# Convergence Classes ofL-Filters in L,M-Fuzzy Topological Spaces

Authors: Ting Yang; Sheng-Gang Li; William Zhu; Xiao-Fei Yang; Ahmed Mostafa Khalil
Journal: Journal of Mathematics
 (2021)

---

## Abstract

AnL,M-fuzzy topological convergence structure on a set X is a mapping which defines a degree in M for any L-filter (of crisp degree) on X to be convergent to a molecule in LX. By means of L,M-fuzzy topological neighborhood operators, we show that the category of L,M-fuzzy topological convergence spaces is isomorphic to the category of L,M-fuzzy topological spaces. Moreover, two characterizations of L-topological spaces are presented and the relationship with other convergence spaces is concretely constructed.


---

## Body

## 1. Introduction

A convergence space of filters (as a generalization of a topological space based on the concept of convergence of filters as fundamental) is a pairX,Conv, where X is a set, Conv is a subset of FX×X, and FX is the set of filters on X. The pair ℱ,x∈Conv means that ℱ is said to converge to x∀ℱ∈FX. The convergence theory of filters is an important part in topology. It was proved that the category Conv of convergence spaces is a quasitopos which may be thought of as a nice category of spaces that includes Top (the category topological spaces) as a full subcategory (see [1–3] for details).The corresponding convergence theory of filters in the fuzzy setting is also studied by many authors. E. Lowen and R. Lowen [4] gave a one-to-one correspondence between the set of limit functions on X and the set of stratified I-topologies on X based on I-filters (filters in the lattice 0,1X [5]). Jäger [6] introduced another fuzzy convergence structure (called stratified L-generalized convergence structure for L, a frame which was extended to the case of complete residual lattices by Yao [7] later on). Here, the degree of convergence of a stratified L-fuzzy filter (called also stratified L-filter in [8]) is from L but it converges to a crisp point.Jäger also gave some kinds of characterizations of stratifiedL-topological spaces (see [6, 9–11]). With the help of Jäger’s work, Li [12] proved that there is a one-to-one correspondence between the set of all convergence functions of specific L-fuzzy filters on X and the set of all specific L-topologies on X for L, a complete residual lattice. Following papers [13, 14], Güloǧlu and Çoker [15] proved that there exists a one-to-one correspondence between the set of I-fuzzy topological convergence structures on X and the set of I-fuzzy topologies on X. As a generalization, Pang and Fang [16] proved that there is a one-to-one correspondence between the set of topological L-fuzzy Q-convergence structures on X and the set of L-fuzzy topologies on X when L is a completely distributive complete lattice with an order-reversing involution ′. Here, the L-fuzzy filter converges to a fuzzy point but the degree of convergence is from 0,1. Furthermore, Pang [17, 18] discussed categorical properties in some fuzzy convergence spaces, such as L-fuzzifying convergence spaces and stratified L-convergence tower spaces.In [19], the convergence theory of molecular nets in L,M-fuzzy topological spaces was discussed. As a generalization, Yao [20] presented a definition of L,M-fuzzy nets and established a Moore–Smith convergence in L,M-fuzzy topology. It is well known that every molecular net can induce an L-filter (a filter in lattice LX) and an L-filter can induce a molecular net (see [21]). In addition, the degree of fuzzy topology, openness degree, and quotient degree [22] are interesting. Taking these in mind, we will use L-filters different from that of [6, 8, 16] to give some axioms of filter-theoretical convergence classes in L,M-fuzzy topological spaces. Such a convergence class on a set X will be called an L,M-fuzzy topological convergence structure, where the L-filter converges to a molecular in LX and the degree of convergence is an element in M. This kind of convergence class is proved to have nice properties.The present paper is arranged as follows. The rest of this section contains some basic definitions and notions which will be used in this paper. In Section2, we prove that there exists a one-to-one correspondence between L,M-fuzzy topological convergence structures on a set X and L,M-fuzzy topologies on X. In Section 3, we give two approaches to L,2-fuzzy topological convergence structures. In Section 4, we discuss the relation between L,2-fuzzy topological convergence structures and Li’s structures in [12].


