(L,⊙)-Fuzzy (K,E)-Soft Filter

Abstract

In the present study, we preface the notion of the $(L,\odot )$-fuzzy $(K,E)$-soft filter and probe many of its characteristics. Using certain operations, we induce $(L,\odot )$-fuzzy $(K,E)$-soft topologies from $(L,\odot )$-fuzzy $(K,E)$-soft filters. Finally, we introduce the products of the $(L,\odot )$-fuzzy $(K,E)$-soft filters and $(L,\odot )$-fuzzy $(K,E)$-soft topologies on strictly two-sided commutative quantale lattices $(L,\odot )$ and $(L,\ast )$.

1. Introduction

The theory of soft sets was presented by Molodtsov [1]. This theory was is considered a novel mathematical tool to simplify dealing with uncertainty in modeling problems in many scientific fields, such as computer science, physics, medical sciences and engineering. Molodtsov devoted his efforts to developing his theory and used it in many fields [1,2,3]. Molodtsov et al. [4] were able to introduce some applications of their new theory.

Maji et al. [5,6] defined the notion of a fuzzy soft set and studied several basics of its characteristics. Maji et al. [7] and Roy and Maji [8] used soft sets and fuzzy soft sets to study some problems in decision making. Many authors have developed the theory of fuzzy soft sets [9,10,11,12].

Shabir and Naz [13] introduced the concept of soft topological spaces and studied their numerous properties. Aygünoǧlu et al. [14] presented the notion of $(L,\odot )$-fuzzy $(K,E)$-soft topology in the sense of Šostak [15]. Many concepts in fuzzy topological spaces in the sense of Chang [16] and in the sense of Šostak [15] were extended to fuzzy soft theory (see [17,18,19,20,21,22,23,24]).

The concept of L-filters was presented and investigated by Höhle and Šostak [25]. L-filters were used to introduce many types of convergence spaces ([26,27,28,29]). The L-filter is a significant tool for investigating the properties of L-fuzzy topological spaces ([30,31]). L-filter has been developed in different directions ([32,33,34,35]).

The intent of our study was to extend the notion of the L-filter into the theory of fuzzy soft sets and study their characteristics. This motivated us to use it for studying more properties of $(L,\odot )$-fuzzy $(K,E)$-soft topology. The structure of the whole paper is as follows: In Section 2, we recall some essential concepts and properties which we need for our study. In Section 3, we preface the concept of a $(L,\odot )$-fuzzy $(K,E)$-soft filter and study its various properties. In Section 4, by using certain operations, we induce $(L,\odot )$-fuzzy $(K,E)$-soft topologies from $(L,\odot )$-fuzzy $(K,E)$-soft filters. In Section 5, we present the products of the $(L,\odot )$-fuzzy $(K,E)$-soft filters and $(L,\odot )$-fuzzy $(K,E)$-soft topologies and study some of their properties.

2. Preliminaries

Let $L=(L,\le ,\vee ,\wedge ,0_L,1_L)$ be a completely distributive lattice where $0_L$ is the least element and $1_L$ is the greatest element. In this manuscript, X is denoted as an initial universe; and E and K are the sets of all parameters for X. An L-fuzzy set on X is a mapping $\lambda :X\to L$ [36]. $L^X$ is the set of all L-fuzzy sets on X.

Definition 1 

([37,38]). A tripartite $(L,\le ,\odot )$ is said to be a strictly two-sided commutative quantale (for short, stsc-quantale) if it holds the following conditions:

(L1) 

$L=(L,\le ,0_L,1_L)$ is a complete lattice;

(L2) 

$(L,\odot )$ is a commutative semigroup;

(L3) 

$u=u\odot 1_L$, ∀$u\in L$;

(L4) 

$({\bigvee }_{i\in \Gamma }{u}_i)\odot v={\bigvee }_{i\in \mathrm{\Gamma }}(u_i\odot v).$

Definition 2 

([25]). If $(L,\ast )$ and $(L,\odot )$ are two stsc-quantales, the operation ⊙ dominates * if it holds:

$$(u_1\ast v_1)\odot (u_2\ast v_2)\ge (u_1\odot u_2)\ast (v_1\odot v_2).$$

Lemma 1 

([39]). Let $(L,\odot )$ and $(L,\ast )$ be stsc-quantales which induced two implications $u\to v=\bigvee \{w:u\odot w\le v\}$ and $u\Rightarrow v=\bigvee \{w:u\ast w\le v\}$, respectively. Let ⊙ dominates *. For each $u,v,w,,z,u_i,v_i\in L$, we have:

(i) 

If $v\le w$, then $u\odot v\le u\odot w$ and $u\ast v\le u\ast w$.

(ii) 

$u\odot v\le w$ iff $u\le v\to w$. Moreover, $u\ast v\le w$ iff $u\le v\Rightarrow w$.

