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

Authors: A. A. Abd El-Latif; H. Aygün; V. Çetkin
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

We give in this paper the definitions of(L,M)-double fuzzy filter base and (L,M)-double fuzzy filter structures where L and M are strictly two-sided commutative quantales, and we also investigate the relations between them. Moreover, we propose second-order image and preimage operators of (L,M)-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 (L,M)-DFIL of (L,M)-double fuzzy filter spaces and filter maps between these spaces is a topological category over the categorySET.


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## Body

## 1. Introduction

Kubiak [1] and Šostak [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. 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 latticeL. 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 ofL-filter was introduced by Höhle and Šostak [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, letX be a nonempty set. Let L=(L,≤,∨,∧) be a complete lattice with the least element 0L and the greatest element 1L. For α∈L,  α_(x)=α for all x∈X. The second lattice belonging to the context of our work is denoted by M and M0=M-{0M} and M1=M-{1M}.A complete latticeL=(L,≤,∧,∨) is called completely distributive, if for any family {{ai,j:j∈Ji}:i∈I} in L the following identity holds:(1)CD⋀i∈I⋁j∈Jiai,j=⋁φ∈∏i∈IJi⋀i∈Iai,φi.Definition 1 (see [24, 31–33]).
A tripleL=(L,≤,⊙) is called a strictly two-sided commutative quantale (stsc-quantale, for short) iff it satisfies the following properties:(L1)
(L,⊙) is a commutative semigroup.(L2)
x⊙1L=x, for all x∈L.(L3)
⊙ is distributive over arbitrary joins:(2)x⊙⋁i∈Iyi=⋁i∈Ix⊙yi,∀x∈L,∀yii∈I⊆L.An stsc-quantaleL=(L,≤,⊙) is an ⋀-distributive quantale (or stsc-biquantale [34]) if ⊙ is distributive over nonempty meets: (3)x⊙⋀i∈Iyi=⋀i∈Ix⊙yi,∀x∈L,∀yii∈I⊆L.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],≤,∧,0,1) is an ⋀-distributive quantale.
(2) Every left-continuous t-normT on [0,1],([0,1],≤,T) is an stsc-quantale.
(3) Every continuous t-normT on [0,1],([0,1],≤,T) is an ⋀–distributive quantale.
(4) Every GL-monoid is an stsc-quantale.
(5) Let(L,≤,⊙) be an stsc-quantale. For each x,y∈L, we define (4)x⟼y=⋁z∈L∣x⊙z≤y.
Then, it satisfies Galois correspondence; that is,(5)x⊙z≤y⟺z≤x⟼y,∀x,y,z∈L.Definition 3 (see [1, 7, 24, 29, 31, 33, 35–38]).
Let(L,≤,⊙) be an stsc-quantale. A mapping ⋆:L→L is called an order-reversing involution if it satisfies the following conditions:
(1)x⋆⋆=x, for each x∈L.
(2) Ifx≤y, then y⋆≤x⋆, for each x,y∈L.
An stsc-quantale is called a Girard monoid [37] if (x↦0L)↦0L=x,∀x∈L.
Hence, in caseL is a Girard monoid, residuation ↦ induces an order-reversing involution ⋆:L→L. In this paper, we always assume that (L,≤,⊙,⊕,⋆) (resp., (M,≤,⊙~,⊕~,⋆)) is a Girard monoid with an order-reversing involution ⋆, and the operation ⊕ is defined by (6)x⊕y=x⋆⊙y⋆⋆,x⋆=x⟼0Lunless otherwise specified, where ⊙~,⊕~ denote the quantale operations on M.Remark 4 (see [39]).
When the underlying latticeL 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⋆=x↦0).Lemma 5 (see [34]).
LetL be a Girard monoid. For each x,y,z,xi,yi∈L, one has the following properties:
(1) Ify≤z, then x⊙y≤x⊙z, x⊕y≤x⊕z,x↦y≤x↦z, and y↦x≥z↦x.
(2)x⊙y≤x∧y≤x∨y≤x⊕y.
LetL be a complete lattice and φ:X→Y be a function. The Zadeh image and preimage operators φ→:LX→LY and φ←:LY→LX are defined by (7)φ→λy=⋁λx∣y=φx,φ←μx=μφx,∀x∈X,y∈Y.Lemma 6 (see [40]).
Let(L,≤,⊙) be an stsc-quantale and φ:X→Y be a function. For each λ,μ∈LX and λi∈LY, one has the following properties:(1)
φ→(λ⊙μ)≤φ→(λ)⊙φ→(μ) with equality if φ is injective.(2)
φ←(⊙i=1nλi)=⊙i=1nφ←(λi).Definition 7 (see [40]).
Basic scheme for second-order image operators: letφ:X→Y be a function.Case 1.
Consider(8)φL→L→:LLX⟶LLY.
This is the Zadeh image operator of the Zadeh image operator. We denote it byφ1⇉; that is, for all U∈LLX and μ∈LY, (9)φ1⇉Uμ=φL→L→Uμ=⋁Uλ:μ=φL→λ.Case 2.
Consider(10)φL←L←:LLX⟶LLY.
This is the Zadeh preimage operator of the Zadeh preimage operator. We denote it byφ2⇉; that is, for all U∈LLX and μ∈LY, (11)φ2⇉Uμ=φL←L←Uμ=U∘φL←μ.
Basic scheme for second-order preimage operators: letφ:X→Y be a function.Case 1.
Consider(12)φL←L→:LLX⟵LLY.
This is the Zadeh image operator of the Zadeh preimage operator. We denote it byφ1⇇; that is, for all V∈LLY and λ∈LX, (13)φ1⇇Vλ=φL←L→Vλ=⋁Vμ:λ=φL←μ.Case 2.
Consider(14)φL→L←:LLX⟵LLY.
This is the Zadeh preimage operator of the Zadeh image operator. We denote it byφ2⇇; that is, for all V∈LLY and λ∈LX, (15)φ2⇇Vλ=φL→L←Vλ=V∘φL→λ.
In this paper, we consider additional operators as follows.
Define the operatorφ1∗⇇:MLX←MLY as φ1∗⇇(V)(λ)=⋀{V(μ):λ=φL←(μ)}, for all V∈MLY and λ∈LX.
Define the operatorφ1∗⇉:MLX→MLY as φ1∗⇉(V)(μ)=⋀{V(λ):μ=φL→(λ)}, for all V∈MLX and μ∈LY.
All algebraic operations onL can be extended pointwise to the sets LX and MLX as follows: for all x∈X,λ,μ∈LX, and U,V∈MLX;(1)
λ≤μ iff λ(x)≤μ(x).(2)
(λ⊙μ)(x)=λ(x)⊙μ(x).(3)
U≤V iff U(λ)≤V(λ).Definition 8 (see [41]).
The pair(T,T∗) of maps T,T∗:LX→M is called an (L,M)-double fuzzy topology on X if it satisfies the following conditions:
(LO1)T(λ)≤(T∗(λ))⋆, for each λ∈LX,
(LO2)T(0_)=T(1_)=1M,  T∗(0_)=T∗(1_)=0M,
(LO3)T(λ1⊙λ2)≥T(λ1)⊙~T(λ2) and T∗(λ1⊙λ2)≤T∗(λ1)⊕~T∗(λ2), for each λ1,λ2∈LX,
(LO4)T(⋁i∈Δλi)≥⋀i∈ΔT(λi) and T∗(⋁i∈Δλi)≤⋁i∈ΔT∗(λi), for each λi∈LX,i∈Δ.
The triplet(X,T,T∗) is called an (L,M)-double fuzzy topological space ((L,M)-dfts, for short). T and T∗ may be interpreted as gradation of openness and gradation of nonopenness, respectively.
Let(U,U∗) and (T,T∗) be (L,M)-double fuzzy topologies on X. We say that (U,U∗) is finer than (T,T∗) ((T,T∗) is coarser than (U,U∗)) if T(λ)≤U(λ) and T∗(λ)≥U∗(λ) for all λ∈LX.
Let(X,T,T∗) and (Y,U,U∗) be (L,M)-dfts’s. A function φ:X→Y is called LF-continuous iff U(λ)≤T(φ←(λ)) and U∗(λ)≥T∗(φ←(λ)), for all λ∈LY.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.