## 2. Preliminaries

Now, we review some basic notions and results which will be used in this paper. Unless other explanations are given,L always stands for a complete lattice with a smallest element 0 and a largest element 1 in this paper. Obviously, for every set X, LX (the set of all L-subsets of X) is also a complete lattice with the pointwise order. We will denote the L-set taking constant value a on X by aX, and we write ↑μ=ν∈LX|μ≤ν for each μ∈LX (↓μ, ↑a, and ↓a can be defined analogously for a∈L). For the sake of convenience, we denote the orders of L and LX and their restrictions by the same symbol ≤. a∈L−0 is called a join irreducible element if, for any finite subset J⊆L satisfying a=∨J (the supremum of J), there exists a j∈J such that a=j. a∈L−0 is called a coprime element if, for any finite subset J⊆L satisfying a≤∨J, there exists a j∈J such that a≤j. p∈L−1 is called a prime element of L iff it is a coprime element of Lop, where Lop denotes the opposite lattice of L. The set of all join irreducible elements (resp., all coprime elements and all prime elements) of L will be denoted by JL (resp., Copr (L) and Pr (L)). Clearly, CoprLX=xa|x∈X,a∈CoprL, where xa∈LX is the L-subset taking value a at x and 0 elsewhere, and we call the members of PtLX=xa|x∈X,a∈L−0L-points. It is well known that JL=CoprL if L is a distributive lattice. L is called a completely distributive complete lattice if it satisfies the following completely distributive laws:(i)
(CD1)∨i∈I∧j∈Jiai,j=∧f∈∏i∈IJi∨i∈Iai,fi,∀ai,j|i∈I,j∈Ji∈2L(ii)
(CD2)∧i∈I∨j∈Jiai,j=∨f∈∏i∈IJi∧i∈Iai,fi∀ai,j|i∈I,j∈Ji∈2LIt is also well known thata=∨⇓a∩CoprL∀a∈L when L is a completely distributive complete lattice, where ⇓a=b∈L|b⊲a (⇑a, ⇓μ, and ⇑μ can be defined analogously for μ∈LX), and ⊲⊆L×L is called the wedge-below relation on L which is defined as follows: a⊲b iff for any S⊆L satisfying b≤∨S, there exists an s∈S such that a≤s (⊳ can be defined analogously).An order-reversing involution ′ onL is a self-map on L such that, for any a,b∈L, the following hold: (1) a≤b implies b′≤a′; (2) a″=a. The following hold for any subset ai|i∈I⊆L:(1)
∨i∈Iai′=∧i∈Iai′(2)
∧i∈Iai′=∨i∈Iai′For each mappingf:X⟶Y, we have a mapping (called L-forward powerset operator) fL⟶:LX⟶LY which is defined by fL⟶μy=∨μx|fx=y∀μ∈LX,∀y∈Y. The right adjoint to fL⟶ (called L-backward powerset operator) is denoted by fL← (therefore, the pair fL⟶,fL← is a Galois connection on LX,≤ and LY,≤). It can be easily verified that fL⟶ preserves arbitrary supremal, fL← preserves arbitrary supremal and arbitrary infimal (and also complements if L has an order-reversing involution ′), and fL←ν=∨μ|fL⟶μ≤ν=ν∘f∀ν∈LY.For other undefined notions, refer to [8, 23].