(iii) 

If $v\le w$, then $u\to v\le u\to w$ and $w\to u\le v\to u$ for $\to \in \{\to ,\Rightarrow \}$.

(iv) 

$u\ast v\le u\odot v$, $u\to v\le u\Rightarrow v$ and $u\ast (v\odot w)\le (u\ast v)\odot w$.

(v) 

$(u\Rightarrow v)\odot (w\Rightarrow z)\le (u\odot w)\Rightarrow (v\odot z)$.

(vi) 

$(v\Rightarrow w)\le (u\odot v)\Rightarrow (u\odot w)$.

(vii) 

$(v\to w)\le (u\Rightarrow v)\to (u\Rightarrow w)$ and $(v\Rightarrow u)\le (u\to w)\to (v\Rightarrow w)$.

(viii) 

$u_i\to v_i\le ({\bigwedge }_{i\in \Gamma }{u}_i)\to ({\bigwedge }_{i\in \Gamma }{v}_i)$.

(ix) 

$u_i\to v_i\le ({\bigvee }_{i\in \Gamma }{u}_i)\to ({\bigvee }_{i\in \Gamma }{v}_i)$.

(x) 

$(w\Rightarrow u)\ast (v\to z)\le (u\to v)\to (w\Rightarrow z)$.

(xi) 

$u\le v$ iff $u\to v=1_L$.

Definition 3 

([14]). h is said to be an L-fuzzy soft sets on X, where $h:E\to L^X$ is a mapping from a parameter set E into $L^X$, i.e., $h_e:=h(e)\in L^X$, $\forall e\in E$. ${(L^X)}^E$ denotes the collection of L-fuzzy soft sets on X. Let $h_1,h_2\in {(L^X)}^E$. Then:

(i) 

$h_1$ is a subset of $h_2$, and write $h_1⊑h_2$ if ${(h_1)}_e\le {(h_2)}_e$, $\forall e\in E$. $h_1=h_2$ iff $h_1⊑h_2$ and $h_2⊑h_1$.

(ii) 

The intersection of $h_1$ and $h_2$ is an L-fuzzy soft set $h=h_1\sqcap h_2$, where $h_e={(h_1)}_e\wedge {(h_2)}_e$, $\forall e\in E$.

(iii) 

The union of $h_1$ and $h_2$ is an L-fuzzy soft set $h=h_1\bigsqcup h_2$, where $h_e={(h_1)}_e\vee {(h_2)}_e$, $\forall e\in E$.

(iv) 

An L-fuzzy soft set $h=h_1\odot h_2$ is defined as $h_e={(h_1)}_e\odot {(h_2)}_e$, $\forall e\in E$.

(v) 

$h^{\ast }$ is the complement of h and is defined by $h^{\ast }$, where $h^{\ast }:E\to L^X$ is a mapping obtained by $h_e^{\ast }={(h_e)}^{\ast }$, $\forall e\in E$.

(vi) 

$0_X$ is a null L-fuzzy soft set, if ${(0_X)}_e(x)=0_L$, $\forall e\in E,x\in X$.

(vii) 

$1_X$ is absolute L-fuzzy soft set, if ${(1_X)}_e(x)=1_L$, $\forall e\in E,x\in X$.

Definition 4 

([14]). Let $E_1$ and $E_2$ be two parameters sets for the crisp sets X and Y, respectively; and let $\varphi :X\to Y$ and $\psi :E_1\to E_2$ be two mappings. Then ${\varphi }_{\psi }:{(L^X)}^{E_1}\to {(L^Y)}^{E_2}$ is called a fuzzy soft mapping.

(i) 

For $f\in {(L^X)}^{E_1}$, ${\varphi }_{\psi }{(f)}_{e_2}(y)={\bigvee }_{x\in {\psi }^{-1}(\left\{y\right\})}({\bigvee }_{e_1\in {\psi }^{-1}(\left\{e_2\right\})}{f}_{e_1}(x))$, $\forall k\in K,\forall y\in Y$,

(ii) 

For $g\in {(L^Y)}^{E_2}$, ${\varphi }_{\psi }^{\leftarrow }{(g)}_e(x)=g_{\psi (e)}(\varphi (x)),\forall e\in E,\forall x\in X.$

(iii) 

${\varphi }_{\psi }:{(L^X)}^{E_1}\to {(L^Y)}^{E_2}$ is called one to one (resp. onto, bijective) if ϕ and ψ are both one to one (resp. onto, bijective).