LetX={x,y} be a set, L=M=[0,1] and x⊙y=max⁡x+y-1,0,x⊕y=min⁡{x+y,1}. Then, ([0,1],≤,⊙) is a left-continuous t-norm (Lukasiewicz t-norm) with strong induced negation x↦0=min⁡{1-x,1}. Let μ,ρ∈[0,1]X be defined as follows: μ(x)=0.6,μ(y)=0.3,ρ(x)=0.5,ρ(y)=0.7. Define T,T⋆:[0,1]X→[0,1] as follows:(16)Tλ=1,ifλ=0_,1_;0.8,ifλ=μ;0.3,ifλ=ρ;0.7,ifλ=μ∨ρ;0.2,ifλ=μ∧ρ;0,otherwise,T∗λ=0,ifλ=0_,1_;0.2,ifλ=μ;0.7,ifλ=ρ;0.3,ifλ=μ∨ρ;0.8,ifλ=μ∧ρ;1,otherwise.
Then, the pair(T,T∗) is a ([0,1],[0,1])-dft on X.Remark 10.
(1) If(L=M=[0,1],⊙=∧,⊕=∨) with an order-reversing involution ⋆,(a⋆=1-a)(L,M)-dfts is the concept of Mondal and Samanta [18].
(2) IfL and M are frames with 0 and 1, (L,M)-dfts is the concept of Gutiérrez García and Rodabaugh [19].
(3) If⊙=∧, (L,M)-dfts is the concept of Abd El-latif [42].Definition 11 (see [29, 30]).
A mapF:LX→L is called an L-filter if it fulfills the following conditions:
(LF1)F(0_)=0M and F(1_)=1M.
(LF2)F(λ⊙μ)≥F(λ)⊙F(μ), for each λ,μ∈LX.
(LF3) Ifλ≤μ, then F(λ)≤F(μ).
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∗) of maps F,F∗:LX→M is called an (L,M)-double fuzzy filter (briefly, (L,M)-dff) on X if it fulfills the following axioms:
(DFF1)F(λ)≤(F∗(λ))⋆, for each λ∈LX.
(DFF2)F(0_)=0M,  F(1_)=1M and F∗(0_)=1M,  F∗(1_)=0M.
(DFF3)F(λ⊙μ)≥F(λ)⊙~F(μ) and F∗(λ⊙μ)≤F∗(λ)⊕~F∗(μ), for each λ,μ∈LX.
(DFF4) Ifλ≤μ, then F(λ)≤F(μ) and F∗(λ)≥F∗(μ).
The triplet(X,F,F∗) is called an (L,M)-double fuzzy filter space (briefly, (L,M)-dffs).
If(F1,F1∗) and (F2,F2∗) are two (L,M)-dffs on X, we say that (F1,F1∗) is finer than (F2,F2∗) (or (F2,F2∗) is coarser than (F1,F1∗)), denoted by (F2,F2∗)≤(F1,F1∗) if and only if F2(λ)≤F1(λ) and F2∗(λ)≥F1∗(λ), for each λ∈LX.Definition 13.
Let(X,F1,F1∗) and (Y,F2,F2∗) be two (L,M)-dffs’s. Then, a map φ:X→Y is said to be
(i) a filter map if and only ifF2≤F1∘φL← and F2∗≥F1∗∘φL←;
(ii) a filter preserving map if and only ifF1≤F2∘φL→ and F1∗≥F2∗∘φL→.
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) LetF be an L-filter on X and F∗:LX→L defined by F∗(λ)=(F(λ))⋆. Then, the pair (F,F∗) is an (L,L)-dff on X. Therefore, (L,M)-dff is a generalization of L-filter due to Höhle and Šostak [7, 29].
(ii) If⊙=∧, 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∗) produces an (L,M)-dfts (X,TF,TF∗∗).Proof.
When contrasting the axioms of(L,M)-dff and (L,M)-dft, we find (DFF4) implying (DFT4).
Let{λi:i∈Γ}⊆LX. Then, λi≤⋁i∈Γλi for all i∈Γ; due to (DFF4), we have that F(λi)≤F(⋁i∈Γλi) and F∗(λi)≥F∗(⋁i∈Γλi) for all i∈Γ. So,(17)F⋁i∈Γλi≥Fλi≥⋀i∈ΓFλi,F∗⋁i∈Γλi≤F∗λi≤⋁i∈ΓF∗λi.
Then, we can get an(L,M)-dft (TF,TF∗∗) defined by(18)TFλ=Fλ,ifλ≠0_1M,ifλ=0_,TF∗∗λ=F∗λ,ifλ≠0_0M,ifλ=0_.Theorem 16.
Let(X,F1,F1∗) and (Y,F2,F2∗) be (L,M)-dffs’s. If φ:(X,F1,F1∗)→(Y,F2,F2∗) is a filter map, then φ:(X,TF1,TF1∗∗)→(Y,TF2,TF2∗∗) is an LF-continuous map.Proof.
Letμ∈LY. If μ=0Y or φL←(μ)=0X, then the proof is easy. Let μ≠0Y and φL←(μ)≠0X. Then, from the definition of double filter map and Theorem 15, we have (19)TF1φL←μ=F1φL←μ≥F2μ=TF2μ,TF1∗∗φL←μ=F1∗φL←μ≤F2∗μ=TF2∗∗μ.Corollary 17.
The functionF:(L,M)-DFIL→(L,M)-DFTOP defined by F(X,F,F∗)=(X,TF,TF∗∗) and F(φ)=φ is a functor.Notation 18.
LetB,B∗:LX→M be two maps and λ∈LX. Then, 〈B〉 and 〈B∗〉 are defined as follows: (20)Bλ=⋁μ≤λBμ,B∗λ=⋀μ≤λB∗μ.Definition 19.
The pair(B,B∗) of maps B,B∗:LX→M is called an (L,M)-double fuzzy filter base (briefly, (L,M)-dffb) on X if it fulfills the following axioms:
(DFFB1)B(λ)≤(B∗(λ))⋆, for each λ∈LX.
(DFFB2)B(0_)=0M,  B(1_)=1M and B∗(0_)=1M,  B∗(1_)=0M.
(DFFB3)〈B〉(λ⊙μ)≥B(λ)⊙~B(μ) and 〈B∗〉(λ⊙μ)≤B∗(λ)⊕~B∗(μ), for each λ,μ∈LX.
If(B1,B1∗) and (B2,B2∗) are two (L,M)-dffb’s on X, we say (B1,B1∗) is finer than (B2,B2∗) (or (B2,B2∗) is coarser than (B1,B1∗)) denoted by (B2,B2∗)≤(B1,B1∗) if and only if 〈B2〉(λ)≤〈B1〉(λ) and 〈B2∗〉(λ)≥〈B1∗〉(λ), for each λ∈LX.Remark 20.
(i) An(L,M)-dffb is a generalization of L-filter base due to Kim and Ko [40].
(ii) If(F,F∗) is an (L,M)-dff, then (F,F∗) is an (L,M)-dffb with 〈F〉=F and 〈F∗〉=F∗.
(iii) If(B,B∗) is an (L,M)-dffb, then, by (DFFB3), λ⊙μ=0_ implies B(λ)⊙~B(μ)=0M and B∗(λ)⊕~B∗(μ)=1M.Theorem 21.
If(B,B∗) is an (L,M)-dffb, then (〈B〉,〈B∗〉) is the coarsest (L,M)-dff which satisfies B≤〈B〉 and B∗≥〈B∗〉.Proof.
(DFF1) For eachλ∈LX, (21)Bλ=⋁μ≤λBμ≤⋁μ≤λB∗μ⋆=⋀μ≤λB∗μ⋆=B∗λ⋆.
(DFF2) and (DFF4) are easily checked.
