continuous t-norm T on [0,1], ([0,1], [less than or equal to], T) is an stsc-quantale.

(3) Every continuous t-norm T on [0,1], ([0,1], [less than or equal to], T) is an [??]-distributive quantale.

(4) Every GL-monoid is an stsc-quantale.

(5) Let (L, [less than or equal to], [dot encircle]) be an stsc-quantale. For each x, y [member of] L, we define

[mathematical expression not reproducible]. (4)

Then, it satisfies Galois correspondence; that is,

[mathematical expression not reproducible]. (5)

Definition 3 (see [1, 7, 24, 29, 31, 33, 35-38]). Let (L, [less than or equal to], [dot encircle]) be an stsc-quantale. A mapping * : L [right arrow] L is called an order-reversing involution if it satisfies the following conditions:

(1) [x.sup.**] = X, for each x [member of] L.

(2) If x [less than or equal to] y, then [y.sup.*] [less than or equal to] [x.sup.*], for each x, y [member of] L.

An stsc-quantale is called a Girard monoid [37] if (x [??] [0.sub.L]) [??] [0.sub.L] = x, [for all]x [member of] L.

Hence, in case L is a Girard monoid, residuation [??] induces an order-reversing involution * : L [right arrow] L. In this paper, we always assume that (L, [less than or equal to], [dot encircle], [direct sum], *) (resp., (M, [less than or equal to], [??], [??], *)) is a Girard monoid with an order-reversing involution *, and the operation [direct sum] is defined by

x [direct sum] y = [([x.sup.*] [dot encircle] [y.sup.*]).sup.*],

[x.sup.*] = x [??] [0.sub.L] (6)

unless otherwise specified, where [??], [??] denote the quantale operations on M.

Remark 4 (see [39]). When the underlying lattice L is the unit interval [0,1] of the real numbers, the notion of a Girard monoid coincides with the notion of a left-continuous t-norm with strong induced negation *, ([x.sup.*] = x [??] 0).

Lemma 5 (see [34]). Let L be a Girard monoid. For each x, y, z, [x.sub.i], [y.sub.i] [member of] L, one has the following properties:

(1) If y [less than or equal to] z, then x [dot encircle] y [less than or equal to] x [dot encircle] z, x [direct sum] y [less than or equal to] x [direct sum] z, x [??] y [less than or equal to] x [??] z, and y [??] x [greater than or equal to] z [??] x.

(2) x [dot encircle] y [less than or equal to] x [conjunction] y [less than or equal to] x [disjunction] y [less than or equal to] x [direct sum] y.

Let L be a complete lattice and [phi] : X [right arrow] Y be a function. The Zadeh image and preimage operators [[phi].sup.[right arrow]] : [L.sup.X] [right arrow] [L.sup.Y] and [[phi].sup.[left arrow]] : [L.sup.Y] [right arrow] [L.sup.X] are defined by

[[phi].sup.[right arrow]] ([lambda]) (y) = [??] {[lambda](x) | y = [phi](x)},

[[phi].sup.[left arrow]] ([mu]) (x) = [mu]([phi] (x)), (7)

[for all]x [member of] X, y [member of] Y.

Lemma 6 (see [40]). Let (L, [less than or equal to], [dot encircle]) be an stsc-quantale and [phi] : X [right arrow] Y be a function. For each [lambda], [mu] [member of] [L.sup.X] and [[lambda].sub.i] [member of] [L.sup.Y], one has the following properties:

(1) [[phi].sup.[left arrow]] ([lambda] [dot encircle] [mu]) [less than or equal to] [[phi].sup.[left arrow]] ([lambda]) [dot encircle] [[phi].sup.[left arrow]] ([mu]) with equality if [phi] is injective.

(2) [[phi].sup.[left arrow]] ([[dot encircle].sup.n.sub.i=1] [[lambda].sub.i]) = [[dot encircle].sup.n.sub.i=1] [[phi].sup.[left arrow]] ([[lambda].sub.i]).

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

Case 1. Consider

[mathematical expression not reproducible]. (8)

This is the Zadeh image operator of the Zadeh image operator. We denote it by [[phi].sup.[??].sub.1]; that is, for all [mathematical expression not reproducible] and [mu] [member of] [L.sup.Y],

[[phi].sup.[??].sub.1] (U)([mu]) = [[[[phi].sup.[right arrow].sub.L].sup.[right arrow].sub.L] (U)([mu])

= [??]{U ([lambda]) = [mu] = [[phi].sup.[right arrow].sub.L] ([lambda])}. (9)

Case 2. Consider

[mathematical expression not reproducible]. (10)

This is the Zadeh preimage operator of the Zadeh preimage operator. We denote it by [[phi].sup.[??].sub.2]; that is, for all [mathematical expression not reproducible] and [mu] [member of] [L.sup.Y],

[[phi].sup.[??].sub.2] (U) ([mu]) = [[[[phi].sup.[left arrow].sub.L].sup.[left arrow].sub.L] (U) ([mu]) = U [omicron] [[phi].sup.[left arrow].sub.L] ([mu]). (11)

Basic scheme for second-order preimage operators: let [phi]: X [right arrow] Y be a function.

Case 1. Consider

[mathematical expression not reproducible]. (12)

This is the Zadeh image operator of the Zadeh preimage operator. We denote it by [[phi].sup.[??].sub.1]; that is, for all [mathematical expression not reproducible] and [lambda] [member of] [L.sup.X],

[[phi].sup.[??].sub.1] (V) ([lambda]) = [[[[phi].sup.[left arrow].sub.L].sup.[left arrow].sub.L] (V)([lambda])

= [??] (V ([mu]): [lambda] = [[phi].sup.[left arrow].sub.L] ([mu])}. (13)

Case 2. Consider

[mathematical expression not reproducible]. (14)

This is the Zadeh preimage operator of the Zadeh image operator. We denote it by [[phi].sup.[??].sub.2]; that is, for all [mathematical expression not reproducible] and [lambda] [member of] [L.sup.X],

[[phi].sup.[??].sub.2] (V) ([lambda]) = [[[[phi].sup.[right arrow].sub.L].sup.[left arrow].sub.L] (V) ([lambda]) = V [omicron] [[phi].sup.[right arrow].sub.L] ([lambda]). (15)

In this paper, we consider additional operators as follows.

Define the operator [mathematical expression not reproducible] as [[phi].sup.*[??].sub.1] (V)([lambda]) = [??] {V([mu]) : [lambda] = [[phi].sup.[left arrow].sub.L]([mu])}, for all [mathematical expression not reproducible] and [lambda] [member of] [L.sup.X].

Define the operator [mathematical expression not reproducible] as [[phi].sup.*[??].sub.1] (V)([mu]) = [??] {V([lambda]) : [mu] = [[phi].sup.[left arrow].sub.L]([lambda])}, for all [mathematical expression not reproducible] and [mu] [member of] [L.sup.Y].

All algebraic operations on L can be extended pointwise to the sets [L.sup.X] and [mathematical expression not reproducible] as follows: for all x [member of] X, [lambda], [mu] [member of] [L.sup.X], and [mathematical expression not reproducible];

(1) A [less than or equal to] [mu] iff [lambda](x) [less than or equal to] [mu](x).

(2) ([lambda] [dot encircle] [mu])(x) = [lambda](x) [dot encircle] [mu](x).

(3) U [less than or equal to] V iff U([lambda]) [less than or equal to] V([lambda]).

Definition 8 (see [41]). The pair (T, [T.sup.*]) of maps T, [T.sup.*] : [L.sup.X] [right arrow] M is called an (L, M)-double fuzzy topology on X if it satisfies the following conditions:

(LO1) T([lambda]) [less than or equal to] [([T.sup.*]([lambda])).sup.*], for each [lambda] [member of] [L.sup.X],

(LO2) T(0) = T([1.bar]) = [1.sub.M], [T.sup.*]([0.bar]) = [T.sup.*]([1.bar]) = [0.sub.M],

(LO3) T([[lambda].sub.1] [dot encircle] [[lambda].sub.2]) [greater than or equal to] T([[lambda].sub.1]) [??] T([[lambda].sub.2]) and [T.sup.*]([[lambda].sub.1] [dot encircle] [[lambda].sub.2]) [less than or equal to] [T.sup.*]([[lambda].sub.1]) [??] [T.sup.*] ([[lambda].sub.2]), for each [[lambda].sub.1], [[lambda].sub.2] [member of] [L.sup.X],

(LO4) [mathematical expression not reproducible], for each [[lambda].sub.i] [member of] [L.sup.X], i [member of] [DELTA].