## 3. Relationship betweenL,M-Fuzzy Topological Spaces and L,M-Fuzzy Topological Convergence Spaces

In this section, we assume thatL (resp., M) is a completely distributive complete lattice with an order-reversing involution ′ (resp., ∗); in this case, JL=CoprL, JM=CoprM, and JLX=CoprLX. Our main task is to give a one-to-one correspondence between L,M-fuzzy topological convergence structures (resp., L,M-fuzzy principle convergence structures) on a set X and L,M-fuzzy topologies (resp., L,M-fuzzy preinterior operators) on X, which gives rise to an isomorphism between the category of L,M-fuzzy topological convergence spaces (resp., the category of L,M-fuzzy principle convergence spaces) and the category of L,M-fuzzy topological spaces (resp., the category of L,M-fuzzy preinterior spaces).Definition 1 (See [13, 14]).
(1)
(See [13, 14]). An L-fuzzy filter on X is a mapping F:LX⟶L, which satisfies the following conditions:(i)
(FF1)F1X=1, F0X=0(ii)
(FF2) IfA,B∈LX and A≤B, then FA≤FB(iii)
(FF3)FA∧FB≤FA∧B∀A,B∈LX(iv)
AnL-fuzzy filter is said to be stratified [8] if it satisfies the following condition: (Fs) a≤FaX∀a∈L.The set of allL-fuzzy filters (resp., stratified L-fuzzy filters) on X is denoted as FX,L,L (resp., FSX,L,L). Apparently, the mapping x:LX⟶L, defined by xA=Ax, is a stratified L-fuzzy filter (∀x∈X).(2)
(See [5]). An L-filter on X is a family F⊆LX which satisfies the following conditions:(i)
(F1)1X∈F, 0X∉F(ii)
(F2) Ifμ∈F and μ≤ν, then ν∈F(iii)
(F3) Ifμ,ν∈F, then μ∧ν∈FThe set of allL-filters on X is denoted as FX,L,2 (where 2=0,1). Obviously, every L-filter may be looked upon as an L-fuzzy filter.Remark 1.
Apparently,↑xa∈FX,L,2,∀x∈X,∀a∈L. Moreover, for every mapping f:X⟶Y and every L-filter F on X, define(1)fL⟹F=∩G∈FX,L,2|fL⟶F⊆G,and then fL⟹F∈FX,L,2 and fL⟹F=μ∈LY|fL←μ∈F=μ∈LY|μ≥fL⟶νfor someν∈F, where fL⟶F=fL⟶ν|ν∈F.Definition 2 (see [24]).
(1)
AnL,M-fuzzy topology on X is a mapping T:LX⟶M, which satisfies the following conditions:(i)
(LMFT1)T0X=T1X=1(ii)
(LMFT2)Tμ∧Tν≤Tμ∧ν∀μ,ν∈LX(iii)
(LMFT3)∧j∈JTμj≤T∨j∈Jμj,∀μjj∈J⊆LXIn this case,Tμ can be interpreted as the degree for μ to be an open set ∀μ∈LX, and the pair X,T is called an L,M-fuzzy topological space. An L,2-fuzzy topology is also called L-topology.(2)
A continuous mapping from oneL,M-fuzzy topological space X,TX to another L,M-fuzzy topological space Y,TY is a mapping f:X⟶Y, which satisfies TYμ≤TXfL←μ∀μ∈LY. The category of L,M-fuzzy topological spaces and continuous mappings between them is denoted by L,M-FTop.Definition 3 (see [25]).
(1)
AnL,M-fuzzy neighborhood operator on a set X is a mapping N:JLX⟶MLX, which satisfies the following conditions:(i)
(LMFN1)Nxa1X=1, Nxa0X=0,∀xa∈JLX(ii)
(LMFN2)Nxaμ=0xa≤μ(iii)
(LMFN3)Nxaμ∧ν=Nxaμ∧Nxaν,∀μ,ν∈LX,∀xa∈JLX(iv)
(LMFN4)Nxaμ=∨ν∈↓μ∩↑xa∧yb∈JLX∩⇓νNybν∀μ∈LX,∀xa∈JLXIn this case,X,N is called an L,M-fuzzy neighborhood space.(2)
A mappingf:X,NX⟶Y,NY (where X,NX and Y,NY are both L,M-fuzzy neighborhood spaces) is said to be continuous if NYfxaμ≤NXxafL←μ holds for any μ∈LY and any xa∈JLX . The category of L,M-fuzzy neighborhood spaces and continuous mappings between them will be denoted by L,M-FNS. In [25], Shi proved that L,L-FNS is isomorphic to L,L-FTop. Furthermore, it is easily proved that L,M-FNS is isomorphic to L,M-FTop.Definition 4.
(1)
AnL,M-fuzzy preinterior operator on X is a mapping I:LX⟶MJLX, which satisfies the following conditions:(i)
(LMPI1)I1Xxa=1,∀xa∈JLX(ii)
(LMPI2)Iμxa=0,xa≤μ(iii)
(LMPI3)Iμ∧Iν=Iμ∧ν,∀μ,ν∈LXIn this case,X,I is called an L,M-fuzzy preinterior space.(2)
A mappingf:X,IX⟶Y,IY (where X,IX and Y,IY are both L,M-fuzzy preinterior spaces) is said to be continuous if IYμfxa≤IXfL←μxa holds for each μ∈LY and each xa∈JLX. The category of L,M-fuzzy preinterior spaces and continuous mappings between them will be denoted by L,M-FPIS.Definition 5.
(1)
AnL,M-fuzzy convergence structure on X is a mapping Conv:FX,L,2⟶MJLX, which satisfies the following conditions:(i)
(LM1)Conv↑xaxa=1∀xa∈JLX(ii)
(LM2) IfF⊆G, then ConvF≤ConvGIn this case,X,Conv is called an L,M-fuzzy convergence space.(2)
A mappingf:X,ConvX⟶Y,ConvY (where X,ConvX and Y,ConvY are both L,M-fuzzy convergence spaces) is said to be continuous if ConvXFxa≤ConvYfL⟹Ffxa for any F∈FX,L,2 and any xa∈JLX. The category of L,M-fuzzy convergence spaces and continuous mappings between them will be denoted by L,M-FCS.(3)
For anL,M-fuzzy convergence space X,Conv, we define a mapping NConv:JLX⟶MLX as follows:(2)NConvxaμ=∧μ∉FConvFxa∗,∀xa∈JLX,∀μ∈LX.Definition 6.
(1)
AnL,M-fuzzy convergence structure Conv on X is said to be an L,M-fuzzy principle convergence structure if it satisfies the following condition:(LM3)ConvFxa=∧μ∉FNConvxaμ∗,∀xa∈JLXIn this case,X,Conv is called an L,M-fuzzy pretopological convergence space.(2)