(DFF3) Suppose that there existλ,μ∈LX such that (22)B∗λ⊙μ≰B∗λ⊕~B∗μ.By the definition of 〈B∗〉 and (L4′), there exist λ1,μ1∈LX with λ1≤λ and μ1≤μ such that(23)B∗λ⊙μ≰B∗λ1⊕~B∗μ1.Since (B,B∗) is an (L,M)-dffb, (24)B∗λ1⊙μ1≤B∗λ1⊕~B∗μ1.Since λ1⊙μ1≤λ⊙μ, we have (25)B∗λ⊙μ≤B∗λ1⊙μ1≤B∗λ1⊕~B∗μ1.It is a contradiction. Thus, 〈B∗〉(λ⊙μ)≤〈B∗〉(λ)⊕~〈B∗〉(μ), for each λ,μ∈LX. Similarly, 〈B〉(λ⊙μ)≥〈B〉(λ)⊙~〈B〉(μ), for each λ,μ∈LX.
Let(F,F∗) be another (L,M)-dff which is finer than (B,B∗), that is, B≤F and B∗≥F∗. Then, we have (26)Bλ=⋁μ≤λBμ≤⋁μ≤λFμ=Fλ,B∗λ=⋀μ≤λB∗μ≥⋀μ≤λF∗μ=F∗λ.Theorem 22.
H , H ∗ : L X → M are maps fulfilling the following conditions:
(C1)H(λ)≤(H∗(λ))⋆, for each λ∈LX,
(C2)H(1_)=1M and H∗(1_)=0M and for each finite index set K, if ⨀i∈Kλi=0_, then ⨀i∈K~H(λi)=0M and ⨁i∈K~H∗(λi)=1M.
We define the mapsBH,BH∗∗:LX→M as (27)
B
H
λ
=
⋁
⨀
i
∈
K
~
H
λ
i
:
λ
=
⨀
i
∈
K
λ
i
,
B
H
∗
∗
λ
=
⋀
⨁
i
∈
K
~
H
∗
λ
i
:
λ
=
⨀
i
∈
K
λ
i
,where ⋁ and ⋀ are taken for every finite index set K such that λ=⨀i∈Kλi, respectively. Then, the following properties are satisfied:
(i)(BH,BH∗∗) is an (L,M)-dffb on X.
(ii) IfH≤B, H∗≥B∗ and (B,B∗) is an (L,M)-dffb on X, then 〈BH〉≤〈B〉 and 〈BH∗∗〉≥〈B∗〉.Proof.
(i) (DFFB1) For eachλ∈LX, the following is valid:(28)
B
H
λ
=
⋁
⨀
i
∈
K
~
H
λ
i
:
λ
=
⨀
i
∈
K
λ
i
≤
⋁
⨀
i
∈
K
~
H
∗
λ
i
⋆
:
λ
=
⨀
i
∈
K
λ
i
=
⋀
⨁
i
∈
K
~
H
∗
λ
i
:
λ
=
⨀
i
∈
K
λ
i
⋆
=
B
H
∗
∗
λ
⋆
.
(DFFB2) It is clear by condition (C2).
(DFFB3) For eachλ,μ∈LX and for any two finite index sets K,J with λ=⨀k∈Kλk and μ=⨀j∈Jμi, since λ⊙μ=(⨀k∈Kλk)⊙(⨀j∈Jμi), by the definition of BH and BH∗∗, we get (29)
B
H
λ
⊙
μ
≥
B
H
λ
⊙
μ
≥
⨀
k
∈
K
~
H
λ
k
⊙
~
⨀
j
∈
J
~
H
μ
j
,
B
H
∗
∗
λ
⊙
μ
≤
B
H
∗
∗
λ
⊙
μ
≤
⨁
k
∈
K
~
H
∗
λ
k
⊕
~
⨁
j
∈
J
~
H
∗
μ
j
.
If supremum and infimum are taken over finite index setK, respectively, then by (2) and (3), (30)BHλ⊙μ≥BHλ⊙~BHμ,BH∗∗λ⊙μ≤BH∗∗λ⊕~BH∗∗μ.Thus, (BH,BH∗∗) is an (L,M)-dffb on X.
(ii) For any finite family{λi:⨀i∈Kλi≤λ}, the following are true: (31)
B
λ
≥
B
⨀
i
∈
K
λ
i
≥
⨀
i
∈
K
~
B
λ
i
≥
⨀
i
∈
K
~
B
λ
i
≥
⨀
i
∈
K
~
H
λ
i
,
B
∗
λ
≤
B
∗
⨀
i
∈
K
λ
i
≤
⨁
i
∈
K
~
B
∗
λ
i
≤
⨁
i
∈
K
~
B
∗
λ
i
≤
⨁
i
∈
K
~
H
∗
λ
i
.Then, 〈BH〉≤〈B〉 and 〈BH∗∗〉≥〈B∗〉.Theorem 23.
Let(B1,B1∗) and (B2,B2∗) be two (L,M)-dffb’s on X and Y, respectively, and φ:X→Y be a function. Then, one has the following properties:
(i)φ:(X,〈B1〉,〈B1∗〉)→(Y,〈B2〉,〈B2∗〉) is a filter map if and only if B2≤〈B1〉∘φL← and B2∗≥〈B1∗〉∘φL←.
(ii)φ:(X,〈B1〉,〈B1∗〉)→(Y,〈B2〉,〈B2∗〉) is a filter preserving map if and only if B1≤〈B2〉∘φL→ and B1∗≥〈B2∗〉∘φL→.
(iii) IfB2≤B1∘φL← and B2∗≥B1∗∘φL←, then φ:(X,〈B1〉,〈B1∗〉)→(Y,〈B2〉,〈B2∗〉) is a filter map.
(iv) IfB1≤B2∘φL→ and B1∗≥B2∗∘φL→, then φ:(X,〈B1〉,〈B1∗〉)→(Y,〈B2〉,〈B2∗〉) is a filter preserving map.Proof.
Proving condition (i) is enough since the other conditions are similarly proved.
(i) (⇒:) Since B2(μ)≤〈B2〉(μ) and B2∗(μ)≥〈B2∗〉(μ), for each μ∈LY, it is trivial.
(⇐:) Let B2(μ)≤〈B1〉(φL←(μ)) and B2∗(μ)≥〈B1∗〉(φL←(μ)), for each μ∈LY. We will show that φ is a filter map. For arbitrary μ∈LY, we have (32)B2μ=⋁ν≤μB2ν≤⋁φL←ν≤φL←μB1φL←ν≤B1φL←μ,B2∗μ=⋀ν≤μB2∗ν≥⋀φL←ν≤φL←μB1∗φL←ν≥B1∗φL←μ.Thus, φ is a filter map.Example 24.
LetX={x,y} be a set, L=M=[0,1] be the stsc-quantale with Lukasiewicz t-norm, and μ,ν∈[0,1]X be defined by μ(x)=0.6,μ(y)=0.5,ν(x)=0.1,ν(y)=0. Define the maps Bi,Bi∗:LX→Mi=1,2,3 as follows:(33)B1λ=1,ifλ=1_;0.6,ifλ=μ;0.3,ifλ=ν;0,otherwise,B1∗λ=0,ifλ=1_;0.4,ifλ=μ;0.7,ifλ=ν;1,otherwise,B2λ=1,ifλ=1_;0.6,ifλ=μ;0.5,ifλ=μ⊙μ;0,otherwise,B2∗λ=0,ifλ=1_;0.4,ifλ=μ;0.5,ifλ=μ⊙μ;1,otherwise,B3λ=1,ifλ=1_;0.4,ifλ=μ;0.3,ifλ=μ⊙μ;0,otherwise,B3∗λ=0,ifλ=1_;0.6,ifλ=μ;0.7,ifλ=μ⊙μ;1,otherwise.It can be seen by easy computation that(1)(B1,B1∗) and (B3,B3∗) are not (L,M)-double fuzzy filters but they are (L,M)-double fuzzy filter bases, so they generate (L,M)-double fuzzy filters (〈B1〉,〈B1∗〉) and (〈B3〉,〈B3∗〉).(2)(B2,B2∗) is not an (L,M)-double fuzzy filter base and it does not satisfy condition (C2) of Theorem 22.(3) Since(〈B3〉,〈B3∗〉)≤(〈B1〉,〈B1∗〉), idX:(X,〈B1〉,〈B1∗〉)→(X,〈B3〉,〈B3∗〉) is a filter map and idX:(X,〈B3〉,〈B3∗〉)→(X,〈B1〉,〈B1∗〉) is a filter preserving map though B3≰B1,B3∗≱B1∗ and B1≰B3,B1∗≱B3∗.We also note that ifL=M=[0,1] is considered as a frame, then (B1,B1∗) is an (L,M)-double fuzzy filter base.