The triplet (X, T, [T.sup.*]) is called an (L, M)-double fuzzy topological space ((L, M)-dfts, for short). T and [T.sup.*] may be interpreted as gradation of openness and gradation of nonopenness, respectively.

Let (U, [U.sup.*]) and (T, [T.sup.*]) be (L, M)-double fuzzy topologies on X. We say that (U, [U.sup.*]) is finer than (T, [T.sup.*]) ((T, [T.sup.*]) is coarser than (U, [U.sup.*])) if T([lambda]) [less than or equal to] U([lambda]) and [T.sup.*]([lambda]) [greater than or equal to] [U.sup.*] ([lambda]) for all [lambda] [member of] [L.sup.X].

Let (X, T, [T.sup.*]) and (Y, U, [U.sup.*]) be (L, M)-dfts's. A function [phi] : X [right arrow] Y is called LF-continuous iff U([lambda]) [less than or equal to] T([[phi].sup.[left arrow]]([lambda])) and [U.sup.*] ([lambda]) [greater than or equal to] [T.sup.*] ([[phi].sup.[left arrow]] ([lambda])), for all [lambda] [member of] [L.sup.Y].

Thus, we have the category (L, M)-DFTOP where the objects are (L, M)-dfts's and the morphisms are LF-continuous maps between these spaces.

Example 9. Let X = {x, y} be a set, L = M = [0,1] and x [dot encircle] y = max{x + y - 1,0}, x [direct sum] y = min{x + y, 1}. Then, ([0,1], [less than or equal to], [dot encircle]) is a left-continuous t-norm (Lukasiewicz t-norm) with strong induced negation x [??] 0 = min{1 - x, 1}. Let [mu], [rho] [member of] [[0,1].sup.X] be defined as follows: [mu](x) = 0.6, [mu](y) = 0.3, [rho](x) = 0.5, [rho](y) = 0.7. Define T, [T.sup.*] : [[0,1].sup.X] [right arrow] [0,1] as follows:

[mathematical expression not reproducible]. (16)

Then, the pair (T, [T.sup.*]) is a ([0,1], [0,1])-dft on X.

Remark 10. (1) If (L = M = [0,1], [dot encircle] = [conjunction], [direct sum] = [disjunction]) with an order-reversing involution *, ([a.sup.*] = 1 - a) (L, M)-dfts is the concept of Mondal and Samanta [18].

(2) If L and M are frames with 0 and 1, (L, M)-dfts is the concept of Gutierrez Garcia and Rodabaugh [19].

(3) If [dot encircle] = [conjunction], (L, M)-dfts is the concept of AbdEl-latif [42].

Definition 11 (see [29, 30]). A map F : [L.sup.X] [right arrow] L is called an L-filter if it fulfills the following conditions:

(LF1) F(0) = [0.sub.M] and F([1.bar]) = [1.sub.M].

(LF2) F([lambda] [dot encircle] [mu]) [greater than or equal to] F([lambda]) [dot encircle] F([mu]), for each [lambda], [mu] [member of] [L.sup.X].

(LF3) If [lambda] [less than or equal to] [mu], then F([lambda]) [less than or equal to] F([mu]).

The pair, (XF), is called an L-filter space.

3. (L,M)-Double Fuzzy Filters and (L,M)-Double Fuzzy Filter Bases

Definition 12. The pair (F, [F.sup.*]) of maps F, [F.sup.*] : [L.sup.X] [right arrow] M is called an (L, M)-double fuzzy filter (briefly, (L, M)-dff) on X if it fulfills the following axioms:

(DFF1) F([lambda]) [less than or equal to] [([F.sup.*]([lambda])).sup.*], for each [lambda] [member of] [L.sup.X].

(DFF2) F([0.bar]) = [0.sub.M], F([1.bar]) = [1.sub.M] and [F.sup.*]([0.bar]) = [1.sub.M], [F.sup.*]([1.bar]) = [0.sub.M].

(DFF3) F([lambda] [dot encircle] [mu]) [greater than or equal to] F([lambda]) [??] F([mu]) and [F.sup.*]([lambda] [dot encircle] [mu]) [less than or equal to] [F.sup.*] ([lambda]) [??] [F.sup.*] ([mu]), for each [lambda], [mu] [member of] [L.sup.X].

(DFF4) If [lambda] [less than or equal to] [mu], then F([lambda]) [less than or equal to] F([mu]) and [F.sup.*]([lambda]) [greater than or equal to] [F.sup.*]([mu]).

The triplet (X, F, [F.sup.*]) is called an (L, M)-double fuzzy filter space (briefly, (L, M)-dffs).

If ([F.sub.1], [F.sup.*.sub.1]) and ([F.sub.2], [F.sup.*.sub.2]) are two (L, M)-dffs on X, we say that ([F.sub.1], [F.sup.*.sub.1]) is finer than ([F.sub.2], [F.sup.*.sub.2]) (or ([F.sub.2], [F.sup.*.sub.2]) is coarser than ([F.sub.1], [F.sup.*.sub.1])), denoted by ([F.sub.2], [F.sup.*.sub.2]) [less than or equal to] ([F.sub.1], [F.sup.*.sub.1]) if and only if [F.sub.2]([lambda]) [less than or equal to] [F.sub.1]([lambda]) and [F.sup.*.sub.2]([lambda]) [greater than or equal to] [F.sup.*.sub.1]([lambda]), for each [lambda] [member of] [L.sup.X].

Definition 13. Let (X, ([F.sub.1], [F.sup.*.sub.1])) and (Y, ([F.sub.2], [F.sup.*.sub.2])) be two (L, M)-dffs's. Then, a map [phi] : X [right arrow] Y is said to be

(i) a filter map if and only if [F.sub.2] [less than or equal to] [F.sub.1] [omicron] [[phi].sup.[left arrow].sub.L] and [F.sup.*.sub.2] [greater than or equal to] [F.sup.*.sub.1] [omicron] [[phi].sup.[left arrow].sub.L];

(ii) a filter preserving map if and only if [F.sub.1] [less than or equal to] [F.sub.2] [omicron] [[phi].sup.[right arrow].sub.L] and [F.sup.*.sub.1] [greater than or equal to] [F.sup.*.sub.2] [omicron] [[phi].sup.[right arrow].sub.L].

Normally, the composition of filter maps (resp., filter preserving maps) is a filter map (resp., filter preserving map).

Hence, we get to the category (L, M)-DFIL with objects of all the (L, M)-dffs's and the morphisms are filter maps between these spaces.

Remark 14. (i) Let F be an L-filter on X and [F.sup.*] : [L.sup.X] [right arrow] L defined by [F.sup.*][([lambda]) = (F([lambda])).sup.*]. Then, the pair (F, [F.sup.*]) is an (L, L)-dff on X. Therefore, (L, M)-dff is a generalization of L-filter due to Hohle and Sostak [7, 29].

(ii) If [dot encircle] = [conjunction], the definition of (L, M)-dff coincides with the definition of a proper (L, M)-intuitionistic fuzzy filter due to Abd El-latif [42].

Theorem 15. Each (L, M)-dff (X, F, [F.sup.*]) produces an (L, M)-dfts [mathematical expression not reproducible].

Proof. When contrasting the axioms of (L, M)-dff and (L, M)-dft, we find (DFF4) implying (DFT4).

Let {[[lambda].sub.i] : i [member of] [GAMMA]} [subset or equal to] [L.sup.X]. Then, [[lambda].sub.i] [less than or equal to] [[??].sub.i[member of][GAMMA]] [[lambda].sub.i] for all i [member of] [GAMMA]; due to (DFF4), we have that F([[lambda].sub.i]) < F([[??].sub.i[member of][GAMMA]] [[lambda].sub.i]) and [F.sup.*]([[lambda].sub.i]) [greter than or equal to] [F.sup.*]([[??].sub.i[member of][GAMMA]] [[lambda].sub.i]) for all i [member of] [GAMMA]. So,

[mathematical expression not reproducible]. (17)

Then, we can get an (L, M)-dft [mathematical expression not reproducible] defined by

[mathematical expression not reproducible]. (18)

Theorem 16. Let (X, [F.sub.1], [F.sup.*.sub.1]) and (Y [F.sub.2], [F.sup.*.sub.2]) be (L, M)-dffs's. If [phi] : (X, [F.sub.1], [F.sup.*.sub.1]) [right arrow] (Y, [F.sub.2], [F.sup.*.sub.2]) is a filter map, then [mathematical expression not reproducible] is an LF-continuous map.