The category ofL,M-fuzzy pretopological convergence spaces and continuous mappings (see Definition 5) between them will be denoted by L,M-FPCS.Definition 7.
(1)
AnL,M-fuzzy topological convergence structure on X is an L,M-fuzzy principle convergence structure Conv, which satisfies the following condition:(LM4)NConvxaμ=∨ν∈↓μ∩↑xa∧yb∈⇓ν∩JLXNConvybν,∀μ∈LXIn this case,X,Conv is called an L,M-fuzzy topological convergence space.(2)
The category ofL,M-fuzzy topological convergence spaces and continuous mappings (see Definition 5) between them will be denoted by L,M-FTCS.
In the rest of this section, we will show thatL,M-FNS is isomorphic to L,M-FTCS (thus, L,M-FTop is isomorphic to L,M-FTCS by [25]) and that L,M-FPIS is isomorphic to L,M-FPCS.Proposition 1.
For anL,M-fuzzy topological convergence structure Conv, the mapping NConv:JLX⟶MLX is an L,M-fuzzy neighborhood operator on X.Proof.
SinceF∈FX,L,2|1X∉F=∅, we have NConvxa1X=∧∅=1. ↑xa is an L-filter on X and 0X∉↑xa, NConvxa0X≤Conv↑xaxa∗=0 by (LM1). Thus, (LMFN1) is true. Again for each μ∈LX and each xa∈JLX−Jμ, we have NConvxaμ≤Conv↑xaxa∗=0 since μ∉↑xa. (LMFN2) is also true.
Letμ,ν∈LX and xa∈JLX. Then, F∈FX,L,2|μ∧ν∉F=F∈FX,L,2|μ∉F∪F∈FX,L,2|ν∉F holds. Thus, by definition of NConvxa,(3)NConvxaμ∧ν=∧ConvFxa∗|F∈FX,L,2,μ∧ν∉F=∧ConvFxa∗|μ∉F∧∧ConvFxa∗|ν∉F=NConvxaμ∧NConvxaν,which means that (LMFN3) is true. (LMFN4) follows from (LM4).Proposition 2.
For eachL,M-fuzzy neighborhood operator N on X, the mapping ConvN:FX,L,2⟶MJLX, defined by(4)ConvNFxa=∧Nxaμ∗|μ∈LX,μ∉F,∀F∈FX,L,2,∀xa∈JLX,is an L,M-fuzzy topological convergence structure on X.Proof.
For anyμ∈LX and any xa∈JLX−Jμ, we have Nxaμ=0 by (LMFN2). Hence, ConvN↑xaxa=1, which means that (LM1) is true.
LetF,G∈FX,L,2 and F⊆G. It can be easily checked that μ∈LX|μ∉G⊆μ∈LX|μ∉F. Thus, ConvNFxa=∧Nxaμ∗|μ∈LX,μ∉F≤∧Nxaμ∗|μ∈LX,μ∉G=ConvNGxaμ∉G for any xa∈JLX, which means that (LM2) is true.
SinceConvN satisfies (LM1) and (LM2), the mapping NConvN:JLX⟶MLX is well-defined (see Definition 5). In order to prove (LM3), it suffices to prove the equality NConvNxaμ=Nxaμ,∀xa∈JLX,∀μ∈LX by definition of ConvN. By Definition 5 (3),(5)NConvNxaμ=∧μ∉FConvNFxa∗=∧μ∉F∨ν∉FNxaν.
For eachF∈FX,L,2 satisfying μ∉F, we have ∨Nxaν|ν∈LX,ν∉F≥Nxaν. As F is arbitrary, the inequality NConvNxaμ≥Nxaμ holds. To prove the other inequality NConvNxaμ≤Nxaμ, let I=F∈FX,L,2|μ∉F and JF=ν∈LX|ν∉F. As M is a completely distributive complete lattice,(6)NConvNxaμ=∨g∈∏F∈IJF∧F∈INxagF.
First, we show thatFb∈I, where Fb=γ∈LX|Nxaγ≰b∀b∈⇑Nxaμ∩PrM. As μ∉Fb is obvious, we only need to show that Fb is an L-filter.(i)
(F1) AsNxa1X=1≰b and Nxa0X=0≤b, 1X∈Fb, and 0X∉Fb.(ii)
(F2) Assume thatγ∈Fb and γ≤ν; then Nxaγ≤Nxaν. Further, Nxaγ≰b, so we have Nxaν≰b, and thus, ν∈Fb.(iii)
(F3) Letγ,ν∈Fb. Then we declare that Nxaγ∧ν=Nxaγ∧Nxaν≰b (i.e., γ∧ν∈Fb). Otherwise, Nxaγ≤b or Nxaν≤b since b∈PrM, which is a contradiction.
Next, it follows fromFb∈I that gFb∉Fb∀b∈⇑Nxaμ∩PrM,∀g∈∏F∈IJF, and thus, NxagFb≤b (particularly, ∧NxagF|F∈I≤b). As b is arbitrary and M is a completely distributive complete lattice, ∧NxagF|F∈I≤Nxaμ. As g is arbitrary, NConvNxaμ≤Nxaμ.
SinceNConvNxaμ=Nxaμ and N is an L,M-fuzzy neighborhood operator, (LM4) holds.Proposition 3.
Iff:X,NX⟶Y,NY is an L,M-FNS morphism, then f:X,ConvNX⟶Y,ConvNY is an L,M-FTCS morphism.Proof.
For anyF∈FX,L,2 and any xa∈JLX, as f is an L,M-FNS morphism, NYfxaν≤NXxafL←ν. Further, ∗ is an order-reversing mapping; NXxafL←ν∗≤NYfxaν∗∀ν∈LY. It follows that(7)ConvNXFxa=∧μ∉FNXxaμ∗≤∧ν∉fL⟹FNXxafL←ν∗≤∧ν∉fL⟹FNYfxaν∗=ConvNYfL⟹Ffxa.
Therefore,f:X,ConvNX⟶Y,ConvNY is an L,M-FTCS morphism.Proposition 4.
Iff:X,ConvX⟶Y,ConvY is an L,M-FTCS morphism, then f:X,NConvX⟶Y,NConvY is an L,M-FNS morphism.Proof.
For eachG∈FX,L,2 and xa∈JLX, as f is an L,M-FTCS morphism, ConvXGxa≤ConvYfL⟹Gfxa. Further, ∗ is an order-reversing mapping; ConvYfL⟹Gfxa∗≤ConvXGxa∗. For all μ∈LY, it follows that(8)NConvYfxaμ=∧μ∉FConvYFfxa∗≤∧fL←μ∉GConvYfL⟹Gfxa∗≤∧fL←μ∉GConvXGxa∗=NConvXxafL←μ.
The first inequality holds sinceF∈FY,L,2|μ∉F⊇fL⟹G|G∈FX,L,2,fL←μ∉G. Therefore, f:X,NConvX⟶Y,NConvY is an L,M-FNS morphism.Proposition 5.
For anyL,M-fuzzy topological convergence structure Conv on X, one has ConvNConv=Conv.Proof.
LetF∈FX,L,2 and xa∈JLX. We need to prove ConvNConvFxa=ConvFxa. By Proposition 2 and Definition 5,(9)ConvNConvFxa=∧μ∉F∨μ∉GConvGxa.
For eachμ∈LX with μ∉F, we have(10)∨μ∉GConvGxa≥ConvFxa,and then the inequality ConvNConvFxa≥ConvFxa holds since μ is arbitrary. To show the other inequality ConvNConvFxa≤ConvFxa, put I=μ∈LX|μ∉F and Jμ=G∈FX,L,2|μ∉G. As M is a completely distributive complete lattice,(11)ConvNConvFxa=∨g∈∏μ∈IJμ∧μ∈IConvgμxa.