## 4. The Types(φ1⇇,φ1∗⇇), (φ2⇉,φ2⇉) of Preimages and Images of (L,M)-Double Fuzzy Filter Bases

Theorem 25.
Letφ:X→Y be a function and (B,B∗) be an (L,M)-dffb on Y. Then, the following properties are satisfied.
(i) IfφL←(μ)=0_ implies B(μ)=0M and B∗(μ)=1M, then (φ1⇇(B),φ1∗⇇(B∗)) is an (L,M)-dffb on X and (〈ϕ1⇇(B)〉,〈φ1∗⇇(B∗)〉) is the coarsest (L,M)-dff on X for which φ:(X,〈ϕ1⇇(B)〉,〈φ1∗⇇(B∗)〉)→(Y,〈B〉,〈B∗〉) is a filter map.
(ii) Ifφ is surjective, then (φ1⇇(B),φ1∗⇇(B∗)) is an (L,M)-dffb.
(iii) IfφL←(μ)=0_ implies B(μ)=0M and B∗(μ)=1M, φ is injective, and (B,B∗) is an (L,M)-dff on Y, then (φ1⇇(B),φ1∗⇇(B∗)) is an (L,M)-dff on X.Proof.
(i) (DFFB1) For eachλ∈LX, we have(34)φ1⇇Bλ=⋁Bμ:λ=φL←μ≤⋁B∗μ⋆:λ=φL←μ=⋀B∗μ:λ=φL←μ⋆=φ1∗⇇B∗λ⋆.
(DFFB2) SinceφL←(1_)=1_, then φ1⇇(B)(1_)=1M and φ1∗⇇(B∗)(1_)=0M. By assumption, φ1⇇(B)(0_)=0M and φ1∗⇇(B∗)(0_)=1M.
(DFFB3) Suppose that there existλ1,λ2∈LX such that (35)φ1∗⇇B∗λ1⊙λ2≰φ1∗⇇B∗λ1⊕~φ1∗⇇B∗λ2.By the definition of φ1∗⇇(B∗) and (L4′), there exist μ1,μ2∈LY with λ1=φL←(μ1) and λ2=φL←(μ2) such that (36)φ1∗⇇B∗λ1⊙λ2≰B∗μ1⊕~B∗μ2.
Since(B,B∗) is an (L,M)-dffb, the following is valid: (37)B∗μ1⊙μ2≤B∗μ1⊕~B∗μ2.Then, (38)φ1∗⇇B∗λ1⊙λ2≰B∗μ1⊙μ2.By the definition of 〈B∗〉, there exists ν∈LY with ν≤μ1⊙μ2 such that (39)φ1∗⇇B∗λ1⊙λ2≰B∗ν.
On the other hand, since(40)λ1⊙λ2=φL←μ1⊙φL←μ2=φL←μ1⊙μ2≥φL←ν,then 〈φ1∗⇇(B∗)〉(λ1⊙λ2)≤φ1∗⇇(B∗)(φL←(ν))≤B∗(ν).
This contradicts the assumption. Thus,(41)φ1∗⇇B∗λ1⊙λ2≤φ1∗⇇B∗λ1⊕~φ1∗⇇B∗λ2,for  eachλ1,λ2∈LX.
Similarly,〈φ1⇇(B)〉(λ1⊙λ2)≥φ1⇇(B)(λ1)⊙~φ1⇇(B)(λ2),foreachλ1,λ2∈LX.
Hence,(φ1⇇(B),φ1∗⇇(B∗)) is an (L,M)-dffb on X.
Let(F,F∗) be another (L,M)-dff on X such that φ:(X,F,F∗)→(Y,〈B〉,〈B∗〉) is a filter map. Then, for each λ∈LX, the following inequalities are valid:(42)φ1⇇Bλ=⋁Bμ:φL←μ≤λ≤⋁FφL←μ:φL←μ≤λ≤Fλ,φ1∗⇇B∗λ=⋀B∗μ:φL←μ≤λ≥⋀F∗φL←μ:φL←μ≤λ≥F∗λ.
(ii) Sinceφ is surjective, φL←(μ)=0_ implies μ=0_. So, B(μ)=0M and B∗(μ)=1M. Then, by (i), (φ1⇇(B),φ1∗⇇(B∗)) is an (L,M)-dffb on X.
(iii) (DFF1)–(DFF3) are obvious.
(DFF4) Letλ1≤λ2, for λ1,λ2∈LX. Since φ is injective, there exists ν∈LY with φL←(ν)=λ1 and λ2=φL←(ν∨φL→(λ2)). It implies (43)φ1⇇Bλ2≥Bν∨φL→λ2≥Bν,φ1∗⇇B∗λ2≤B∗ν∨φL→λ2≤B∗ν.
If supremum and infimum are taken over{λ1∣φL←(ν)=λ1}, respectively, then it is clear that φ1⇇(B)(λ2)≥φ1⇇(B)(λ1) and φ1∗⇇(B∗)(λ2)≤φ1∗⇇(B∗)(λ1).Theorem 26.
Let{φi:X→Xi}i∈Γ be a family of functions and {(Bi,Bi∗)}i∈Γ be a family of (L,M)-dffb’s on Xi satisfying the following condition:
(C) For each finite subsetK of Γ, if ⨀i∈K(λi∘φi)=0_, then ⨀i∈KBi(λi)=0M and ⨁i∈KBi∗(λi)=1M.
We define the maps⨆i∈Γ(φi)1⇇(Bi),⨅i∈Γ(φi∗)1⇇(Bi∗):LX→M as(44)
⨆
i
∈
Γ
φ
i
1
⇇
B
i
λ
=
⋁
⨀
i
∈
K
~
B
i
λ
i
:
λ
=
⨀
i
∈
K
λ
i
∘
φ
i
,
⨅
i
∈
Γ
φ
i
∗
1
⇇
B
i
∗
λ
=
⋀
⨁
i
∈
K
~
B
i
∗
λ
i
:
λ
=
⨀
i
∈
K
λ
i
∘
φ
i
,where ⋁ and ⋀ are taken for every finite index subset K of Γ such that λ=⨀i∈K(λi∘φi).
LetB=⨆i∈Γ(φi)1⇇(Bi) and B∗=⨅i∈Γ(φi∗)1⇇(Bi∗) be given. Then, the following properties are satisfied.
(i)(B,B∗) is an (L,M)-dffb on X and (〈B〉,〈B∗〉) is the coarsest (L,M)-dff for which φi:(X,〈B〉,〈B∗〉)→(Xi,〈Bi〉,〈Bi∗〉) is a filter map for each i∈Γ.
(ii) A functionφ:(Y,F,F∗)→(X,〈B〉,〈B∗〉) is a filter map if and only if, for each i∈Γ,  φi∘φ:(Y,F,F∗)→(Xi,〈Bi〉,〈Bi∗〉) is a filter map.
(iii)〈⨆i∈Γ(φi)1⇇(Bi)〉=〈⨆i∈Γ(φi)1⇇(〈Bi〉)〉 and 〈⨅i∈Γ(φi∗)1⇇(Bi∗)〉=〈⨅i∈Γ(φi∗)1⇇(〈Bi∗〉)〉.Proof.
(i) (DFFB1) Letλ∈LX with λ=⨀i∈K(λi∘φi). Then, the following inequality is valid:(45)
B
λ
=
⋁
⨀
i
∈
K
~
B
i
λ
i
:
λ
=
⨀
i
∈
K
λ
i
∘
φ
i
≤
⋁
⨀
i
∈
K
~
B
i
∗
λ
i
⋆
:
λ
=
⨀
i
∈
K
λ
i
∘
φ
i
=
⋀
⨁
i
∈
K
~
B
i
∗
λ
i
:
λ
=
⨀
i
∈
K
λ
i
∘
φ
i
⋆
=
B
∗
λ
⋆
.
(DFFB2) By condition (C),B(0_)=0M and B∗(0_)=1M. Since, 1_=1_∘φi, then B(1_)=1M and B∗(1_)=0M.