Proof. Let [mu] [member of] [L.sup.Y]. If [mu] = [0.sub.Y] or [[phi].sup.[left arrow].sub.L]([mu]) = [0.sub.X], then the proof is easy. Let [mu] [not equal to] [0.sub.Y] and [[phi].sup.[left arrow].sub.L]([mu]) [not equal to] [0.sub.X]. Then, from the definition of double filter map and Theorem 15, we have

[mathematical expression not reproducible]. (19)

Corollary 17. The function F : (L, M)-DFIL [right arrow] (L, M)-DFTOP defined by [mathematical expression not reproducible] and F([phi]) = [phi] is a functor.

Notation 18. Let B, [B.sup.*] : [L.sup.X] [right arrow] M be two maps and [lambda] [member of] [L.sup.X]. Then, <B> and <[B.sup.*]> are defined as follows:

[mathematical expression not reproducible]. (20)

Definition 19. The pair (B, [B.sup.*]) of maps B, [B.sup.*] : [L.sup.X] [right arrow] M is called an (L, M)-double fuzzy filter base (briefly, (L, M)-dffb) on X if it fulfills the following axioms:

(DFFB1) B([lambda]) [less than or equal to] [([B.sup.*]([lambda])).sup.*], for each [lambda] [member of] [L.sup.X].

(DFFB2) B([0.bar]) = [0.sub.M], B([1.bar]) = [1.sub.M] and [B.sup.*]([0.bar]) = [1.sub.M], [B.sup.*] ([1.bar]) = [0.sub.M].

(DFFB3) <B> ([lambda] [dot encircle] [mu]) [greater than or equal to] B([lambda]) [??] B([mu]) and <[B.sup.*]>([lambda] [dot encircle] [mu]) [less than or equal to] [B.sup.*]([lambda]) [??] [B.sup.*]([mu]), for each [lambda], [mu] [member of] [L.sup.X].

If ([B.sub.1], [B.sup.*.sub.1]) and ([B.sub.2], [B.sup.*.sub.2]) are two (L, M)-dffb's on X, we say ([B.sub.1], [B.sup.*.sub.1]) is finer than ([B.sub.2], [B.sup.*.sub.2]) (or ([B.sub.2], [B.sup.*.sub.2]) is coarser than ([B.sub.1], [B.sup.*.sub.1])) denoted by ([B.sub.2], [B.sup.*.sub.2]) [less than or equal to] ([B.sub.1], [B.sup.*.sub.1]) if and only if <[B.sub.2]>([lambda]) [less than or equal to] <[B.sub.1]>([lambda]) and <[B.sup.*.sub.2]>)([lambda]) [greater than or equal to] <[B.sup.*.sub.2]>)([lambda]), for each [lambda] [member of] [L.sup.X].

Remark 20. (i) An (L, M)-dffb is a generalization of L-filter base due to Kim and Ko [40].

(ii) If (F, [F.sup.*]) is an (L, M)-dff, then (F, [F.sup.*]) is an (L, M)-dffb with <F> = F and <[F.sup.*]> = [F.sup.*].

(iii) If (B, [B.sup.*]) is an (L, M)-dffb, then, by (DFFB3), [lambda] [dot encircle] [mu] = [0.bar] implies B([lambda]) [??] B([mu]) = [0.sub.M] and [B.sup.*] ([lambda]) [??] [B.sup.*] ([mu]) = [1.sub.M].

Theorem 21. If (B, [B.sup.*]) is an (L, M)-dffl>, then (<B> <[B.sup.*]>) is the coarsest (L, M)-dff which satisfies B [less than or equal to] <B> and [B.sup.*] [greater than or equal to] <[B.sup.*]>.

Proof. (DFF1) For each [lambda] [member of] [L.sup.X],

[mathematical expression not reproducible]. (21)

(DFF2) and (DFF4) are easily checked.

(DFF3) Suppose that there exist [lambda], [mu] [member of] [L.sup.X] such that

[mathematical expression not reproducible]. (22)

By the definition of <[B.sup.*]> and (L4'), there exist [[lambda].sub.1], [[mu].sub.1] [member of] [L.sup.X] with [[lambda].sub.1] [less than or equal to] [lambda] and [[mu].sub.1] [less than or equal to] [mu] such that

[mathematical expression not reproducible]. (23)

Since (B, [B.sup.*]) is an (L, M)-dffb,

<[B.sup.*]>([[lambda].sub.1] [dot encircle] [[mu].sub.1]) [less than or equal to] [B.sup.*] ([[lambda].sub.1]) [??] [B.sup.*] ([[mu].sub.1]). (24)

Since [[lambda].sub.1] [dot encircle] [[mu].sub.1] [less than or equal to] [lambda] [dot encircle] [mu], we have

<[B.sup.*]> ([lambda] [dot encircle] [mu]) [less than or equal to] <B.sup.*]> ([[lambda].sub.1]) [dot encircle] [[mu].sub.1])

[less than or equal to] [B.sup.*] ([[lambda].sub.1]) [??] [B.sup.*] ([[mu].sub.1]). (25)

It is a contradiction. Thus, <[B.sup.*]>([lambda] [dot encircle] [mu]) [less than or equal to] <[B.sup.*]>([lambda] [??]<[B.sup.*]> [mu]), for each [lambda], [mu] [member of] [L.sup.X]. Similarly, <B?([lambda] [dot encircle] [mu]) [greater than or equal to] <B> ([lambda]) [??] <B>([mu]), for each [lambda], [mu] [member of] [L.sup.X].

Let (F, [F.sup.*]) be another (L, M)-dff which is finer than (B, [B.sup.*]), that is, B [less than or equal to] F and [B.sup.*] [greater than or equal to] [F.sup.*]. Then, we have

[mathematical expression not reproducible]. (26)

Theorem 22. H, [H.sup.*] : [L.sup.X] [right arrow] M are maps fulfilling the following conditions:

(C1) H([lambda]) [less than or equal to] [([H.sup.*](X)).sup.*], for each [lambda] [member of] [L.sup.X],

(C2) H([1.bar]) = [1.sub.M] and [H.sup.*]([1.bar]) = [0.sub.M] and for each finite index set K, if [[??].sub.i[member of]K][[lambda].sub.i] = [0.bar], then [mathematical expression not reproducible] and [mathematical expression not reproducible].

We define the maps [mathematical expression not reproducible] as

[mathematical expression not reproducible], (27)

where [??] and [??] are taken for every finite index set K such that [lambda] = [[??].sub.i[member of]K][[lambda].sub.i], respectively. Then, the following properties are satisfied:

(i) [mathematical expression not reproducible] is an (L, M)-dffb on X.

(ii) If H [less than or equal to] B, [H.sup.*] [greater than or equal to] [B.sup.*] and (B, [B.sup.*]) is an (L,M)-dffb on X, then <[B.sub.H]> [less than or equal to] <B> and [mathematical expression not reproducible].

Proof. (i) (DFFB1) For each [lambda] [member of] [L.sup.X], the following is valid:

[mathematical expression not reproducible]. (28)

(DFFB2) It is clear by condition (C2).

(DFFB3) For each [lambda], [mu] [member of] [L.sup.X] and for any two finite index sets K, J with [lamda] = [[??].sub.k[member of]K][[lambda].sub.k] and [mu] = [[??].sub.j[member of]J][[mu].sub.i], since [lambda] [dot encircle] [mu] = ([[??].sub.k[member of]K][[lambda].sub.k]) [dot enricle] ([[??].sub.j[member of]J][[mu].sub.i]), by the definition of [B.sub.H] and [mathematical expression not reproducible], we get

[mathematical expression not reproducible]. (29)

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

[mathematical expression not reproducible]. (30)

Thus, [mathematical expression not reproducible] is an (L, M)-dffb on X.

(ii) For any finite family {[[lambda].sub.i] : [[??].sub.i[member of]K][[lambda].sub.i] [less than or equal to] [lambda]}, the following are true:

[mathematical expression not reproducible]. (31)

Then, <[B.sub.H]> [less than or equal to] <B> and [mathematical expression not reproducible].

Theorem 23. Let ([B.sub.1], [B.sup.*.sub.1]) and ([B.sub.2], [B.sup.*.sub.2]) be two (L, M)-dffbs on X and Y, respectively, and [phi] : X [right arrow] Y fee a function. Then, one has the following properties:

(i) [phi] : (X <[B.sub.1]>, <[B.sup.*.sub.1]>) [right arrow] (Y <[B.sub.2]>, <[B.sup.*.sub.2]>) is a filter map if and only if [B.sub.2] [less than or equal to] <[B.sub.1]> [omicron] [[phi].sup.[left arrow].sub.L] and [B.sup.*.sub.2] [greater than or equal to] <[B.sup.*.sub.1]> [omicron] [[phi].sup.[left arrow].sub.L].