It suffices to prove(12)∧μ∈IConvgμxa≤ConvFxa,∀g∈∏μ∈IJμ.
For eachg∈∏μ∈IJμ, by (LM3), we have ∧μ∈IConvgμxa=∧μ∈I∧ν∉gμNConvxaν∗=∧ν∉∩μ∈IgμNConvxaν∗=Conv∩μ∈Igμxa. As ∩μ∈Igμ⊆ℱ, from (LM2), we have Conv∩μ∈Igμxa≤ConvFxa, which means(13)∧μ∈IConvgμxa≤ConvFxa,∀g∈∏μ∈IJμ.
By Propositions1–5, we have the following.Theorem 1.
(1)
L,M-FNS is isomorphic to L,M-FTCS.Similar to [25], it is easy to check that L,M-FNS is isomorphic to L,M-FTop(2)
ThusL,M-FTop is isomorphic to L,M-FTCS.Remark 2.
(1)
IfConv is an L,M-fuzzy topological convergence structure on X, then it satisfies the following: if xa≤xb, then ConvFxa≤ConvFxb (∀xa,xb∈JLX,F∈FX,L,2).(2)
From Proposition5, the following is true due to (LM3):(14)∧i∈IConvFixa=Conv∩i∈IFixa.(3)
Now we turn our attention to the Moore–Smith convergence theory; for basic notions, refer to [19, 21, 23].
For a molecule netS=Sm|m∈D (Sm∈JLX, and D is a directed set), we define an L-filter FS associated with the net S as follows: FS=μ|Sis eventually inμ. Conversely, for an L-filter F, define a molecule net SF=Sm|m∈D, where D=xa,A|xa∈JLX,A∈F,xa≤A is a directed set, on which the order is equipped with the relation ≤: xa,A≤yb,B if and only if A≥B, and the mapping is Sxa,A=xa.
Now define convergence ofS as(15)ConvSxa=∧Nxaμ∗|Sis not eventually inμ.
In anL,M-fuzzy topological space X,T, the following hold. The proof is simple and is missing:(i)
(1)ConvSxa=ConvFSxa for each molecule S and xa∈JLX.(ii)
(2)ConvFxa=ConvSFxa for each F∈FX,L,2 and xa∈JLX.(iii)
(4) In [16], Pang and Fang proposed the concept of topological L-fuzzy Q-convergence spaces (the corresponding category is denoted by L-QFTCS). In these convergence spaces, the value of an L-fuzzy filter converging to a fuzzy point is from 0,1. Thus, Pang and Fang’s spaces are different from ours. But L-QFTCS is isomorphic to L,L-FTCS when L is a completely distributive lattice with an order-reversing involution.(iv)
(5) For a given setX, let TopConvX,L,M be a set of all L,M-fuzzy topological convergence structures on X, and the relation ≤ on TopConvX,L,M is defined by Conv1≤Conv2 if and only if Conv1Fxa≤Conv2Fxa∀xa∈JLXandF∈FX,L,2.
The set of allL,M-fuzzy neighborhood operators on X is written as NOX,L,M, and the relation ≤ on NOX,L,M is defined by N1≤N2 if and only if N1xaμ≥N2xaμ∀xa∈JLXandμ∈LX.
From Propositions2 and 5, we have NConvN=N and for each N∈OPX,L,M, ConvNConv=Conv for each Conv∈TopConvX,L,M. This implies that there exists a bijection between TopConvX,L,M,≤ and OPX,L,M,≤. Similar to paper [26], it is easily checked that TopConvX,L,M,≤ and OPX,L,M,≤ are complete lattices and they are isomorphic.
By Theorem1 (1), there exists an isomorphic functor between L,M-FNS and L,M-FTCS. Furthermore, the restriction of this functor to OPX,L,M,≤ is an order isomorphism.
Now we prove thatL,M-FPIS is isomorphic to L,M-FPCS. Since its proof is similar to Theorem 1, we only give some propositions as follows.Proposition 6.
For eachL,M-fuzzy preinterior operator I on X, the mapping ConvI:FX,L,2⟶MJLX, defined by(16)ConvIFxa=∧μ∉FIμxa∗,∀F∈FX,L,2,xa∈JLX,is an L,M-fuzzy principal convergence structure on X.Proposition 7.
For eachL,M-fuzzy principal convergence structure Conv on X, the mapping IConv:LX⟶MJLX, defined by(17)IConvμxa=∧μ∉FConvFxa∗,∀μ∈LX,xa∈JLX,is an L,M-fuzzy preinterior operator on X.Proposition 8.
For anyL,M-fuzzy principal convergence structure Conv on X, one has ConvIConv=Conv.Proposition 9.
For anyL,M-fuzzy preinterior operator I on X, one has IConvI=I.Proposition 10.
Iff:X,IX⟶Y,IY is an L,M-FPIS morphism, then f:X,ConvIX⟶Y,ConvIY is an L,M-FPCS morphism.Proposition 11.
Iff:X,ConvX⟶Y,ConvY is an L,M-FPCS morphism, then f:X,IConvX⟶Y,IConvY is an L,M-FPIS morphism.
By Propositions6–11, we have the following.Theorem 2.
L,M-FPIS is isomorphic to L,M-FPCS.
Based on the above facts, we give the following example.Example 1.
LetX=x be a single set, L=1,0.7,0.5,0.3,0 and M=1,0.5,0, where unary operation ∗ on M is defined as 1∗=0, 0.5∗=0.5, and 0∗=1. Define T(0X) = T(1X) = 1; otherwise, TA=0.5, A∈LX. Then(1)
X,T is an L,M-fuzzy topological space.(2)
DefineNxaA=∨TB|xa≤B≤A. It follows from Theorem 3.2 in [25] that X,N is an L,M-fuzzy neighborhood space. In this way, Nx0.5, which is defined by Nx0.5x1=1, Nx0.5x0.7=Nx0.5x0.5=0.5, and Nx0.5x0.3=Nx0.5x0=0, is the L,M-fuzzy neighborhood of x0.5.(3)
LetF be an L-filter, ConvFxa=∧NxaA∗|A∉F. In this way, supposing F=x1,x0.7, we can see that ConvFx0.5=Nx0.5x0.5∗∧Nx0.5x0.3∗∧Nx0.5x0∗=0.5, which means that the degree of F convergence to x0.5 is 0.5.