(DFFB3) Suppose that there existλ,μ∈LX such that 〈B∗〉(λ⊙μ)≰B∗(λ)⊕~B∗(μ).
By the definition ofB∗(λ), and (L4′), there exists a finite subset K of Γ with λ=⨀k∈K(λk∘φk) such that 〈B∗〉(λ⊙μ)≰(⨁k∈K~Bk∗(λk))⊕~B(μ).
Again, by the definition ofB∗(μ) and (L4′), there exists a finite subset J of Γ with μ=⨀j∈J(μj∘φj) such that(46)
B
∗
λ
⊙
μ
≰
⨁
k
∈
K
~
B
k
∗
λ
k
⊕
~
⨁
j
∈
J
~
B
j
∗
μ
j
.
Putm∈(K∪J) such that(47)ρm=λm,ifm∈K∖K∩Jμm,ifm∈J∖K∩Jλm⊙μm,ifm∈K∩J.
Since, for eachm∈K∩J,〈Bm∗〉(λm⊙μm)≤Bm∗(λm)⊕~Bm∗(μm), we have (48)
B
∗
λ
⊙
μ
≰
⨁
m
∈
K
∪
J
-
K
∩
J
~
B
m
∗
ρ
m
⊕
~
⨁
m
∈
K
∩
J
~
B
m
∗
λ
m
⊙
μ
m
.
From the definition of〈Bm∗〉, there exists νm∈LXm with νm≤λm⊙μm such that(49)
B
∗
λ
⊙
μ
≰
⨁
m
∈
K
∪
J
-
K
∩
J
~
B
m
∗
ρ
m
⊕
~
⨁
m
∈
K
∩
J
~
B
m
∗
ν
m
.
On the other hand, since(50)λ⊙μ=⨀k∈Kλk∘φk⊙⨀j∈Jμj∘φj≥⨀m∈K∪J-K∩Jρm∘φm⊙⨀m∈K∩Jνm∘ϕmand since K∪J is finite, we have (51)
B
∗
λ
⊙
μ
≤
⨁
m
∈
K
∪
J
-
K
∩
J
~
B
m
∗
ρ
m
⊕
~
⨁
m
∈
K
∩
J
~
B
m
∗
ν
m
.
This contradicts the assumption. Then,〈B∗〉(λ⊙μ)≤B∗(λ)⊕~B∗(μ), for each λ,μ∈LX. Similarly, 〈B〉(λ⊙μ)≥B(λ)⊙~B(μ), for each, λ,μ∈LX. Hence, (B,B∗) is an (L,M)-dffb on X.
SinceB(λi∘φi)≥Bi(λi) and B∗(λi∘φi)≤Bi∗(λi), for each i∈Γ, by Theorem 23(iii), φi:(X,〈B〉,〈B∗〉)→(Xi,〈Bi〉,〈Bi∗〉) is a filter map.
Let(F,F∗) be an (L,M)-dff on X such that, for each i∈Γ, the map φi:(X,F,F∗)→(Xi,〈Bi〉,〈Bi∗〉) is a filter map. Then,(52)Fνi∘φi≥Biνi,F∗νi∘φi≤Bi∗νi,for  eachi∈Γ,νi∈LXi.For any finite subset K of Γ with ν≥⨀k∈K(νk∘φk), since F(νk∘φk)≥〈Bk〉(νk) and F∗(νk∘φk)≤〈Bk∗〉(νk), for each k∈K, we have(53)
F
λ
≥
F
⨀
k
∈
K
ν
k
∘
φ
k
≥
⨀
k
∈
K
~
F
ν
k
∘
φ
k
≥
⨀
k
∈
K
~
B
k
ν
k
≥
⨀
k
∈
K
~
B
k
ν
k
,
F
∗
λ
≤
F
∗
⨀
k
∈
K
ν
k
∘
φ
k
≤
⨁
k
∈
K
~
F
∗
ν
k
∘
φ
k
≤
⨁
k
∈
K
~
B
k
ν
k
≤
⨁
k
∈
K
~
B
k
ν
k
.
Hence, by the definition of〈B〉 and 〈B∗〉, it is obvious that (〈B〉,〈B∗〉)≤(F,F∗).
(ii) Necessity of the composition condition is obvious.
Conversely, for every finite subsetK of Γ with ν≥⨀k∈K(νk∘φk), since, for each k∈K,  φk∘φ:(Y,F,F∗)→(Xk,〈Bk〉,〈Bk∗〉) is a filter map, that is, 〈Bk〉(νk)≤F(νk∘(φk∘φ)) and 〈Bk∗〉(νk)≥F∗(νk∘(φk∘φ)). Since ν∘φ≥⨀k∈K((νk∘φk)∘φ), we have(54)
F
ν
∘
φ
≥
⨀
k
∈
K
~
F
ν
k
∘
φ
k
∘
φ
≥
⨀
k
∈
K
~
B
k
ν
k
≥
⨀
k
∈
K
~
B
k
ν
k
,
F
∗
ν
∘
φ
≤
⨁
k
∈
K
~
F
∗
ν
k
∘
φ
k
∘
φ
≤
⨁
k
∈
K
~
B
k
∗
ν
k
≤
⨁
k
∈
K
~
B
k
∗
ν
k
.
By the definition of〈B〉 and 〈B∗〉, we have 〈B〉(ν)≤F(ν∘φ) and 〈B∗〉(ν)≥F∗(ν∘φ).
(iii) PutF=⨆i∈Γ(φi)1⇇(〈Bi〉) and F∗=⨅i∈Γ(φi∗)1⇇(〈Bi∗〉); by applying (i) to both (〈B〉,〈B∗〉) and (〈F〉,〈F∗〉), the desired equality is obtained.The following corollaries are the direct results of Theorem26.Corollary 27.
Let{(Bi,Bi∗)}i∈Γ be a family of (L,M)-dffb’s on X satisfying the following condition:
(C) For any finite subsetK of Γ, if ⨀i∈Kλi=0_, then ⨀i∈K~Bi(λi)=0M and ⨁i∈K~Bi∗(λi)=1M.
We define the maps⨆i∈ΓBi,⨅i∈ΓBi∗:LX→M as(55)
⨆
i
∈
Γ
B
i
λ
=
⋁
⨀
i
∈
K
~
B
i
λ
i
:
λ
=
⨀
i
∈
K
λ
i
,
⨅
i
∈
Γ
B
i
∗
λ
=
⋀
⨁
i
∈
K
~
B
i
∗
λ
i
:
λ
=
⨀
i
∈
K
λ
i
,where ⋁ and ⋀ are taken for every finite index subset K of Γ such that λ=⨀i∈Kλi. Then, (⨆i∈ΓBi,⨅i∈ΓBi∗) is an (L,M)-dffb on X and (〈⨆i∈ΓBi〉,〈⨅i∈ΓBi∗〉) is the coarsest (L,M)-dff which is finer than (〈Bi〉,〈Bi∗〉) for each i∈Γ.Example 28.
LetX={x,y} be a set and L=M=[0,1] be an stsc-quantale with ⊙ (Lukasiewicz t-norm). We define maps Bi,Bi∗:LX→M as follows (i=1,2):(56)B1λ=1,ifλ=1_;0.4,ifλ=0.6·1_;0.5,ifλ=0.1·1_;0,otherwise,B1∗λ=0,ifλ=1_;0.6,ifλ=0.6·1_;0.5,ifλ=0.1·1_;1,otherwise,B2λ=1,ifλ=1_;0.6,ifλ=0.7·1_;0.3,ifλ=0.3·1_;0,otherwise,B2∗λ=0,ifλ=1_;0.4,ifλ=0.7·1_;0.7,ifλ=0.3·1_;1,otherwise.Each(Bi,Bi∗) for i=1,2 is an (L,M)-double fuzzy filter base but (B1⊔B2,B1∗⊓B2∗) is not.Corollary 29.
Letπi:X→Xi be projection maps, for all i∈Γ, where X=Πi∈ΓXi is the product set. Let {(Bi,Bi∗)}i∈Γ be a family of (L,M)-dffb’s on Xi satisfying the following condition:
(C) For any finite subsetK of Γ, if ⨀i∈K(λi∘πi)=0_, then ⨀i∈K~Bi(λi)=0M and ⨁i∈K~Bi∗(λi)=1M.