Lemma 2 

([40]). Let ${\varphi }_{\psi }:{(L^X)}^{E_1}\to {(L^Y)}^{E_2}$. Then, for $f,f_i\in {(L^X)}^{E_1}$ and $g,g_i\in {(L^Y)}^{E_2}$, we have

(i) 

$g⊒{\varphi }_{\psi }({\varphi }_{\psi }^{\leftarrow }(g))$ and $g={\varphi }_{\psi }({\varphi }_{\psi }^{\leftarrow }(g))$ if ${\varphi }_{\psi }$ is onto.

(ii) 

$f⊑{\varphi }_{\psi }^{\leftarrow }({\varphi }_{\psi }(f))$ and $f={\varphi }_{\psi }^{\leftarrow }({\varphi }_{\psi }(f))$ if ${\varphi }_{\psi }$ is one to one.

(iii) 

If ${\varphi }_{\psi }$ is one to one,

$${\varphi }_{\psi }{(f)}_{e_2}(y)=\left\{\begin{array}{cc}{f}_{e_1}(x),\hfill & \mathrm{if}x\in {\varphi }^{-1}(y),e_1\in {\psi }^{-1}(e_2)\hfill \\ 0_L,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(iv) 

${\varphi }_{\psi }^{\leftarrow }(g^{\ast })={({\varphi }_{\psi }^{\leftarrow }(g))}^{\ast }$.

(v) 

${\varphi }_{\psi }^{\leftarrow }({\bigsqcup }_{i\in \Gamma }{g}_i)={\bigsqcup }_{i\in \Gamma }{\varphi }_{\psi }^{\leftarrow }(g_i)$.

(vi) 

${\varphi }_{\psi }^{\leftarrow }({\sqcap }_{i\in \Gamma }{g}_i)={\sqcap }_{i\in \Gamma }{\varphi }_{\psi }^{\leftarrow }(g_i)$.

(vii) 

${\varphi }_{\psi }({\bigsqcup }_{i\in \Gamma }{f}_i)={\bigsqcup }_{i\in \Gamma }{\varphi }_{\psi }(f_i)$.

(viii) 

${\varphi }_{\psi }({\sqcap }_{i\in \Gamma }{f}_i)⊑{\sqcap }_{i\in \Gamma }{\varphi }_{\psi }(f_i)$ with equality if ${\varphi }_{\psi }$ is one to one.

(ix) 

${\varphi }_{\psi }^{\leftarrow }(g_1\odot g_2)={\varphi }_{\psi }^{\leftarrow }(g_1)\odot {\varphi }_{\psi }^{\leftarrow }(g_2)$.

(x) 

${\varphi }_{\psi }(f_1\odot f_2)⊑{\varphi }_{\psi }(f_1)\odot {\varphi }_{\psi }(f_2)$ with equality if ${\varphi }_{\psi }$ is one to one.

Definition 5 

([14]). An $(L,\odot )$-fuzzy $(K,E)$-soft topology (briefly, $(L_{\odot },K_E)$-fs-topology) on a set X is a map $\mathcal{T}:K\to L^{{(L^X)}^E}$ (where ${\mathcal{T}}_k:=\mathcal{T}(k):{(L^X)}^E\to L$ is a map for each $k\in K$) that satisfies the following conditions for each $k\in K$:

(SO1) 

${\mathcal{T}}_k(0_X)={\mathcal{F}}_k(1_X)=1_L$,

(SO2) 

${\mathcal{T}}_k(f\odot g)\ge {\mathcal{T}}_k(f)\odot {\mathcal{T}}_k(g)$, for each $f,g\in {(L^X)}^E$,

(SO3) 

${\mathcal{T}}_k({⨆}_{i\in \Gamma }{f}_i)\ge {\bigwedge }_{i\in \Gamma }{\mathcal{T}}_k(f_i),\forall f_i\in {(L^X)}^E,i\in \Gamma$

An $(L_{\odot },K_E)$-fs-topology is enriched if

(SR) 

${\mathcal{T}}_k(\beta \odot g)\ge {\mathcal{T}}_k(g)$, $\forall g\in {(L^X)}^E$ and $\beta \in L$.

$(X,\mathcal{T})$ is said to be an $(L,\odot )$-fuzzy $(K,E)$-soft topological space (briefly, $(L_{\odot },K_E)$-fs-topological space). Let $(X,{\mathcal{T}}^1)$ be an $(L_{\odot },K_{1E_1})$-fs-topological space and $(Y,{\mathcal{T}}^2)$ be an $(L_{\odot },K_{2E_2})$-fs-topological space. If $\varphi :X\to Y$, $\psi :E_1\to E_2$ and $\eta :K_1\to K_2$ are mappings, then ${\varphi }_{\psi ,\eta }:(X,{\mathcal{T}}^1)\to (Y,{\mathcal{T}}^2)$ is L-fuzzy soft continuous if ${\mathcal{T}}_k^1({\varphi }_{\psi }^{\leftarrow }(g))\ge {\mathcal{T}}_{\eta (k)}^2(g)$, $\forall g\in {(L^Y)}^{E_2},k\in K_1$.