## 4. Two Approaches toL-Topological Convergence Structures

In this section, we restrict our attention to the caseM=2, and thus, we identify a set Y with its characteristic function χY. We write F⟶xa whenever ConvFxa=1 and replace Conv by ⟶ in order to emphasize the convergence. In this manner, the notion of an L,2-fuzzy topological convergence space and the notion of an L,2-fuzzy neighborhood space can be easily given as follows.Definition 8.
(1)
A pairX,⟶ is an L-convergence space (⟶ is called an L-convergence structure on X) if it satisfies the following conditions:(i)
(L1)↑xa⟶xa∀xa∈JLX(ii)
(L2) IfF⊆G and F⟶xa, then G⟶xa∀xa∈JLX,∀F,G∈FX,L,2(Order) IfF⟶xa and xa≤xb, then F⟶xb∀xa,xb∈JLX,∀F∈FX,L,2.(2)
A pairX,⟶ is an L-topological convergence space(⟶ is called an L-topological convergence structure on X) if it satisfies the following conditions:(i)
(L1)↑xa⟶xa∀xa∈JLX(ii)
(L2) IfF⊆G and F⟶xa, then G⟶xa∀xa∈JLX,∀F,G∈FX,L,2(iii)
(L3)Nxa⟶xa, where Nxa=∩F⟶xaF∀xa∈JLX(iv)
(L4) Ifμ∈Nxa, then there exists ν∈LX such that ν∈↑xa∩↓μ and ν∈Nyb for each yb∈JLX∩⇓ν∀μ∈LX,∀xa∈JLXObviously, anL-topological convergence space is an L-convergence space.(3)
(See [25]) A pair X,N is an L-neighborhood space (N is called an L-neighborhood system on X) if it satisfies the following conditions:(i)
(LN0)1X∈Nxa and 0X∉Nxa (∀xa∈JLX)(ii)
(LN1) Ifxa≰μ, then μ∉Nxa∀xa∈JLX,∀μ∈LX(iii)
(LN2) Ifμ≤ν and μ∈Nxa, then ν∈Nxa∀xa∈JLX,∀μ,ν∈LX(iv)
(LN3) Ifμ,ν∈Nxa, then μ∧ν∈Nxa∀xa∈JLX,∀μ,ν∈LX(v)
(LN4) Ifμ∈Nxa, then there exists ν∈LX such thatν∈↑xa∩↓μ and ν∈Nyb for each yb∈JLX∩⇓ν∀xa∈JLX,∀μ∈LXRemark 3.
(see [25]). For a given set X, let TX,L be the set of all L-topologies on X and ℕX,L the set of all L-topological neighborhood systems on X. Then, the mapping φ12:TX,L⟶ℕX,L defined by NTxa=ν∈LX|∃μ∈T,μ∈↑xa∩↓ν∀T∈TX,L,∀xa∈JLX is a one-to-one correspondence, whose inverse φ21:ℕX,L⟶TX,L is given by TN=ν∈LX|ν∈Nyb∀yb∈JLX∩⇓ν,∀N∈ℕX,L. Thus, by Theorem 1, there exists a one-to-one correspondence between the set of all L-topological convergence structures on X and the set of L-topologies on X.
In [9], Jäger gave a generalization of Kowalsky’s diagonal condition. Based on different tools, we give another form of Kowalsky’s diagonal condition [2] in fuzzy setting. In the presence of this form, we give a characterization of an L-topological convergence structure.Theorem 3.
X,⟶ is an L-topological convergence space if and only if it satisfies (L1)–(L3), (order) in Definition 8, and the following:(i)
LK4 For any G∈FX,L,2 and any subfamily Fybyb∈JLX⊆FX,L,2 which satisfies(18)LFDFya⊇ℱyb,wheneverya≤yb,if G⟶xa and Fyb⟶yb, then GF−⟶xa (xa,yb∈JLX), where the mapping F−:LX⟶LX and the subfamily GF−⊆LX are defined as follows:(19)F−μ=∨xa|μ∈Fxa,∀μ∈LXGF−=μ∈LX|F−μ∈G.
To prove Theorem3, we need several lemmas.Lemma 1.
LetG∈FX,L,2 and Gxaxa∈JLX⊆FX,L,2 satisfy (LFD). Then GF−=μ∈LX|F−μ∈G is an L-filter on X.Proof.
As1X∈Fxa∀xa∈JLX, F−1X=∨JLX=1X∈G, which means 1X∈GF−. Since xa∈JLX|0X∈Fxa=∅, we have F−0X=0X∉G; that is, 0X∈GF−. Thus, GF− satisfies (F1). Obviously, GF− satisfies (F2).
For anyμ,ν∈GF−, put Jμ=xa∈JLX|μ∈Fxa and Jν=xa∈JLX|ν∈Fxa. Then F−μ=∨Jμ, F−ν=∨Jν and F−μ∧F−ν∈G by definition of GF−. We first show the inequality F−μ∧F−ν≤∨Jμ∩Jν. Suppose xa∈JLX and xa⊲F−μ∧F−ν, and then xa⊲F−μ=∨Jμ. There exists an xb∈Jμ (which implies μ∈Fxb) such that xa≤xb, and thus, Fxb⊆Fxa since Fxaxa∈JLX satisfies (LFD). Therefore, μ∈Fxa (i.e., xa∈Jμ). Similarly, xa∈Jν. Consequently xa≤∨Jμ∩Jν, which implies F−μ∧F−ν≤∨Jμ∩Jν since LX is a completely distributive lattice. Notice that Jμ∩Jν⊆xa∈JLX|μ∧ν∈Fxa, we have F−μ∧F−ν≤F−μ∧ν, and thus, F−μ∧ν∈G, which means μ∧ν∈GF−. Hence, GF− also satisfies (F3).Lemma 2.