We define the maps⨆i∈Γ(πi)1⇇(Bi),⨅i∈Γ(πi∗)1⇇(Bi∗):LX→M as (57)
⨆
i
∈
Γ
π
i
1
⇇
B
i
λ
=
⋁
⨀
i
∈
K
~
B
i
λ
i
:
λ
=
⨀
i
∈
K
λ
i
∘
π
i
,
⨅
i
∈
Γ
π
i
∗
1
⇇
B
i
∗
λ
=
⋀
⨁
i
∈
K
~
B
i
∗
λ
i
:
λ
=
⨀
i
∈
K
λ
i
∘
π
i
,where ⋁ and ⋀ are taken for every finite subset K of Γ such that λ=⨀i∈K(λi∘πi).
LetB=⨆i∈Γ(πi)1⇇(Bi) and B∗=⨅i∈Γ(πi∗)1⇇(Bi∗) be given. Then, the following properties are satisfied:
(i)(B,B∗) is an (L,M)-dffb on X and (〈B〉,〈B∗〉) is the coarsest (L,M)-dff on X for which πi:(X,〈B〉,〈B∗〉)→(Xi,〈Bi〉,〈Bi∗〉) is a filter map.
(ii) A mapφ:(Y,F,F∗)→(X,〈B〉,〈B∗〉) is a filter map if and only if, for each i∈Γ, πi∘φ:(Y,F,F∗)→(Xi,〈Bi〉,〈Bi∗〉) is a filter map.In Corollary29, the structure (〈⨆i∈Γ(πi)1⇇(Bi)〉,〈⨅i∈Γ(πi∗)1⇇(Bi∗)〉) is called a product of (L,M)-dffs’s on X.Theorem 30.
Letφ:X→Y be an injective function and (B,B∗) be an (L,M)-dffb on X. Then, the following properties are satisfied:
(i)(φ2⇉(B),φ2⇉(B∗)) is an (L,M)-dffb on Y, and (〈φ2⇉(B)〉,〈φ2⇉(B∗)〉) is the coarsest (L,M)-dff, for which the function φ:(X,〈B〉,〈B∗〉)→(Y,〈φ2⇉(B)〉,〈φ2⇉(B∗)〉) is a filter preserving map.
(ii)(φ1⇇(φ2⇉(B)),φ1∗⇇(φ2⇉(B∗))) is an (L,M)-dffb on X with φ1⇇(φ2⇉(B))=B and φ1∗⇇(φ2⇉(B∗))=B∗.Proof.
(i) (DFFB1) For eachν∈LY, we have (58)φ2⇉Bν=BφL←ν≤B∗φL←ν⋆=φ2⇉B∗ν⋆.
(DFFB2) It is straightforward from the definition.
(DFFB3) Suppose that there existν1,ν2∈LY such that(59)φ2⇉B∗ν1⊙ν2≰φ2⇉B∗ν1⊕~φ2⇉B∗ν2.By the definition of φ2⇉(B∗), we have(60)φ2⇉B∗ν1⊙ν2≰B∗ν1∘φ⊕~B∗ν2∘φ.
Since(B,B∗) is an (L,M)-dffb, the following is obtained:(61)B∗ν1⊙ν2∘φ=B∗ν1∘φ⊙ν2∘φ≤B∗ν1∘φ⊕~B∗ν2∘φ.
Thus,(62)φ2⇉B∗ν1⊙ν2≰B∗ν1⊙ν2∘φ.
By the definition of〈B∗〉, there exists λ∈LX with λ≤(ν1⊙ν2)∘φ such that (63)φ2⇉B∗ν1⊙ν2≰B∗λ.
SinceφL→(λ)≤φL→(φL←(ν1⊙ν2))≤ν1⊙ν2 and φ is injective,(64)φ2⇉B∗ν1⊙ν2≤φ2⇉B∗φL→λ=B∗φL←φL→λ=B∗λ.
This contradicts the assumption. Then,(65)φ2⇉B∗ν1⊙ν2≤φ2⇉B∗ν1⊕~φ2⇉B∗ν2,for  eachν1,ν2∈LY.Similarly, it can be verified that(66)φ2⇉Bν1⊙ν2≥φ2⇉Bν1⊙~φ2⇉Bν2,for  eachν1,ν2∈LY.Hence, (φ2⇉(B),φ2⇉(B∗)) is an (L,M)-dffb on Y.
For eachλ∈LX, we have (67)φ2⇉BφL→λ≥φ2⇉BφL→λ=BφL←φL→λ=Bλ,φ2⇉B∗φL→λ≤φ2⇉B∗φL→λ=B∗φL←φL→λ=B∗λ.
Hence, by Theorem23(ii), φ:(X,〈B〉,〈B∗〉)→(Y,〈φ2⇉(B)〉,〈φ2⇉(B∗)〉) is a filter preserving map.
Let(F,F∗) be another (L,M)-dff such that φ:(X,〈B〉,〈B∗〉)→(Y,F,F∗) is a filter preserving map. So, for each μ∈LY, the following is valid:(68)φ2⇉B∗μ=⋀φ2⇉B∗ν:ν≤μ=⋀B∗φL←ν:ν≤μ≥⋀F∗φL→φL←ν:ν≤μ≥⋀F∗ν:ν≤μ≥F∗μ.
Hence,〈φ2⇉(B∗)〉≥F∗. Similarly, it can be proved that 〈φ2⇉(B)〉≤F.
(ii) IfφL←(μ)=0_, then φ2⇉(B)(μ)=B(φL←(μ))=B(0_)=0M, and φ2⇉(B∗)(μ)=B∗(φL←(μ))=B∗(0_)=1M. By Theorem 25(i), (φ1⇇(φ2⇉(B)),φ1∗⇇(φ2⇉(B∗))) is an (L,M)-dffb on X. For each ν∈LX, following equalities are obtained:(69)φ1⇇φ2⇉Bν=⋁φ2⇉Bμ:ν=φL←μ=⋁BφL←μ:ν=φL←μ=Bν,φ1∗⇇φ2⇉B∗ν=⋀φ2⇉B∗μ:ν=φL←μ=⋀B∗φL←μ:ν=φL←μ=B∗ν.Example 31.
LetX={a,b},Y={x,y} be sets and L=M=[0,1] be the stsc-quantale with Lukasiewicz t-norm ⊙. Let φ:X→Y be a function defined by φ(a)=φ(b)=x and μ1,μ2∈[0,1]X be defined by μ1(a)=μ1(b)=0.6,μ2(a)=0.1,μ2(b)=0. We define maps B,B∗:[0,1]X→[0,1] as follows:(70)Bλ=1,ifλ=1_;0.6,ifλ=μ1;0.3,ifλ=μ2;0,otherwise,B∗λ=0,ifλ=1_;0.4,ifλ=μ1;0.7,ifλ=μ2;1,otherwise.
Then,(B,B∗) is an (L,M)-double fuzzy filter base but (φ2⇉(B),φ2⇉(B∗)) is not an (L,M)-double fuzzy filter base.Theorem 32.
Let{φi:Xi→X}i∈Γ be a family of injective functions and {(Bi,Bi∗)}i∈Γ be a family of (L,M)-dffb’s on Xi satisfying the following condition:
(C) For any finite subsetK of Γ, if ⨀i∈Kλi=0_, then ⨀~i∈K(φi)2⇉(Bi)(λi)=0M, and ⨁~i∈K(φi)2⇉(Bi∗)(λi)=1M.
We define the mapsB,B∗:LX→M as (71)
B
λ
=
⋁
⨀
i
∈
K
~
φ
i
2
⇉
B
i
λ
i
:
λ
=
⨀
i
∈
K
λ
i
,
B
∗
λ
=
⋀
⨁
i
∈
K
~
φ
i
2
⇉
B
i
∗
λ
i
:
λ
=
⨀
i
∈
K
λ
i
,where ⋁ and ⋀ are taken for every finite index subset K of Γ. Then, the following properties are satisfied:
(i)(B,B∗) is an (L,M)-dffb on X and (〈B〉,〈B∗〉) is the coarsest (L,M)-dff for which φi:(Xi,〈Bi〉,〈Bi∗〉)→(X,〈B〉,〈B∗〉) is a filter preserving map.