(ii) [phi] : (X, <[B.sub.1]>, <[B.sup.*.sub.1]>) [right arrow] (Y, <[B.sub.2]>, <[B.sup.*.sub.2]>) is a filter preserving map if and only if [B.sub.1] [less than or equal to] <[B.sub.2]> [omicron] [[phi].sup.[right arrow].sub.L] and [B.sup.*.sub.1] [greater than or equal to] <[B.sup.*.sub.2]> [omicron] [[phi].sup.[right arrow].sub.L].

(iii) If [B.sub.2] [less than or equal to] [B.sub.1] [omicron] [[phi].sup.[left arrow].sub.L] and [B.sup.*.sub.2] [greater than or equal to] [B.sup.*.sub.1] [omicron] [[phi].sup.[left arrow].sub.L], then [phi] : (X, <[B.sub.1]>, <[B.sup.*.sub.1]>) [right arrow] (Y <[B.sub.2]>, <[B.sup.*.sub.2]>) is a filter map.

(iv) If [B.sub.1] [less than or equal to] [B.sub.2] [omicron] [[phi].sup.[right arrow].sub.L] and [B.sup.*.sub.1] [greater than or equal to] [B.sup.*.sub.2] [omicron] [[phi].sup.[right arrow].sub.L], then [phi] : (X, <[B.sub.1]>, <[B.sup.*.sub.1]>) [right arrow] (Y <[B.sub.2]>, <[B.sup.*.sub.2]>) is a filter preserving map.

Proof. Proving condition (i) is enough since the other conditions are similarly proved.

(i) ([??]:) Since [B.sub.2]([mu]) [less than or equal to] <[B.sub.2]>([mu]) and [B.sup.*.sub.2]([mu]) [greater than or equal to] <[B.sup.*.sub.2]>([mu]), for each [mu] [member of] [L.sup.Y], it is trivial.

([??]:) Let [B.sub.2]([mu]) [less than or equal to] <[B.sub.2]>([[phi].sup.[left arrow].sub.L][([mu])) and [B.sup.*.sub.2]([mu]) [greater than or equal to] <[B.sup.*.sub.1]>([[phi].sup.[left arrow].sub.L][([mu])), for each [mu] [member of] [L.sup.Y]. We will show that [phi] is a filter map. For arbitrary [mu] [member of] [L.sup.Y], we have

[mathematical expression not reproducible]. (32)

Thus, [phi] is a filter map.

Example 24. Let X = [x,y] be a set, L = M = [0,1] be the stsc-quantale with Lukasiewicz t-norm, and [mu], v [member of] [[0,1].sup.X] be defined by [mu](x) = 0.6, [mu](y) = 0.5, v(x) = 0.1, v(y) = 0. Define the maps [B.sub.i], [B.sup.*.sub.i] : [L.sup.X] [right arrow] M I = 1, 2, 3 as follows:

[mathematical expression not reproducible]. (33)

It can be seen by easy computation that

(1) ([B.sub.1], [B.sup.*.sub.1]) and ([B.sub.3], [B.sup.*.sub.3]) are not (L, M)-double fuzzy filters but they are (L, M)-double fuzzy filter bases, so they generate (L,M)-double fuzzy filters (<[B.sub.1]>), (<[B.sup.*.sub.1]>)) and (<[B.sub.3]>), (<[B.sup.*.sub.3]>)).

(2) ([B.sub.2], [B.sup.*.sub.2]) is not an (L, M)-double fuzzy filter base and it does not satisfy condition (C2) of Theorem 22.

(3) Since (<[B.sub.3]>), (<[B.sup.*.sub.3]>)) [less than or equal to] (<[B.sub.1]>), (<[B.sup.*.sub.1]>), [id.sub.X] : (X, <[B.sub.1]>, <[B.sup.*.sub.1]>) [right arrow] (X <[B.sub.3]>, <[B.sup.*.sub.3]>) is afiltermap and [id.sub.X] : (X, (<[B.sub.3]>), (<[B.sup.*.sub.3]>) [right arrow] (X, <[B.sub.1]>), (<[B.sup.*.sub.1]>) is a filter preserving map though [mathematical expression not reproducible] and [mathematical expression not reproducible].

We also note that if L = M = [0,1] is considered as a frame, then ([B.sub.1], [B.sup.*.sub.1]) is an (L, M)-double fuzzy filter base.

4. The Types ([[phi].sup.[??].sub.1], [[phi].sup.*[??].sub.1]), ([[phi].sup.[??].sub.2], [[phi].sup.[??].sub.2]) of Preimages and Images of (L,M)-Double Fuzzy Filter Bases

Theorem 25. Let [phi]: X [right arrow] Y be a function and (B, [B.sup.*]) be an (L,M)-dffb on Y. Then, the following properties are satisfied.

(i) If [[phi].sup.[left arrow].sub.L]([mu]) = 0 implies B([mu]) = [0.sub.M] and [B.sup.*]([mu]) = [1.sub.M], then ([[phi].sup.[??].sub.1](B), [[phi].sup.*[??].sub.1]([B.sup.*])) is an (L,M)-dffb on X and (<[[phi].sup.[??].sub.1](B)>, <[[phi].sup.*[??].sub.1]([B.sup.*])>) is the coarsest (L,M)-dff on X for which [phi] : (X, <[[phi].sup.[??].sub.1](B)>, <[[phi].sup.*[??].sub.1]([B.sup.*])>) [right arrow] (Y, <B>, <[B.sup.*]>) is a filter map.

(ii) If [phi] is surjective, then ([[phi].sup.[??].sub.1](B)>, [[phi].sup.*[??].sub.1]([B.sup.*])) is an (L, M)- dffb.

(iii) If [[phi].sup.[left arrow].sub.L]([mu]) = 0 implies B([mu]) = [0.sub.M] and [B.sup.*]([mu]) = [1.sub.M], [phi] is injective, and (B, [B.sup.*]) is an (L, M)-dff on Y, then ([[phi].sup.[??].sub.1](B), [[phi].sup.*[??].sub.1](B)) is an (L, M)-dff on X.

Proof. (i) (DFFB1) For each [lambda] [member of] [L.sup.X], we have

[mathematical expression not reproducible]. (34)

(DFFB2) Since [[phi].sup.[left arrow].sub.L]([1.bar]) = 1, then [[phi].sup.[??].sub.1](B)([1.bar]) = [1.sub.M] and [[phi].sup.*[??].sub.1]([B.sup.*])([1.bar]) = [0.sub.M]. By assumption, [[phi].sup.[??].sub.1](B)([0.bar]) = [0.sub.M] and [[phi].sup.*[??].sub.1]([B.sup.*])([0.bar]) = [1.sub.M].

(DFFB3) Suppose that there exist [[lambda].sub.1], [[lambda].sub.2] [member of] [L.sup.X] such that

[mathematical expression not reproducible]. (35)

By the definition of [[phi].sup.*[??].sub.1]([B.sup.*]) and (L4'), there exist [[mu].sub.1], [[mu].sub.2] [member of] [L.sup.Y] with [[lambda].sub.1] = [[phi].sup.[left arrow].sub.L]([[mu].sub.1]) and [[lambda].sub.2] = [[phi].sup.[left arrow].sub.L]([[mu].sub.2]) such that

[mathematical expression not reproducible]. (36)

Since (B, [B.sup.*]) is an (L, M)-dffb, the following is valid:

<[B.sup.*]> ([[mu].sub.1] [dot encircle] [[mu].sub.2]) [less than or equal to] [B.sup.*] ([[mu].sub.1]) [??] [B.sup.*] ([[mu].sub.2]). (37)

Then,

[mathematical expression not reproducible]. (38)

By the definition of <[B.sup.*]>, there exists v [member of] [L.sup.Y] with v [less than or equal to] [[mu].sub.1] [dot encircle] [[mu].sub.2] such that

[mathematical expression not reproducible]. (39)

On the other hand, since

[[mu].sub.1] [dot encircle] [[mu].sub.2] = [[phi].sup.[left arrow].sub.L] ([[mu].sub.1]) [dot encircle] [[phi].sup.[left arrow].sub.L] ([[mu].sub.2]) = [[phi].sup.[left arrow].sub.L] ([[mu].sub.1] [dot encircle] [[mu].sub.2])

[greater than or equal to] [[phi].sup.[left arrow].sub.L] (v), (40)

then <[[phi].sup.*[??].sub.1] ([B.sup.*])> ([[lambda].sub.1] [dot encircle] [[lambda].sub.2]) [less than or equal to] [[phi].sup.*[??].sub.1]([B.sup.*])([[[phi].sup.[left arrow].sub.L](v)) [less than or equal to] [B.sup.*](v).