3. $(\mathit{L},\odot )$-Fuzzy $(\mathit{K},\mathit{E})$-Soft Filter

In this section, let L be a stsc-quantale and order-dense chain lattice.

Definition 6.

An $(L,\odot )$-fuzzy $(K,E)$-soft filter (briefly, $(L_{\odot },K_E)$-fs-filter) on a set X is a map $\mathcal{F}:K\to L^{{(L^X)}^E}$ (where ${\mathcal{F}}_k:=\mathcal{F}(k):{(L^X)}^E\to L$ is a map for each $k\in K$) satisfies the following conditions for each $k\in K$:

(SF1) 

${\mathcal{F}}_k(0_X)=0_L$ and ${\mathcal{F}}_k(1_X)=1_L$,

(SF2) 

${\mathcal{F}}_k(f\odot g)\ge {\mathcal{F}}_k(f)\odot {\mathcal{F}}_k(g)$, for each $f,g\in {(L^X)}^E$,

(SF3) 

If $f⊑g$, then ${\mathcal{F}}_k(f)\le {\mathcal{F}}_k(g)$.

An $(L_{\odot },K_E)$-fs-filter is stratified if,

(SS) 

${\mathcal{F}}_k(\beta \odot g)\ge \beta \odot {\mathcal{F}}_k(g)$, $\forall g\in {(L^X)}^E$ and $\beta \in L$.

$(X,\mathcal{F})$ is said to be an $(L,\odot )$-fuzzy $(K,E)$-soft filtered set (briefly, $(L_{\odot },K_E)$-fs-filtered set).

Let ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ be two $(L_{\odot },K_E)$-fs-filters on X. We indicate that ${\mathcal{F}}_1$ is finer than ${\mathcal{F}}_2$ (or ${\mathcal{F}}_2$ is coarser than ${\mathcal{F}}_1$), denoted by ${\mathcal{F}}_2⊑{\mathcal{F}}_1$ iff ${({\mathcal{F}}_2)}_k(f)\le {({\mathcal{F}}_1)}_k(f)$, for each $f\in {(L^X)}^E$, $k\in K$.

Definition 7.

Let $(X,\mathcal{F})$ be an $(L_{\odot },K_{1E_1})$-fs-filtered set and $(Y,\mathcal{G})$ be an $(L_{\odot },K_{2E_2})$-fs-filtered set. Let $\varphi :X\to Y$, $\psi :E_1\to E_2$ and $\eta :K_1\to K_2$ be mappings. Then the mapping ${\varphi }_{\psi ,\eta }:(X,\mathcal{F})\to (Y,\mathcal{G})$ is called L-fuzzy soft filter if ${\mathcal{F}}_k({\varphi }_{\psi }^{\leftarrow }(f))\ge {\mathcal{G}}_{\eta (k)}(f)$, for each $f\in {(L^Y)}^{E_2}$, $k\in K_1$.

Theorem 1.

If ${\left\{{\mathcal{F}}_i\right\}}_{i\in \Gamma }$ is a family of $(L_{\odot },K_E)$-fs-filtered sets in a fixed set X. Define the mapping $\mathcal{F}={\sqcap }_{i\in \Gamma }{\mathcal{F}}_i:K\to L^{{(L^X)}^E}$, where $\mathcal{F}:={\mathcal{F}}_k={\bigwedge }_{i\in \Gamma }{({\mathcal{F}}_i)}_k:{(L^X)}^E\to L$ defined by: ${\mathcal{F}}_k(f)={\bigwedge }_{i\in \Gamma }{({\mathcal{F}}_i)}_k(f)$, for each $f\in {(L^X)}^E$, $k\in K$. Then $\mathcal{F}$ is an $(L_{\odot },K_E)$-fs-filter on X.

Proof. 

(SF1) $\forall k\in K$, we get ${\mathcal{F}}_k(0_X)={\bigwedge }_{i\in \Gamma }{({\mathcal{F}}_i)}_k(0_X)={\bigwedge }_{i\in \Gamma }{0}_L=0_L$ and ${\mathcal{F}}_k(1_X)={\bigwedge }_{i\in \Gamma }{({\mathcal{F}}_i)}_k(1_X)={\bigwedge }_{i\in \Gamma }{1}_L=1_L$.