For a pairX,N satisfying conditions (LN0)-(LN3) in Definition 8 (3), the following are equivalent:(1)
(LN4)(2)
Nxaxa∈JLX satisfies (LFD) and Nxa⊆NxaN−∀xa∈JLXProof.
1⟹2. Since X,N satisfies conditions (LN0), (LN2), and (LN3), Nxaxa∈JLX⊆FX,L,2. For any xa,xb∈JLX satisfying xa≤xb and any μ∈Nxb=NTNxb, there exists a ν∈TN such that xb≤ν≤μ since X,N satisfies conditions (LN0)–(LN4) (and thus, N=NTN; see [25] or Remark 3). It follows from xa≤xb that μ∈NTNxa=Nxa. Therefore, Nxaxa∈JLX satisfies (LFD), and thus, NxaN− is well-defined from Lemma 1∀xa∈JLX.Suppose thatμ∈Nxa, then xa≤ν≤μ and ν∈Nyc∀yc∈JLX∩⇓ν hold for some ν∈LX by (LN4). It follows that μ∈Nyc∀yc∈JLX∩⇓ν and ν≤N−μ. As ν∈TN (see Remark 3), we have ν∈NTNxa=Nxa, and thus, N−μ∈Nxa (i.e., μ∈NxaN−). Therefore, Nxa⊆NxaN−∀xa∈JLX.2⟹1. Assume μ∈Nxa; then μ∈NxaN− by (2), and then N−μ∈Nxa by definition of NxaN−. We will show that ν (put ν=N−μ) is the required one in (LN4). Firstly, for all yb∈JLX satisfying μ∈Nyb, we have yb≤μ by (LN1) and thus, N−μ≤μ by definition of N−μ. Further, it follows from N−μ∈Nxa that xa≤N−μ. Therefore, ν∈↑xa∩↓μ. Secondly, we may show N−ν=ν. On the one hand, as ν≤μ, N−ν≤N−μ=ν. On the other hand, for each yb∈JLX satisfying μ∈Nyb, we have μ∈NybN− by (2), and thus, N−μ∈Nyb by definition of NybN−, which implies N−μ≤N−N−μ. Therefore, ν≤N−ν. Finally, for each zc⊲ν=N−ν, there exists a zb∈JLX such that zc≤zb and ν∈Nzb. ν∈Nzc since Nxaxa∈JLX satisfies (LFD). That is, for each zc⊲ν, ν∈Nzc holds.Lemma 3.
For a pairX,⟶ satisfying conditions (L1)-(L3) and (order) in Definition 8, the following are equivalent:(1)
(L4)(2)
(LK4)Proof.
(i)
1⟹2. It suffices to prove Nxa⊆GF− by (L3). First, we show GN−⊆GF−. Suppose that μ∈GN− (i.e., N−μ∈G). For each yb∈JLX satisfying μ∈Nyb, we have Nyb⊆Fyb (and thus, μ∈Fyb) for Fyb⟶yb and (L3), which implies N−μ≤F−μ, and thus, F−μ∈G; that is μ∈GF−. Next, we show NxaN−⊆GN−. Since G⟶xa, we obtain Nxa⊆G. Suppose μ∈NxaN−, then N−μ∈Nxa, and thus, N−μ∈G; that is, μ∈GN−. Finally, as Nxa⊆NxaN− (see Lemma 2), we have Nxa⊆GF− by the preceding two conclusions.(ii)
2⟹1. Take G=Nxa and Fyb=Nxb∀yb∈JLX. Since (order) is satisfied, Fybyb∈JLX is a subfamily of FX,L,2 satisfying (LFD). G⟶xa, and Fyb⟶yb (∀yb∈JLX). Thus, GF−⟶xa by (2); that is, Nxa⊆GF−=NxaN−. It follows from Lemma 2 that (L4) holds.Proof of Theorem3.
It follows from Lemma3.
In paper [11], based on stratified L-fuzzy filter, Gähler’s neighborhood condition is studied in stratified L-convergence space. Based on L-filter, Gähler’s neighborhood condition is also studied in L-convergence space.Theorem 4.
X,⟶ is an L-topological convergence space if and only if it satisfies (L1)–(L3), (order) in Definition 8, and the following:(i)
LG4 If F⟶xa, then FN−⟶xa (∀xa∈JLX,F∈FX,L,2)Proof.
SupposeX,⟶ is an L-topological convergence space; we show that X,⟶ satisfies (LG4). If F⟶xa, then, by (L3), Nxa⊆F. It can be easily checked that NxaN−⊆FN−. It follows from Lemma 2 that Nxa⊆FN−; that is, FN−⟶xa. Conversely, it follows from (L3) that Nxa⟶xa, and thus, NxaN−⟶xa by (LG4). This implies Nxa⊆NxaN−. Following Lemma 2, (L4) holds.