This contradicts the assumption. Thus,

[mathematical expression not reproducible], (41)

for each [[lambda].sub.1], [[lambda].sub.2] [member of] [L.sup.X].

Similarly, [mathematical expression not reproducible], for each [[lambda].sub.1], [[lambda].sub.2] [member of] [L.sup.X].

Hence, ([[[phi].sup.[??].sub.1](B), [[[phi].sup.*[??].sub.1]([B.sup.*])) is an (L, M)-dffb on X.

Let (F, [F.sup.*]) be another (L,M)-dff on X such that [phi] : (X, F, [F.sup.*]) [right arrow] (Y, <B>, <[B.sup.*]>) is a filter map. Then, for each [lambda] [member of] [L.sup.X], the following inequalities are valid:

[mathematical expression not reproducible]. (42)

(ii) Since [phi] is surjective, [[phi].sup.[left arrow].sub.L]([mu]) = [0.bar] implies [mu] = [0.bar]. So, B([mu]) = [0.sub.M] and [B.sup.*] ([mu]) = [1.sub.M]. Then, by (i), ([[phi].sup.[??].sub.1](B), ([[phi].sup.*[??].sub.1]([B.sup.*])) is an (L, M)-dffb on X.

(iii) (DFF1)-(DFF3) are obvious.

(DFF4) Let [[lambda].sub.1] [less than or equal to] [[lambda].sub.2], for [[lambda].sub.1], [[lambda].sub.2] [member of] [L.sup.X]. Since [phi] is injective, there exists v [member of] [L.sup.Y] with [[phi].sup.[left arrow].sub.L](v) = [[lambda].sub.1] and [[lambda].sub.2] = [[phi].sup.[left arrow].sub.L](v [disjunction] [[phi].sup.[right arrow].sub.L]([[lambda].sub.2])). It implies

[[phi].sup.[??].sub.1] (B) ([[lambda].sub.2]) [greater than or equal to] B (v [disjunction] [[phi].sup.[right arrow].sub.L] ([[lambda].sub.2])) [greater than or equal to] B (v),

[[phi].sup.*[??].sub.1] ([B.sup.*]) ([[lambda].sub.2]) [less than or equal to] [B.sup.*] (v [disjunction] [[phi].sup.[right arrow].sub.L] ([[lambda].sub.2])) [less than or equal to] [B.sup.*] (v). (43)

If supremum and infimum are taken over {[[lambda].sub.1] | [[phi].sup.[left arrow].sub.L](v) = [[lambda].sub.1]}, respectively, then it is clear that [[phi].sup.[??].sub.1](B)([[lambda].sub.2]) [greater than or equal to] [[phi].sup.[??].sub.1](B)([[lambda].sub.1]) and [[phi].sup.*[??].sub.1]([B.sup.*])([[lambda].sub.2]) [less than or equal to] [[phi].sup.*[??].sub.1]([B.sup.*])([[lambda].sub.1]).

Theorem 26. Let [{[[phi].sub.i] : X [right arrow] [X.sub.i]}.sub.i[member of][GAMMA]] be a family of functions and [{([B.sub.i], [B.sup.*.sub.i])}.sub.i[member of][GAMMA]] be a family of (L, M)-dffb's on [X.sub.i] satisfying the following condition:

(C) For each finite subset K of [GAMMA], if [[??].sub.i[member of]K] ([[lambda].sub.i] [omicron] [[phi].sub.i]) = [0.bar], then [[??].sub.i[member of]K] [B.sub.i] ([[lambda].sub.i]) = [0.sub.M] and [[??].sub.i[member of]K] [B.sup.*.sub.i] ([[lambda].sub.i]) = [1.sub.M]

We define the maps [mathematical expression not reproducible] as

[mathematical expression not reproducible], (44)

where [??] and [??] are taken for every finite index subset K of [GAMMA] such that [lambda] = [[??].sub.i[member of]K] ([[lambda].sub.i] [omicron] [[phi].sub.i]).

Let [mathematical expression not reproducible] be given. Then, the following properties are satisfied.

(i) (B, [B.sup.*]) is an (L,M)-dffb on X and (<B>, <[B.sup.*]>) is the coarsest (L, M)-dff for which [[phi].sub.i] : (X, <B>, <[B.sup.*]>) [right arrow] ([X.sub.i], <[B.sub.i]>), <[B.sup.*.sub.i]>) is a filter map for each i [member of] [GAMMA].

(ii) A function [phi] : (Y, F, [F.sup.*]) [right arrow] (X, <B>, <[B.sup.*]>) is a filter map if and only if, for each i [member of] [GAMMA], [[phi].sub.i] [omicron] [phi] : (Y, F, [F.sup.*]) [right arrow] ([X.sub.i], <[B.sub.i]>, <[B.sup.*.sub.i]>) is a filter map.

(iii) [mathematical expression not reproducible].

Proof. (i) (DFFB1) Let [lambda] [member of] [L.sup.X] with [lambda] = [[??].sub.i[member of]K]([[lambda].sub.i] [omicron] [[phi].sub.i]). Then, the following inequality is valid:

[mathematical expression not reproducible]. (45)

(DFFB2) By condition (C), B([0.bar]) = [0.sub.M] and [B.sup.*] ([0.bar]) = [1.sub.M]. Since, [1.bar] = [1.bar] [omicron] [[phi].sub.i], then B([1.bar]) = [1.sub.M] and [B.sup.*] ([1.bar]) = [0.sub.M].

(DFFB3) Suppose that there exist [lambda], [mu] [member of] [L.sup.X] such that [mathematical expression not reproducible].

By the definition of [B.sup.*] ([lambda]), and (L4'), there exists a finite subset K of [GAMMA] with [lambda] = [[??].sub.k[member of]K] ([[lambda].sub.k] [omicron] [[phi].sub.K]) such that [mathematical expression not reproducible].

Again, by the definition of [B.sup.*]([mu]) and (L4'), there exists a finite subset J of [GAMMA] with [mu] = [[??].sub.j[member of]J] ([[mu].sub.j] [omicron] [[phi].sub.j]) such that

[mathematical expression not reproducible]. (46)

Put m [member of] (K [union] J) such that

[mathematical expression not reproducible]. (47)

Since, for each m [member of] K [intersection] J, <[B.sup.*.sub.m]>([[lambda].sub.m] [dot encircle] [[mu].sub.m]) [less than or equal to] [B.sup.*.sub.m]([[lambda].sub.m]) [??] [B.sup.*.sub.m]([[micro].sub.m]), we have

[mathematical expression not reproducible]. (48)

From the definition of <[B.sup.*.sub.m]>, there exists [mathematical expression not reproducible] with [v.sub.m] [less than or equal to] [[lambda].sub.m] [dot encircle] [[mu].sub.m] such that

[mathematical expression not reproducible]. (49)

On the other hand, since

[mathematical expression not reproducible] (50)

and since K [union] J is finite, we have

[mathematical expression not reproducible]. (51)

This contradicts the assumption. Then, <[B.sup.*]> ([lambda] [omicron] [mu]) [less than or equal to] [B.sup.*]([lambda]) [??] [B.sup.*]([mu]), for each [lambda], [mu] [member of] [L.sup.X]. Similarly, <B>([lambda] [dot encircle] [mu]) [greater than or equal to] B([lambda]) [??] B([mu]), for each, [lambda], [mu] [member of] [L.sup.X]. Hence, (B, [B.sup.*]) is an (L, M)-dffb on X.

Since B([[lambda].sub.i] [omicron] [[phi].sub.i]) [greater than or equal to] [B.sub.i]([[lambda].sub.i]) and [B.sup.*]([[lambda].sub.i] [omicron] [[phi].sub.i]) [less than or equal to] [B.sup.*.sub.i]([[lambda].sub.i]), for each i [member of] [GAMMA], by Theorem 23(iii), [[phi].sub.i] : (X, <B>, <[B.sup.*]>) [right arrow] ([X.sub.i], <[B.sub.i]>), <[B.sup.*.sub.i]>) is a filter map.

Let (F, [F.sup.*]) be an (L, M)-dff on X such that, for each i [member of] [GAMMA], the map [[phi].sub.i] : (X, F, [F.sup.*]) [right arrow] ([X.sub.i], <[B.sub.i]>), <[B.sup.*.sub.i]>) is a filter map. Then,

F ([v.sub.i] [omicron] [[phi].sub.i]) [greater than or equal to] <[B.sub.i]>)([v.sub.i]),

[F.sup.*] ([v.sub.i] [omicron] [[phi].sub.i]) [less than or equal to] <[B.sup.*.sub.i]> ([v.sub.i]), (52)

for each [mathematical expression not reproducible].