(SF2) For each $f,g\in {(L^X)}^E$, we have

$$\begin{array}{cc}{\mathcal{F}}_k(f)\odot {\mathcal{F}}_k(g)& =\underset{i\in \Gamma }{\bigwedge }{({\mathcal{F}}_i)}_k(f)\odot \underset{i\in \Gamma }{\bigwedge }{({\mathcal{F}}_i)}_k(g)\hfill \\ & \le \underset{i\in \Gamma }{\bigwedge }({({\mathcal{F}}_i)}_k(f)\odot {({\mathcal{F}}_i)}_k(g))\hfill \\ & \le \underset{i\in \Gamma }{\bigwedge }{({\mathcal{F}}_i)}_k(f\odot g)\hfill \\ & ={\mathcal{F}}_k(f\odot g).\hfill \end{array}$$

(SF3) If $f⊑g$, then we have ${({\mathcal{F}}_i)}_k(f)\le {({\mathcal{F}}_i)}_k(g)$, $\forall i\in \Gamma ,k\in K$. Therefore,

$${\mathcal{F}}_k(f)=\underset{i\in \Gamma }{\bigwedge }{({\mathcal{F}}_i)}_k(f)\le \underset{i\in \Gamma }{\bigwedge }{({\mathcal{F}}_i)}_k(g)={\mathcal{F}}_k(g).$$

Then $\mathcal{F}$ is an $(L_{\odot },K_E)$-fs-filter on X. □

Theorem 2.

Let $\varphi :X\to Y$, $\psi :E_1\to E_2$ and $\eta :K_1\to K_2$. Let $\mathcal{F}$ be an $(L_{\odot },K_{1E_1})$-fs-filter on X. Define a map ${\varphi }_{\psi ,\eta }(\mathcal{F}):K_2\to L^{{(L^Y)}^{E_2}}$ (where ${({\varphi }_{\psi ,\eta }(\mathcal{F}))}_k:={\mathcal{F}}_{{\eta }^{-1}(k)}({\varphi }_{\psi }^{\leftarrow })(g):{(L^Y)}^{E_2}\to L$ is a map for each $k\in K_2$) by ${({\varphi }_{\psi ,\eta }(\mathcal{F}))}_k(g)={\mathcal{F}}_{{\eta }^{-1}(k)}({({\varphi }_{\psi })}^{\leftarrow }(g))$. Then, ${\varphi }_{\psi ,\eta }(\mathcal{F})$ is an $(L_{\odot },K_{2E_2})$-fs-filter on Y.

Proof. 

(SF1) It is clear.

(SF2) For $k\in K_2$, $f,g\in {(L^Y)}^{E_2}$, we have

$$\begin{array}{cc}{({\varphi }_{\psi ,\eta }(\mathcal{F}))}_k(f\odot g)& ={\mathcal{F}}_{{\eta }^{-1}(k)}({\varphi }_{\psi }^{\leftarrow }(f\odot g))\hfill \\ & ={\mathcal{F}}_{{\eta }^{-1}(k)}({\varphi }_{\psi }^{\leftarrow }(f)\odot {\varphi }_{\psi }^{\leftarrow }(g))\hfill \\ & \ge {\mathcal{F}}_{{\eta }^{-1}(k)}({\varphi }_{\psi }^{\leftarrow }(f))\odot {\mathcal{F}}_{{\eta }^{-1}(k)}({\varphi }_{\psi }^{\leftarrow }(g))\hfill \\ & ={({\varphi }_{\psi ,\eta }(\mathcal{F}))}_k(f)\odot {({\varphi }_{\psi ,\eta }(\mathcal{F}))}_k(g).\hfill \end{array}$$

(SF3) If $f⊑g$, then

$${({\varphi }_{\psi ,\eta }(\mathcal{F}))}_k(f)={\mathcal{F}}_{{\eta }^{-1}(k)}({({\varphi }_{\psi })}^{\leftarrow }(f))\le {\mathcal{F}}_{{\eta }^{-1}(k)}({({\varphi }_{\psi })}^{\leftarrow }(g))={({\varphi }_{\psi ,\eta }(\mathcal{F}))}_k(g).$$

Thus, ${\varphi }_{\psi ,\eta }(\mathcal{F})$ is an $(L_{\odot },K_{2E_2})$-fs-filter on Y. □

We denote ${\mathcal{F}}_k^0=\{f\in {(L^X)}^E:{\mathcal{F}}_k(f)>0_L\}$.

Theorem 3.

Let ${\left\{{\mathcal{F}}_i\right\}}_{i\in \Gamma }$ be a collection of $(L_{\odot },K_E)$-fs-filters on X satisfying the next condition: (C) If $f_i\in {({\mathcal{F}}_i)}_k^0$ for each $i\in \Gamma$, $k\in K$ then we have ${\odot }_{i\in J}{f}_i\ne 0_X$ for each $J\subseteq \Gamma$, J is finite. Define a mapping ${⨆}_{i\in \Gamma }{\mathcal{F}}_i:K\to L^{{(L^X)}^E}$ (where ${({⨆}_{i\in \Gamma }{\mathcal{F}}_i)}_k:=({⨆}_{i\in \Gamma }{\mathcal{F}}_i)(k):{(L^X)}^E\to L$) as:

$${(\underset{i\in \Gamma }{⨆}{\mathcal{F}}_i)}_k(f)=\left\{\begin{array}{cc}\bigvee \left\{{\odot }_{i\in J}{({\mathcal{F}}_i)}_k(f_i)\right\},\hfill & \mathrm{if}f={\odot }_{i\in J}{f}_i,f_i\in {({\mathcal{F}}_i)}_k^0,k\in K\hfill \\ 0_L,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

where ⋁ is taken for each $J\subseteq \Gamma$, J is finite such that $f={\odot }_{i\in J}{f}_i$. Then ${⨆}_{i\in \Gamma }{\mathcal{F}}_i$ is the coarsest $(L_{\odot },K_E)$-fs-filter finer than ${\mathcal{F}}_i$, $\forall i\in \Gamma$.

Proof. 

First: we will prove that, ${\mathcal{H}}_k={({⨆}_{i\in \Gamma }{\mathcal{F}}_i)}_k$ is an $(L_{\odot },K_E)$-fs-filter on X.

(SF1) It is trivial that ${\mathcal{H}}_k(0_X)=0_L$. Since $1_X=1_X\odot 1_X$, we have ${\mathcal{H}}_k(1_X)=1_L$.

(SF2) For every finite index subsets M and J of $\Gamma$ such that $f={\odot }_{m\in M}{f}_m$, $g={\odot }_{j\in J}{g}_j$, we have

$$f\odot g=({\odot }_{m\in M}{f}_m)\odot ({\odot }_{j\in J}{g}_j).$$

Furthermore, for each $n\in M\cup J$, put $f\odot g={\odot }_{n\in M\cup J}{h}_n$, where

$$h_n=\left\{\begin{array}{cc}{f}_n,\hfill & \mathrm{if}n\in M-(M\cap J)\hfill \\ g_n,\hfill & \mathrm{if}n\in J-(M\cap J)\hfill \\ f_n\odot g_n,\hfill & \mathrm{if}n\in M\cap J\hfill \end{array}\right.$$

We have

$$\begin{array}{cc}{\mathcal{H}}_k(f\odot g)& \ge {\odot }_{n\in M\cup J}{({\mathcal{F}}_n)}_k(h_n)\hfill \\ & \ge ({\odot }_{m\in M}{({\mathcal{F}}_m)}_k(f_m))\odot ({\odot }_{j\in J}{({\mathcal{F}}_j)}_k(g_j)).\hfill \end{array}$$

By $(L4)$ of Definition 1, we have

$${\mathcal{H}}_k(f\odot g)\ge {\mathcal{H}}_k(f)\odot {\mathcal{H}}_k(g).$$

(SF3) Suppose that $f⊑g$. By the definition of $\mathcal{H}$, there exists an index $J\subseteq \Gamma$, J is finite and $f={\odot }_{j\in J}{f}_j$ with ${\mathcal{H}}_k(f)\ge {\odot }_{j\in J}{({\mathcal{F}}_j)}_k(f_j)$. On the other hand, since $g=f\bigsqcup g={\odot }_{j\in J}(f_j\bigsqcup g)$, we have

$$\begin{array}{cc}{\mathcal{H}}_k(g)& \ge {\odot }_{j\in J}{({\mathcal{F}}_j)}_k(f_j\bigsqcup g)\hfill \\ & \ge {\odot }_{j\in J}{({\mathcal{F}}_j)}_k(f_j).\hfill \end{array}$$

By (L4), we have ${\mathcal{H}}_k(f)\le {\mathcal{H}}_k(g)$.

Second: we will show that ${\mathcal{H}}_k\ge {({\mathcal{F}}_i)}_k$, for each $i\in \Gamma$, $k\in K$. For each $f\in {(L^X)}^E$, if ${({\mathcal{F}}_i)}_k(f)=0_L$, it is trivial. If ${({\mathcal{F}}_i)}_k(f)\ne 0_L$, for $f=f\odot 1_X$, we have

$${\mathcal{H}}_k(f)\ge {({\mathcal{F}}_i)}_k(f)\odot {({\mathcal{F}}_i)}_k(1_X)={({\mathcal{F}}_i)}_k(f).$$

Third: for each $(L_{\odot },K_E)$-fs-filter $\mathcal{G}$ such that ${\mathcal{G}}_k\ge {({\mathcal{F}}_i)}_k$, for each $i\in \Gamma$. We will show that ${\mathcal{G}}_k\ge {\mathcal{H}}_k$. By the definition of $\mathcal{H}$, there exists an index $J\subseteq \Gamma$, J is finite and $f={\odot }_{j\in J}{f}_j$ with ${\mathcal{H}}_k(f)\ge {\odot }_{j\in J}{({\mathcal{F}}_j)}_k(f)$. On the other hand, since ${\mathcal{G}}_k\ge {({\mathcal{F}}_i)}_k$, for each $i\in J$, we have

$${\mathcal{G}}_k(f)\ge {\odot }_{j\in J}{\mathcal{G}}_k(f_j)\ge {\odot }_{j\in J}{({\mathcal{F}}_j)}_k(f_j).$$

By (L4), we have ${\mathcal{G}}_k(f)\ge {\mathcal{H}}_k(f)$. □

Theorem 4.