## 5. The Relation betweenX,⟶ and Li’s X,lim

AnL-topological stratified L-fuzzy convergence space [6] (where L is a complete Heyting algebra) is a pair X,lim, where X is a set, and lim:FSX,L,L⟶LX (called an L-topological stratified L-fuzzy convergence structure on X) is a mapping which satisfies the following conditions:(i)
L1∗limxx=1∀x∈X.(ii)
L2∗F≤G implies limF≤limG∀F,G∈FSX,L,L.(iii)
(Lp)limFx=∧μ∈LXNxμ⟶Nμ,(iv)
where(20)Nxμ=∧G∈SFX,L,LlimGx⟶Gμ,∀x∈X.(v)
(Lt)Nxμ≤∨Nxν|νy≤Nyμ∀y∈X,∀x∈X,∀μ∈LX.From Proposition 6.4 in [6], there exists a one-to-one correspondence between the set of all L-topological stratified L-fuzzy convergence structures on X and the set of stratified L-topologies on X. Following Jäger’s work, Li [12] proved that there exists a one-to-one correspondence between the set of all L-topological L-fuzzy convergence structures on X and the set of L-topologies on X, where an L-topological L-fuzzy convergence structure is a mapping lim:FX,L,L⟶LX which satisfies L1∗, L2∗, (Lp), and (Lt).The spacesX,⟶ and X,lim are absolutely different in their forms. For the former, a filter is a family of subsets of LX; the point is fuzzy but the convergence is crisp. However, for the latter, a filter is a mapping from LX to L; the point is crisp but the convergence is fuzzy. From Remark 3, there exists a one-to-one correspondence between the set of all L-topological convergence structures on X and the set of L-topologies on X. Thus, these two notions (L-topological L-fuzzy convergence structures [12] and L-topological convergence structures) can be determined reciprocally for L being a completely distributive lattice. To say it more precisely, there exists a one-to-one correspondence φ13 from AX,L (the set of all L-topological convergence structures on X) to BX,L (the set of all L-topological L-fuzzy convergence structures on X), which is defined by(21)φ13⟶Fx=∧μ∈LXNxμ⟶Fμ,∀F∈FX,L,L,∀x∈X,for each ⟶ in AX,L, where Nxμ=∨a∈JL|μ∈Nxa,∀μ∈LX; the inverse mapping φ31 of φ13 is given by φ31lim=⟶lim for each lim ∈BX,L, which satisfies(22)F⟶limxa⟺Nxa⊆F,where Nxa=μ∈LX|Nxμ≥a∀xa∈JLX,F∈FX,L,2.


## 6. Conclusions

In the present paper, we propose the notion ofL,M-fuzzy topological convergence structure (given by L-filters) and show that such a structure can be used to characterize an L,M-fuzzy topology. With convergence theory of molecular nets [19], we give a Moore–Smith convergence theory in L,M-fuzzy topological space. For further issues, we are devoted to L-topological convergence structure and give two conditions, Kowalsky’s diagonal condition and Gähler’s neighborhood condition, in fuzzing setting. From these results, we consider that L-filter is an important and appropriate tool to L,M-fuzzy topological space and also to L-topological space.It is well known thatL,M-fuzzy topological space has bifuzziness: fuzziness of open sets and fuzziness of openness. So we are sure that there exist other kinds of convergence structures such that all these structures can be categorically isomorphic to L,M-fuzzy topologies. For example, with the help of L-filter and idea of Q-neighborhood operator (or L-fuzzy filter and idea of neighborhood operator), we could consider the corresponding convergence structures. This will be our future work. After these efforts, maybe, due to one kind of these convergence structures, we can study other properties of L,M-fuzzy topological spaces and L,M-fuzzy topological group [27] for more convenience.



---

*Source: 8246173-2021-10-01.xml*