For any finite subset K of [GAMMA] with v [greater than or equal to] [[??].sub.k[member of]K]([v.sub.k] [omicron] [[phi].sub.k]), since F([v.sub.k] [omicron] [[phi].sub.k]) [greater than or equal to] <[B.sub.k]>([v.sub.k]) and [F.sup.*] ([v.sub.k] [omicron] [[phi].sub.k]) [less than or equal to] <[B.sup.*.sub.k]>([v.sub.k]), for each k [member of] K, we have

[mathematical expression not reproducible]. (53)

Hence, by the definition of <B> and <[B.sup.*]>, it is obvious that (<B>, <[B.sup.*]>) [less than or equal to] (F, [F.sup.*]).

(ii) Necessity of the composition condition is obvious.

Conversely, for every finite subset K of [GAMMA] with v [greater than or equal to] [[??].sub.k[member of]K]([v.sub.k] [omicron] [[phi].sub.k]), since, for each k [member of] K, [[phi].sub.k] [omicron] [phi] : (Y, F, [F.sup.*]) [right arrow] ([X.sub.k], <[B.sub.k]>, <[B.sup.*.sub.k]>) is a filter map, that is, <[B.sub.k]>, ([v.sub.k]) [less than or equal to] F([v.sub.k] [omicron] ([[phi].sub.k] [omicron] [phi])) and <[B.sup.*.sub.k]> ([v.sub.k]) [greater than or equal to] [F.sup.*]([v.sub.k] [omicron] ([[phi].sub.k] [omicron] [phi])). Since v [omicron] [phi] [greater than or equal to] [[??].sub.k[member of]K] (([v.sub.k] [omicron] [[phi].sub.k]) [omicron] [phi]), we have

[mathematical expression not reproducible]. (54)

By the definition of <B> and <[B.sup.*]>, we have <B>(v) [less than or equal to] F(v [omicron] [phi]) and <[B.sup.*]>(v) [greater than or equal to] [F.sup.*](v [omicron] [phi]).

(iii) Put [mathematical expression not reproducible]; by applying (i) to both (<B>, <[B.sup.*]>) and (<F>, <[F.sup.*]>), the desired equality is obtained.

The following corollaries are the direct results of Theorem 26.

Corollary 27. Let [{([B.sub.i], [B.sup.*.sub.i])}.sub.i[member of][GAMMA]] be a family of (L,M)-dffb's on X satisfying the following condition:

(C) For any finite subset K of [GAMMA], if [[??].sub.i[membe rof]K] [[lambda].sub.i] = [0.bar], then [mathematical expression not reproducible].

We define the maps [mathematical expression not reproducible] as

[mathematical expression not reproducible], (55)

where [??] and [??] are taken for every finite index subset K of [GAMMA] such that [lambda] = [[??].sub.i[member of]K] [[lambda].sub.i]. Then, [mathematical expression not reproducible] is an (L,M)=dffb on X and [mathematical expression not reproducible] is the coarsest (L, M)-dff which is finer than (<[B.sub.i]>, <[B.sup.*.sub.i]>)) for each i [member of] [GAMM].

Example 28. Let X = {x, y} be a set and L = M = [0,1] be an stsc-quantale with [dot encircle]] (Lukasiewicz t-norm). We define maps [B.sub.i], [B.sup.*.sub.i] : [L.sup.X] [right arrow] as follows (i = 1,2):

[mathematical expression not reproducible]. (56)

Each ([B.sub.i], [B.sup.*.sub.i]) for i = 1,2 is an (L, M)-double fuzzy filter base but [mathematical expression not reproducible] is not.

Corollary 29. Let [[pi].sub.i] : X [right arrow] [X.sub.i] be projection maps, for all i [member of] [GAMMA], where [mathematical expression not reproducible] is the product set. Let [{([B.sub.i], [B.sup.*.sub.i])}.sub.i[member of][GAMMA]] be a family of (L, M)-dffb's on [X.sub.i] satisfying the following condition:

(C) For any finite subset K of [GAMMA], if [[??].sub.i[member of]K] ([[lambda].sub.i] [omicron] [[pi].sub.i]) = [0.bar], then [mathematical expression not reproducible] and [mathematical expression not reproducible].

We define the maps [mathematical expression not reproducible] as

[mathematical expression not reproducible], (57)

where [??] and [??] are taken for every finite subset K of [GAMMA] such that [lambda] = [[??].sub.i[member of]K]([[lambda].sub.i] [omicron] [[pi].sub.i]).

Let [mathematical expression not reproducible] be given. Then, the following properties are satisfied:

(i) (B, [B.sup.*]) is an (L,M)-dffb on X and (<B>, <[B.sup.*]>) is the coarsest (L, M)-dff on X for which [[pi].sub.i] : (X, <B>, <[B.sup.*]>) [right arrow] ([X.sub.i], <[B.sub.i]>, <[B.sup.*.sub.i]>) is a filter map.

(ii) A map [phi] : (Y, F, [F.sup.*]) [right arrow] (X, <B>, <[B.sup.*]>) is a filter map if and only if, for each i [member of] [GAMMA], [[pi].sub.i] [omicron] [phi] : (Y, F, [F.sup.*]) [right arrow] ([X.sub.i], <[B.sub.i]>), <[B.sup.*.sub.i]>) is a filter map.

In Corollary 29, the structure [mathematical expression not reproducible] is called a product of (L, M)-dffs's on X.

Theorem 30. Let [phi] : X [right arrow] Y be an injective function and (B, [B.sup.*]) be an (L, M)-dffb on X. Then, the following properties are satisfied:

(i) ([[phi].sup.[??].sub.2](B), [[phi].sup.[??].sub.2]([B.sup.*])) is an (L,M)-dffb on Y, and (<[[phi].sup.[??].sub.2](B)>, <[[phi].sup.[??].sub.2]([B.sup.*])>) is the coarsest (L, M)-dff, for which the function [phi] : (X, <B>, <[B.sup.*]>) [right arrow] (Y, <[[phi].sup.[??].sub.2](B)>, <[[phi].sup.[??].sub.2]([B.sup.*])>) is a filter preserving map.

(ii) [mathematical expression not reproducible] is an (L,M)-dffb on X with [mathematical expression not reproducible].

Proof. (i) (DFFB1) For each v [member of [L.sup.Y], we have

[[phi].sup.[??].sub.2] (B) (v) = B [([[phi].sup.[left arrow].sub.L] (v)) [less than or equal to] ([B.sup.*] ([[phi].sup.[left arrow].sub.L] (v))).sup.*]

= [([[phi].sup.[??].sub.2] ([B.sup.*])(v)).sup.*]. (58)

(DFFB2) It is straightforward from the definition.

(DFFB3) Suppose that there exist [v.sub.1], [v.sub.2] [member of] [L.sup.Y] such that

[mathematical expression not reproducible]. (59)

By the definition of [[phi].sup.[??].sub.2]([B.sup.*]), we have

[mathematical expression not reproducible]. (60)

Since (B, [B.sup.*]) is an (L, M)-dffb, the following is obtained:

<[B.sup.*]>) (([v.sub.1] [dot encircle] [v.sub.2]) [omicron] [phi]) = <[B.sup.*]> (([v.sub.1] [omicron] [phi]) [dot encircle] ([v.sub.2] [omicron] [phi]))

[less than or equal to] [B.sup.*] ([v.sub.1] [omicron] [phi]) [??] [B.sup.*] ([v.sub.2] [omicron] [phi]). (61)

Thus,

[mathematical expression not reproducible]. (62)

By the definition of <[B.sup.*]>, there exists [lambda] [member of] [L.sup.X] with [lambda] [member of] ([v.sub.1] [dot encircle] [v.sub.2]) [omicron] [phi] such that

[mathematical expression not reproducible]. (63)

Since [[phi].sup.[right arrow].sub.L]([lambda]) [less than or equal to] [[phi].sup.[right arrow].sub.L] ([[phi].sup.[left arrow].sub.L] ([v.sub.1] [dot encircle] [v.sub.2])) [less than or equal to] [v.sub.1] [dot encircle] [v.sub.2] and [phi] is injective,

[mathematical expression not reproducible]. (64)

This contradicts the assumption. Then,

[mathematical expression not reproducible]. (65)

for each [v.sub.1], [v.sub.2] [member of] [L.sup.Y].