(ii) A mapφ:(X,〈B〉,〈B∗〉)→(Y,F,F∗) is a filter preserving map if and only if, for each i∈Γ, φ∘φi:(Xi,〈Bi〉,〈Bi∗〉)→(Y,F,F∗) is a filter preserving map.Proof.
(i) By Corollary27 and Theorem 30, (B,B∗) is an (L,M)-dffb on X.
Sinceφi is injective, for each i∈Γ, (72)BφiL→λi≥φi2⇉BiφiL→λi≥BiφiL←φiL→λi=Biλi,B∗φiL→λi≤φi2⇉Bi∗φiL→λi≤Bi∗φiL←φiL→λi=Bi∗λi.
Hence,φi is a filter preserving map, for each i∈Γ.
According to Theorem26(i), other cases are similarly proved.
(ii) It is proved in the same way as Theorem26(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) LetA and B be categories. A functor G:A→B is called topological provided that every G-structured source (fi:B→GAi)i∈Γ has a unique G-initial lift (fi¯:A→Ai)i∈Γ.Proposition 34 (see [43]).
IfG:A→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 functorV:(L,M)-DFIL→SET defined by V(X,F,F∗)=X and V(φ)=φ is topological.Proof.
The proof follows from Definition33, Proposition 34, and Theorem 26.


## 5. The Types(φ1⇉,φ1∗⇉), (φ2⇇,φ2⇇) of Images and Preimages of (L,M)-Double Fuzzy Filter Bases

Theorem 36.
Letφ:X→Y be a surjective function and (B,B∗) be an (L,M)-dffb on X. Then, the following properties are satisfied:
(i)(φ1⇉(B),φ1∗⇉(B∗)) is an (L,M)-dffb on Y.
(ii)(〈φ1⇉(B)〉,〈φ1∗⇉(B∗)〉) is the coarsest (L,M)-dff on Y for which φ:(X,〈B〉,〈B∗〉)→(Y,〈φ1⇉(B)〉,〈φ1∗⇉(B∗)〉) is a filter preserving map.
(iii) If(B,B∗) is an (L,M)-dff, then 〈φ1⇉(B)〉=φ2⇉(B) and 〈φ1∗⇉(B∗))〉=φ2⇉(B∗).Proof.
(i) and (ii) are proved in the same manner as Theorem25(i).
(iii) Let(B,B∗) be an (L,M)-dff. Since φ is surjective, φL→(λ)≤ν is equivalent to λ≤ν∘φ. Then, for each ν∈LY, it is clear that(73)φ1∗⇉B∗ν=⋀φ1∗⇉B∗μ:μ≤ν=⋀B∗λ:φL→λ=μ≤ν=⋀B∗λ:λ≤ν∘φ=B∗ν∘ϕ=φ2⇉B∗ν.
Hence,〈φ1∗⇉(B∗)〉=φ2⇉(B∗). Similarly, 〈φ1⇉(B)〉=φ2⇉(B) is obtained.Remark 37.
Letφ:X→Y be a bijective function, (K,K∗) be an (L,M)-dffb on X, and (B,B∗) be an (L,M)-dffb on Y. Then, the following equalities are clear.(i)
φ1⇉(K)=φ2⇉(K) and φ1∗⇉(K∗)=φ2⇉(K∗).(ii)
φ1⇇(B)=φ2⇇(B) and φ1∗⇇(B∗)=φ2⇇(B∗).Remark 38.
Letφ:X→Y be a bijective function and (B,B∗) be an (L,M)-dffb on Y. Then, it follows from Remark 37(ii) and Theorem 25 that (φ2⇇(B),φ2⇇(B∗)) is an (L,M)-dffb on X and (〈φ2⇇(B)〉,〈φ2⇇(B∗)〉) is the coarsest (L,M)-dff on X for which φ:(X,〈φ2⇇(B)〉,〈φ2⇇(B∗)〉)→(Y,B,B∗) is a filter map.Theorem 39.
Letφ:X→Y be a function and {(Bi,Bi∗)}i∈Γ be a family of (L,M)-dffb’s on X satisfying the following condition:
(C) For every finite subsetK of Γ, if ⨀i∈Kλi=0_, then ⨀~i∈KBi(λi)=0M and ⨁~i∈KBi∗(λi)=1M. Then, the following properties are satisfied:
(i) Ifφ:X→Y is bijective, then φ1⇉(⨆i∈ΓBi)=⨆i∈Γφ1⇉(Bi) and φ1∗⇉(⨅i∈ΓBi∗)=⨅i∈Γφ1∗⇉(Bi∗).
(ii) Ifφ:X→Y is injective, then 〈φ1⇉(⨆i∈ΓBi)〉=⨆i∈Γ〈φ1⇉(Bi)〉 and 〈φ1∗⇉(⨅i∈ΓBi∗)〉=⨅i∈Γ〈φ1∗⇉(Bi∗)〉.Proof.
(i) Let us consider the following condition:
( C 1 ) For every finite subset K of Γ, if ⨀i∈Kνi=0_, then ⨀~i∈Kφ1⇉(Bi)(νi)=0M and ⨁~i∈Kφ1∗⇉(Bi∗)(νi)=1M.
For the proof, it is enough to show that(C1)⇔(C).
( C 1 ) ⇒ ( C ): For any finite subset K of Γ with ⨀i∈Kλi=0_, since φ is injective, by Lemma 6(!),(74)φL→⨀i∈Kλi=⨀i∈KφL→λi=0_.By (C1), we have (75)
0
M
=
⨀
i
∈
K
~
φ
1
⇉
B
i
φ
L
→
λ
i
≥
⨀
i
∈
K
~
B
i
λ
i
,
1
M
=
⨁
i
∈
K
~
φ
1
∗
⇉
B
i
∗
φ
L
→
λ
i
≤
⨁
i
∈
K
~
B
i
∗
λ
i
,
and thus⨀~i∈KBi(λi)=0M and ⨁~i∈KBi∗(λi)=1M is satisfied.
( C ) ⇒ ( C 1 ): Suppose that, for every finite subset K of Γ with ⨀i∈Kνi=0_,  ⨁~i∈Kφ1∗⇉(Bi∗)(νi)≠1M. Then, for each i∈K, there exists λi∈LX with νi=φL→(λi) such that (76)
⨁
i
∈
K
~
φ
1
∗
⇉
B
i
∗
ν
i
≤
⨁
i
∈
K
~
B
i
∗
λ
i
≠
1
M
.By (C), ⨀i∈Kλi≠0_. By Lemma 6(!),(77)φL→⨀i∈Kλi=⨀i∈KφL→λi=⨀i∈Kνi≠0_.
This contradicts the assumption. Thus,⨁~i∈Kφ1∗⇉(Bi∗)(νi)=1M. Similarly, for every finite subset K of Γ, if ⨀i∈Kνi=0_, then ⨀~i∈Kφ1⇉(Bi)(νi)=0M.
Sinceφ is surjective, by Theorem 36, (φ1⇉(Bi),φ1∗⇉(Bi∗)) exists for each i∈Γ. By Corollary 27 and (C1), (⨆i∈Γφ1⇉(Bi),⨅i∈Γφ1∗⇉(Bi∗)) exists.
For each finite subsetK of Γ such that λ=⨀k∈Kλk with φL→(λ)=ν, the following inequalities are satisfied:(78)
⨆
i
∈
Γ
φ
1
⇉
B
i
ν
≥
⨀
k
∈
K
~
φ
1
⇉
B
k
φ
L
→
λ
k
≥
⨀
k
∈
K
~
B
k
λ
k
,
⨅
i
∈
Γ
φ
1
∗
⇉
B
i
∗
ν
≤
⨁
k
∈
K
~
φ
1
∗
⇉
B
k
∗
φ
L
→
λ
k
≤
⨁
k
∈
K
~
B
k
∗
λ
k
.
This implies that(79)⨆i∈ΓBiλ≤⨆i∈Γφ1⇉Biν,⨅i∈ΓBi∗λ≥⨅i∈Γφ1∗⇉Bi∗ν.