Let $\varphi :X\to Y$, $\psi :E_1\to E_2$ and $\eta :K_1\to K_2$ be maps. Let $\mathcal{F}$ be an $(L_{\odot },K_{2E_2})$-fs-filter on Y. We define a fuzzy soft mapping ${\varphi }_{\psi ,\eta }^{\leftarrow }(\mathcal{F}):K_1\to L^{{(L^X)}^{E_1}}$ (where ${({\varphi }_{\psi ,\eta }^{\leftarrow }(\mathcal{F}))}_k:{(L^X)}^{E_1}\to L$ is a map for each $k\in K_1$) by:

$${\varphi }_{\psi ,\eta }^{\leftarrow }({\mathcal{F}}_k)(f)=\left\{\begin{array}{cc}\bigvee \{{\mathcal{F}}_{\eta (k)}(g):f={\varphi }_{\psi }^{\leftarrow }(g)\},\hfill & \mathit{if}g\ne 0_X\hfill \\ 0_L,\hfill & \mathit{if}g=0_X.\hfill \end{array}\right.$$

Then, ${\varphi }_{\psi ,\eta }^{\leftarrow }(\mathcal{F})$ is an $(L_{\odot },K_{1E_1})$-fs-filter on X.

Proof. 

(SF1) Easily proved.

(SF2) For each $f,g\in {(L^X)}^{E_1}$, we have

$$\begin{array}{cc}& {\varphi }_{\psi ,\eta }^{\leftarrow }({\mathcal{F}}_k)(f)\odot {\varphi }_{\psi ,\eta }^{\leftarrow }({\mathcal{F}}_k)(g)\hfill \\ & =(\bigvee \{{\mathcal{F}}_{\eta (k)}(f_1):f={\varphi }_{\psi }^{\leftarrow }(f_1)\})\odot (\bigvee \{{\mathcal{F}}_{\eta (k)}(g_1):g={\varphi }_{\psi }^{\leftarrow }(g_1)\})\hfill \\ & =\bigvee \{{\mathcal{F}}_{\eta (k)}(f_1)\odot {\mathcal{F}}_{\eta (k)}(g_1):f\odot g={\varphi }_{\psi }^{\leftarrow }(f_1)\odot {\varphi }_{\psi }^{\leftarrow }(g_1)\}\hfill \\ & \le \bigvee \{{\mathcal{F}}_{\eta (k)}(f_1\odot g_1):f\odot g={\varphi }_{\psi }^{\leftarrow }(f_1\odot g_1)\}\hfill \\ & \le {\varphi }_{\psi ,\eta }^{\leftarrow }({\mathcal{F}}_k)(f\odot g).\hfill \end{array}$$

(SF3) If $f_1⊑f_2$, we have ${\varphi }_{\psi ,\eta }^{\leftarrow }({\mathcal{F}}_k)(f_1)={\mathcal{F}}_k({\varphi }_{\psi ,\eta }(f_1))\le {\mathcal{F}}_k({\varphi }_{\psi ,\eta }(f_2))={\varphi }_{\psi ,\eta }^{\leftarrow }({\mathcal{F}}_k)(f_2)$.

□

4. $({\mathit{L}}_{\odot },{\mathit{K}}_{\mathit{E}})$-fs-Topologies Induced by $({\mathit{L}}_{\odot },{\mathit{K}}_{\mathit{E}})$-fs-Filters

From an $(L_{\odot },K_E)$-fs-filter $\mathcal{F}:K\to L^{{(L^X)}^E}$, we can obtain an $(L_{\odot },K_E)$-fs-topology on X as in the next theorem:

Theorem 5.

Let $(X,\mathcal{F})$ be an $(L_{\odot },K_E)$-fs-filtered set. Define ${\mathcal{T}}_{\mathcal{F}}:K\to L^{{(L^X)}^E}$ (where ${({\mathcal{T}}_{\mathcal{F}})}_k:={\mathcal{T}}_{\mathcal{F}}(k):{(L^X)}^E\to L$ is a map for each $k\in K$) as:

$${({\mathcal{T}}_{\mathcal{F}})}_k(f)=\left\{\begin{array}{cc}{\mathcal{F}}_k(f),\hfill & \mathit{if}f\ne 0_X\hfill \\ 1_L,\hfill & \mathit{if}f=0_X.\hfill \end{array}\right.$$

Then, $(X,{\mathcal{T}}_{\mathcal{F}})$ is an $(L_{\odot },K_E)$-fs-topological space.