Similarly, it can be verified that

[mathematical expression not reproducible], (66)

for each [v.sub.1], [v.sub.2] [member of] [L.sup.Y].

Hence, ([[phi].sup.[??].sub.2] (B), [[phi].sup.[??].sub.2] ([B.sup.*])) is an (L, M)-dffb on Y.

For each [lambda] [member of] [L.sup.X], we have

[mathematical expression not reproducible]. (67)

Hence, by Theorem 23(ii), [phi] : (X, <B>, <[B.sup.*]>) [right arrow] (Y, <[[phi].sup.[??].sub.2](B)>, <[[phi].sup.[??].sub.2]([B.sup.*])>) is a filter preserving map.

Let (F, [F.sup.*]) be another (E, M)-dff such that [phi] : (X, <B>, <[B.sup.*]>) [right arrow] (Y, F, [F.sup.*]) is a filter preserving map. So, for each [mu] [member of] [L.sup.Y], the following is valid:

[mathematical expression not reproducible]. (68)

Hence, <[[phi].sup.[??].sub.2]([B.sup.*]>) [greater than or equal to] [F.sup.*]. Similarly, it can be proved that <[[phi].sup.[??].sub.2](B)> [less than or equal to] F.

(ii) If [[phi].sup.[left arrow].sub.L]([mu]) = [0.bar], then [[phi].sup.[??].sub.2] (B)([mu]) = B([[phi].sup.[left arrow].sub.L]([mu])) = B([0.bar]) = [0.sub.M], and [[phi].sup.[??].sub.2]([B.sup.*])([mu]) = [B.sup.*]([[phi].sup.[left arrow].sub.L]([mu])) = [B.sup.*]([0.bar]) = [1.sub.M]. By Theorem 25(i), [mathematical expression not reproducible] is an (L,M)-dffb on X. For each v [member of] [L.sup.X], following equalities are obtained:

[mathematical expression not reproducible]. (69)

Example 31. Let X = {a, b}, Y = {x, y} be sets and L = M = [0,1] be the stsc-quantale with Lukasiewicz t-norm [dot encircle]. Let [phi] : X [right arrow] Y be a function defined by [phi](a) = [phi](b) = x and [[mu].sub.1], [[mu].sub.2] [member of] [[0,1].sup.X] be defined by [[mu].sub.1](a) = [[mu].sub.1](b) = 0.6, [[mu].sub.2](a) = 0.1, [[mu].sub.2](b) = 0. We define maps B, [B.sup.*] : [[0,1].sup.X] [right arrow] [0,1] as follows:

[mathematical expression not reproducible]. (70)

Then, (B, [B.sup.*]) is an (L, M)-double fuzzy filter base but ([[phi].sup.[??].sub.2](B), [[phi].sup.[??].sub.2] ([B.sup.*])) is not an (L, M)-double fuzzy filter base.

Theorem 32. Let [{[[phi].sub.i] : [X.sub.i] [right arrow] X}.sub.i[member of][GAMMA]] be a family of injective functions and [{([[B.sub.i], [B.sup.*.sub.i])}.sub.i[member of][GAMMA]] be a family of (E, M)-dffb's on [X.sub.i] satisfying the following condition:

(C) For any finite subset K of [GAMMA], if [[??].sub.i[member of]K][[lambda].sub.i] = 0, then [mathematical expression not reproducible], and [mathematical expression not reproducible].

We define the maps B, [B.sup.*] : [L.sup.X] [right arrow] M as

[mathematical expression not reproducible], (71)

where [??] and [??] are taken for every finite index subset K of [GAMMA]. Then, the following properties are satisfied:

(i) (B, [B.sup.*]) is an (L,M)-dffb on X and (<B>, <[B.sup.*]>) is the coarsest (L,M)-dff for which [[phi].sub.i] : ([X.sub.i], <[B.sub.i]>, <[B.sup.*.sub.i]>) [right arrow] (X, <B>, <[B.sup.*]>) is a filter preserving map.

(ii) A map [phi] : (X, <B>, <[B.sup.*]>) [right arrow] (Y, F, [F.sup.*]) is a filter preserving map if and only if, for each i [member of] [GAMMA], [phi] [omicron] [[phi].sub.i] : ([X.sub.i], <[B.sub.i]>, <[B.sup.*.sub.i]>) [right arrow] (Y, F, [F.sup.*]) is a filter preserving map.

Proof. (i) By Corollary 27 and Theorem 30, (B, [B.sup.*]) is an (L,M)-dffb on X.

Since [[phi].sub.i] is injective, for each i [member of] [GAMMA],

[mathematical expression not reproducible]. (72)

Hence, [[phi].sub.i] is a filter preserving map, for each i [member of] [GAMMA]. 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 (A, U) be a concrete category over X. (A, U) is said to be amnestic provided that its fibres are partially ordered classes; that is, no two different A-objects are equivalent.

(b) Let A and B be categories. A functor G : A [right arrow] B is called topological provided that every G-structured source [([f.sub.i] : B [right arrow] [GA.sub.i]).sub.i[member of][GAMMA]] has a unique G-initial lift [([[bar.f].sub.i] : A [right arrow] [A.sub.i]).sub.i[member of][GAMMA].

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

(1) G is topological.

(2) (A, G) is uniquely transportable.

(3) (A, G) is amnestic.

Theorem 35. The forgetful functor V : (L, M)-DFIL [right arrow] SET defined by V(X, F, [F.sup.*]) = X and V([phi]) = [phi] is topological.

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

5. The Types [mathematical expression not reproducible] of Images and Preimages of (L,M)-Double Fuzzy Filter Bases

Theorem 36. Let [phi] : X [right arrow] Y be a surjective function and (B, [B.sup.*]) be an (L, M)-dffb on X. Then, the following properties are satisfied:

(i) ([[phi].sup.[??].sub.1](B), [[phi].sup.*[??].sub.1] ([B.sup.*])) is an (L, M)-dffb on Y.

(ii) (<[[phi].sup.[??].sub.1] (B)>, <[[phi].sup.*[??].sub.1] ([B.sup.*]>) is the coarsest (L, M)-dff on Y for which [phi] : (X, <B>, <[B.sup.*]>) [right arrow] (Y, <[[??].sup.[??].sub.1](B)>, <[[??].sup.*[??].sub.1]([B.sup.*])>) is a filter preserving map.

(iii) If (B, [B.sup.*]) is an (L, M)-dff, then <[[??].sup.[??].sub.1](B)> = [[??].sup.[??].sub.2](B) and <[[??].sup.*[??].sub.1]([B.sup.*]))> = [[??].sup.[??].sub.1]([B.sup.*]).

Proof. (i) and (ii) are proved in the same manner as Theorem 25(i).

(iii) Let (B, [B.sup.*]) be an (L,M)-dff. Since [phi] is surjective, [[phi].sup.[right arrow].sub.L]([lambda]) [less than or equal to] v is equivalent to [lambda] [less than or equal to] v [omicron] [phi]. Then, for each v [member of] [L.sup.Y], it is clear that

[mathematical expression not reproducible]. (73)

Hence, <[[phi].sup.*[??].sub.1]([B.sup.*])> = [[phi].sup.[??].sub.2]([B.sup.*]. Similarly, <[[phi].sup.[??].sub.1](B)> = [[phi].sup.[??].sub.2](B) is obtained.

Remark 37. Let [phi] : X [right arrow] Y be a bijective function, (K, [K.sup.*]) be an (L, M)-dffb on X, and (B, [B.sup.*]) be an (L, M)-dffb on Y. Then, the following equalities are clear.

(i) [[phi].sup.[??].sub.1] (K) = [[phi].sup.[??].sub.2] (K) and [[phi].sup.*[??].sub.1] ([K.sup.*]) = [[phi].sup.[??].sub.2] ([K.sup.*]).

(ii) [[phi].sup.[??].sub.1] (B) = [[phi].sup.[??].sub.2] (B) and [[phi].sup.*[??].sub.1] ([B.sup.*]) = [[phi].sup.[??].sub.2] ([B.sup.*]).

Remark 38. Let [phi] : X [right arrow] Y be a bijective function and (B, [B.sup.*]) be an (L,M)-dffb on Y. Then, it follows from Remark 37(ii) and Theorem 25 that ([[phi].sup.[??].sub.2] (B), [[phi].sup.[??].sub.2] ([B.sup.*])) is an (L,M)-dffb on X and (<[[phi].sup.[??].sub.2] (B)>, <[[phi].sup.[??].sub.2] ([B.sup.*])>) is the coarsest (L,M)-dffon X for which [phi] : (X, <[[phi].sup.[??].sub.2] (B)>, <[[phi].sup.[??].sub.2] ([B.sup.*])>) [right arrow] (Y, B, [B.sup.*]) is a filter map.