So, the following are clear:(80)φ1⇉⨆i∈ΓBi≤⨆i∈Γφ1⇉Bi,φ1∗⇉⨅i∈ΓBi∗≥⨅i∈Γφ1∗⇉Bi∗.
For any finite subsetJ of Γ with μ=⨀j∈Jμj, there exist ηj∈LX with φL→(ηj)=μj. Thus,(81)
φ
1
⇉
⨆
i
∈
Γ
B
i
μ
≥
⨆
i
∈
Γ
B
i
⨀
j
∈
J
η
j
≥
⨀
j
∈
J
~
B
j
η
j
,
φ
1
∗
⇉
⨅
i
∈
Γ
B
i
∗
μ
≤
⨅
i
∈
Γ
B
i
∗
⨀
j
∈
J
η
j
≤
⨁
j
∈
J
~
B
j
∗
η
j
.
This implies that(82)φ1⇉⨆i∈ΓBi≥⨆i∈Γφ1⇉Bi,φ1∗⇉⨅i∈ΓBi∗≤⨅i∈Γφ1∗⇉Bi∗.From the above inequalities, we have (83)φ1⇉⨆i∈ΓBi=⨆i∈Γφ1⇉Bi,φ1∗⇉⨅i∈ΓBi∗=⨅i∈Γφ1∗⇉Bi∗.
(ii) It is proved by the same method as in (i) and Theorem36(ii).Theorem 40.
Let{φi:Xi→X:i∈Γ} be a family of functions and {(Bi,Bi∗)}i∈Γ be a family of (L,M)-dffbs’ on Xi satisfying the following condition:
(C) For any finite subsetK of Γ, if ⨀i∈K(φi)L→(λi)=0_, then ⨀i∈K~Bi(λi)=0M and ⨁i∈K~Bi∗(λi)=1M.
We define the maps⋃i∈Γ(φi)1⇉(Bi),⋂i∈Γ(φi)1∗⇉(Bi∗):LX→M as(84)
⋃
i
∈
Γ
φ
i
1
⇉
B
i
ν
=
⋁
⨀
i
∈
K
~
B
i
λ
i
:
ν
=
⨀
i
∈
K
φ
i
L
→
λ
i
,
⋂
i
∈
Γ
φ
i
1
∗
⇉
B
i
∗
ν
=
⋀
⨁
i
∈
K
~
B
i
∗
λ
i
:
ν
=
⨀
i
∈
K
φ
i
L
→
λ
i
,where ⋁ and ⋀ are taken for every finite subset K of Γ. Let B=⋃i∈Γ(φi)1⇉(Bi) and B∗=⋂i∈Γ(φi)1∗⇉(Bi∗). Then, the following properties are satisfied:
(i) Ifφj is surjective for some j∈Γ, then (B,B∗) is an (L,M)-dffb on X and (〈B〉,〈B∗〉) is the coarsest (L,M)-dff for which the map φi:(Xi,〈Bi〉,〈Bi∗〉)→(X,〈B〉,〈B∗〉) is a filter preserving map.
(ii) A functionφ:(X,〈B〉,〈B∗〉)→(Y,F,F∗) is a filter preserving map if and only if, for each i∈Γ, φ∘φi:(Xi,〈Bi〉,〈Bi∗〉)→(Y,F,F∗) is a filter preserving map.
(iii) Ifφi are surjective for all i∈Γ, then(85)⋃i∈Γφi1⇉Bi=⨆i∈Γφi1⇉Bi,⋂i∈Γφi1∗⇉Bi∗=⨅i∈Γφi1∗⇉Bi∗.Proof.
(i) (DFFB2) Sinceφj is surjective for some j∈Γ and (C), B(1_)=1M,  B∗(1_)=0M and B(0_)=0M, B∗(0_)=1M.
According to Theorems26(i) and 32(i), other cases are similarly proved.
(ii) The proof is similar to Theorem26(ii).
(iii) Let us consider the following condition:
(C1) For any finite subsetK of Γ, if ⨀i∈Kνi=0_, then ⨀i∈K~(φi)1⇉(Bi)(νi)=0M and ⨁i∈K~(φi)1∗⇉(Bi∗)(νi)=1M.
For the proof, it is enough to show that(C1)⇔(C).
( C 1 ) ⇒ ( C ): For any finite subset K of Γ, if ⨀i∈K(φi)L→(λi)=0_, by (C1), we have (86)
0
M
=
⨀
i
∈
K
~
φ
i
1
⇉
B
i
φ
i
L
→
λ
i
≥
⨀
i
∈
K
~
B
i
λ
i
,
1
M
=
⨁
i
∈
K
~
φ
i
1
∗
⇉
B
i
∗
φ
i
L
→
λ
i
≤
⨁
i
∈
K
~
B
i
∗
λ
i
.
( C ) ⇒ ( C 1 ): Suppose that, for any finite subset K of Γ with ⨀i∈Kνi=0_, we have ⨁i∈K~(φi)1∗⇉(Bi∗)(νi)≠1M. Then, for each i∈K, there exists λi∈LXi with νi=(φi)L→(λi) such that(87)
⨁
i
∈
K
~
φ
i
1
∗
⇉
B
i
∗
ν
i
≤
⨁
i
∈
K
~
B
i
∗
λ
i
≠
1
M
.By (C), (88)⨀i∈KφiL→λi=⨀i∈Kνi≠0_.This is a contradiction. Thus, ⨁i∈K~(φi)1∗⇉(Bi∗)(νi)=1M. Similarly, ⨀i∈K~(ϕi)1⇉(Bi)(νi)=0M.
For any finite index setK with {λi:⨀i∈K(φi)L→(λi)≤ν}, by the definition of
〈 ⨆ i ∈ Γ ( φ i ) 1 ⇉ ( B i ) 〉 and 〈⨅i∈Γ(φi)1∗⇉(Bi∗)〉, the following inequalities are obtained:(89)
⨆
i
∈
Γ
φ
i
1
⇉
B
i
ν
≥
⋁
i
∈
Γ
φ
i
1
⇉
B
i
⨀
i
∈
K
φ
i
L
→
λ
i
≥
⨀
i
∈
K
~
φ
i
1
⇉
B
i
φ
i
L
→
λ
i
by  Corollary
26
≥
⨀
i
∈
K
~
B
i
λ
i
,
⨅
i
∈
Γ
φ
i
1
∗
⇉
B
i
∗
ν
≤
⋀
i
∈
Γ
φ
i
1
∗
⇉
B
i
∗
⨀
i
∈
K
φ
i
L
→
λ
i
≤
⨁
i
∈
K
~
φ
i
1
∗
⇉
B
i
∗
φ
i
L
→
λ
i
by  Corollary
26
≤
⨁
i
∈
K
~
B
i
∗
λ
i
.
Hence,(90)⋃i∈Γφi1⇉Bi≤⨆i∈Γφi1⇉Bi,⋂i∈Γφi1∗⇉Bi∗≥⨅i∈Γφi1∗⇉Bi∗.
For any finite index setJ with {νi:⨀i∈Jνi≤μ}, since φi is surjective, for each i∈J, there exists λi∈LXi with (φi)L→(λi)=νi such that μ≥⨀i∈Jνi=⨀i∈J(φi)L→(λi).
Thus,(91)
⋃
i
∈
Γ
φ
i
1
⇉
B
i
μ
≥
⨀
i
∈
J
~
φ
i
1
⇉
B
i
ν
i
≥
⨀
i
∈
J
~
B
i
λ
i
,
⋂
i
∈
Γ
φ
i
1
∗
⇉
B
i
∗
μ
≤
⨁
i
∈
J
~
φ
i
1
∗
⇉
B
i
∗
ν
i
≤
⨁
i
∈
J
~
B
i
∗
λ
i
.
Hence,(92)⋃i∈Γφi1⇉Bi≥⨆i∈Γφi1⇉Bi,⋂i∈Γφi1∗⇉Bi∗≤⨅i∈Γφi1∗⇉Bi∗.


## 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 overSET. By giving illustrative examples, we considered two types of second-order Zadeh image and preimage operators of (L,M)-double fuzzy filter.



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*Source: 3871282-2018-05-15.xml*