Theorem 39. Let [phi] : X [right arrow] Y be a function and [{([B.sub.i], [B.sup.*.sub.i])}.sub.i[member of][GAMMA]] be a family of (L,M)-dffb's on X satisfying the following condition:

(C) For every finite subset K of [GAMMA], if [[??].sub.i[member of]K][[lambda].sub.i] = 0, then [mathematical expression not reproducible]. Then, the following properties are satisfied:

(i) If [phi] : X [right arrow] Y is bijective, then [mathematical expression not reproducible].

(ii) If [phi] : X [right arrow] Y is injective, then [mathematical expression not reproducible].

Proof. (i) Let us consider the following condition:

(C1) For every finite subset K of [GAMMA], if [[??].sub.i[member of]K][v.sub.i] = [0.bar], then [mathematical expression not reproducible].

For the proof, it is enough to show that (C1) [??] (C).

(C1) [??] (C): For any finite subset K of [GAMMA] with [[??].sub.i[member of]K][[lambda].sub.i] = [0.bar], since [phi] is injective, by Lemma 6(!),

[mathematical expression not reproducible]. (74)

By (C1), we have

[mathematical expression not reproducible], (75)

and thus [mathematical expression not reproducible] and [mathematical expression not reproducible] is satisfied.

(C) [??] (C1): Suppose that, for every finite subset K of [GAMMA] with [mathematical expression not reproducible]. Then, for each i [member of] K, there exists [[lambda].sub.i] [member of] [L.sup.X] with [v.sub.i] = [[phi].sup.[right arrow].sub.L]([[lambda].sub.i]) such that

[mathematical expression not reproducible]. (76)

By (C), [[??].sub.i[member of]K][[lambda].sub.i] [not equal to] [0.bar]. By Lemma 6(!),

[mathematical expression not reproducible]. (77)

This contradicts the assumption. Thus, [mathematical expression not reproducible]. Similarly, for every finite subset K of [GAMMA], if [[??].sub.i[member of]K] [v.sub.i] = [0.bar] then [mathematical expression not reproducible].

Since [phi] is surjective, by Theorem 36, ([[[phi].sup.[??].sub.1]([B.sub.i]), [[phi].sup.*[??].sub.1]([B.sup.*.sub.i])) exists for each i [member of] [GAMMA]. By Corollary 27 and (C1), [mathematical expression not reproducible] exists.

For each finite subset K of [GAMMA] such that [lambda] = [[??].sub.k[member of]K][[lambda].sub.k] with [[phi].sup.[right arrow].sub.L]([lambda]) = v, the following inequalities are satisfied:

[mathematical expression not reproducible]. (78)

This implies that

[mathematical expression not reproducible]. (79)

So, the following are clear:

[mathematical expression not reproducible]. (80)

For any finite subset J of [GAMMA] with [mu] = [[??].sub.j[member of]J][[mu].sub.j], there exist [[eta].sub.j] [member of] [L.sup.X] with [[phi].sup.[right arrow].sub.L]([[eta].sub.j]) = [[mu].sub.j]. Thus,

[mathematical expression not reproducible]. (81)

This implies that

[mathematical expression not reproducible]. (82)

From the above inequalities, we have

[mathematical expression not reproducible]. (83)

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

Theorem 40. Let {[[phi].sub.i] : [X.sub.i] [right arrow] X : i [member of] [GAMMA]} be a family of functions and [{[(B.sub.i], [B.sup.*.sub.i])}.sub.i[member of][GAMMA]] be a family of (L, M)-dffbs' on [X.sub.i] satisfying the following condition:

(C) For any finite subset K of [GAMMA], [[??].sub.i[member of]K] [([[phi].sub.i]).sup.[right arrow].sub.L] ([[lambda].sub.i]) = [0.bar] then [mathematical expression not reproducible] and [mathematical expression not reproducible].

We define the maps [mathematical expression not reproducible] as

[mathematical expression not reproducible], (84)

where [??] and [??] are taken for every finite subset K of [GAMMA]. Let [mathematical expression not reproducible] and [mathematical expression not reproducible]. Then, the following properties are satisfied:

(i) If [[phi].sub.j] is surjective for some j [member of] [GAMMA], then (B, [B.sup.*]) is an (L,M)-dffb on X and (<B>, <[B.sup.*]>) is the coarsest (L,M)-dff for which the map [phi] : ([X.sub.i], <[B.sub.i]>, <[B.sup.*.sub.i]>) [right arrow] (X, <B>, <[B.sup.*]>) is a filter preserving map.

(ii) A function [phi] : (X, <B>, <[B.sup.*]>) [right arrow] (Y, F, [F.sup.*]) is a filter preserving map if and only if, for each i [member of] [GAMMA], [phi] [omicron] [[phi].sub.i] : ([X.sub.i], <[B.sub.i]>, <[B.sup.*.sub.i]>) [right arrow] (Y, F, [F.sup.*]) is a filter preserving map.

(iii) If [[phi].sub.i] are surjective for all i [member of] [GAMMA], then

[mathematical expression not reproducible]. (85)

Proof. (i) (DFFB2) Since [[phi].sub.j] is surjective for some j [member of] [GAMMA] and (C), B([1.bar]) = [1.sub.M], [B.sup.*]([1.bar]) = [0.sub.M] and B([0.bar]) = [0.sub.M], [B.sup.*]([0.bar]) = [1.sub.M].

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 K of [GAMMA], if [[??].sub.i[member of]K][v.sub.i] = 0, then [mathematical expression not reproducible].

For the proof, it is enough to show that (C1) [??] (C).

(C1) [??] (C): For any finite subset K of [GAMMA], if [[??].sub.i[member of]K][([[phi].sub.i]).sup.[right arrow].sub.L] ([[lambda].sub.i]) = [0.bar], by (C1), we have

[mathematical expression not reproducible]. (86)

(C) [??] (C1): Suppose that, for any finite subset K of [GAMMA] with [[??].sub.i[member of]K][v.sub.i] = [0.bar], we have [mathematical expression not reproducible]. Then, for each i [member of] K, there exists [mathematical expression not reproducible] with [v.sub.i] = [([[phi].sub.i]).sup.[right arrow].sub.L]([[lambda].sub.i]) such that

[mathematical expression not reproducible]. (87)

By (C),

[mathematical expression not reproducible]. (88)

This is a contradiction. Thus, [mathematical expression not reproducible]. Similarly, [mathematical expression not reproducible].

For any finite index set K with {[[lambda].sub.i] : [[??].sub.i[member of]K][([[phi].sub.i]).sup.[right arrow].sub.L] ([[lambda].sub.i]) [less than or equal to] v}, by the definition of

[mathematical expression not reproducible] and [mathematical expression not reproducible], the following inequalities are obtained:

[mathematical expression not reproducible]. (89)

Hence,

[mathematical expression not reproducible]. (90)

For any finite index set J with {[v.sub.i] : [[??].sub.i[member of]J][v.sub.i] [less than or equal to] [mu]}, since [[phi].sub.i] is surjective, for each i [member of] J, there exists [mathematical expression not reproducible] with [([[phi].sub.i]).sup.[right arrow].sub.L]([[lambda].sub.i]) = [v.sub.i] such that [mu] [greater than or equal to] [[??].sub.i[member of]J][v.sub.i] = [[??].sub.i[member of]J] [([[phi].sub.i]).sup.[right arrow].sub.L]([[lambda].sub.i]).

Thus,

[mathematical expression not reproducible]. (91)

Hence,

[mathematical expression not reproducible]. (92)

6. Conclusion

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

https://doi.org/10.1155/2018/3871282

Conflicts of Interest

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

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A. A. Abd El-Latif, (1,2) H. Aygun, (3) and V. Cetkin (iD) (3)

(1) Department of Mathematics, Faculty of Science and Arts at Balgarn, P.O. Box 60, University of Bisha, Sabt Al-Alaya 61985, Saudi Arabia

(2) High Institute of Computer King Marriott, P.O. Box 3135, Alexandria, Egypt

(3) Department of Mathematics, Kocaeli University, 41380 Kocaeli, Turkey

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

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

Academic Editor: Kemal Kilic

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Title Annotation: | Research Article |
---|---|

Author: | El-Latif, A.A. Abd; Aygun, H.; Cetkin, V. |

Publication: | Advances in Fuzzy Systems |

Date: | Jan 1, 2018 |

Words: | 11522 |

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