Proof. 

(SO1) and (SO2) are clear. (SO3) Let $\{f_i:i\in \Gamma \}⊑{(L^X)}^E$. Then $f_i⊑{⨆}_{i\in \Gamma }{f}_i$, $\forall i\in \Gamma$. By (SF3), we have

$${\mathcal{F}}_k(f_i)\le {\mathcal{F}}_k(\underset{i\in \Gamma }{⨆}{f}_i),\forall i\in \Gamma .$$

Thus,

$${({\mathcal{T}}_{\mathcal{F}})}_k(f_i)\le {({\mathcal{T}}_{\mathcal{F}})}_k(\underset{i\in \Gamma }{⨆}{f}_i),\forall i\in \Gamma .$$

Thus,

$$\underset{i\in \Gamma }{\bigwedge }{({\mathcal{T}}_{\mathcal{F}})}_k(f_i)\le {({\mathcal{T}}_{\mathcal{F}})}_k(\underset{i\in \Gamma }{⨆}{f}_i).$$

Therefore, each $(L_{\odot },K_E)$-fs-filtered set produces an $(L_{\odot },K_E)$-fs-topological space. □

Theorem 6.

Let $F=\{{\mathcal{F}}^x:x\in X\}$ be a family of $(L_{\odot },K_E)$-fs-filters $\forall x\in X$. An operator ⊙ dominates *, which induces $u\Rightarrow v=\bigvee \{w\in L:u\ast w\le v\}$. For every $x\in X$, $H^x:K\to L^{{(L^X)}^E}$, where $H_k^x:=H^x(k):{(L^X)}^E\to L$ is a map for each $k\in K$ satisfying the following conditions:

(H1) 

$H_k^x(0_X)=0_L$ and $H_k^x(1_X)=1_L$.

(H2) 

$H_k^x(f\odot g)\le H_k^x(f)\odot H_k^x(g)$, $\forall f,g\in {(L^X)}^E$.

(H3) 

$H_k^x({⨆}_{i\in \Gamma }{f}_i)\le {\bigvee }_{i\in \Gamma }{H}_k^x(f_i)$, $\forall f_i\in {(L^X)}^E,i\in \Gamma$.

Define ${\mathcal{T}}_{\mathcal{F}}:K\to L^{{(L^X)}^E}$, where ${({\mathcal{T}}_{\mathcal{F}})}_k:={\mathcal{T}}_{\mathcal{F}}(k):{(L^X)}^E\to L$ is a map, $\forall k\in K$ defined by:

$${({\mathcal{T}}_{\mathcal{F}})}_k(f)=\underset{x\in X}{\bigwedge }(H_k^x(f)\Rightarrow {\mathcal{F}}_k^x(f)).$$

Then,

(i) 

${\mathcal{T}}_{\mathcal{F}}$ is an $(L_{\odot },K_E)$-fs-topology.

(ii) 

If ${\mathcal{F}}^x$ is stratified $(L_{\odot },K_E)$-fs-filter and $H_k^x(\alpha \odot g)\le \alpha \odot H_k^x(g)$, $\forall x\in X$, then ${\mathcal{T}}_{\mathcal{F}}$ is an enriched $(L_{\odot },K_E)$-fs-topology.

Proof. 

For each $k\in K$ we have

(SO1)

${({\mathcal{T}}_{\mathcal{F}})}_k(0_X)={\bigwedge }_{x\in X}(H_k^x(0_X)\Rightarrow {\mathcal{F}}_k^x(0_X))=1_L$. and ${({\mathcal{T}}_{\mathcal{F}})}_k(1_X)={\bigwedge }_{x\in X}(H_k^x(1_X)\Rightarrow {\mathcal{F}}_k^x(1_X))=1_L$.

(SO2)

$\forall g_1,g_2\in {(L^X)}^E$ and $k\in K$, we have

$$\begin{array}{cc}{({\mathcal{T}}_{\mathcal{F}})}_k(g_1\odot g_2)& =\underset{x\in X}{\bigwedge }(H_k^x(g_1\odot g_2)\Rightarrow {\mathcal{F}}_k^x(g_1\odot g_2))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(H_k^x(g_1)\odot H_k^x(g_2)\Rightarrow {\mathcal{F}}_k^x(g_1)\odot {\mathcal{F}}_k^x(g_2))\hfill \\ & \ge \underset{x\in X}{\bigwedge }((H_k^x(g_1)\Rightarrow {\mathcal{F}}_k^x(g_1))\odot (H_k^x(g_2)\Rightarrow {\mathcal{F}}_k^x(g_2))),(\mathrm{by}\mathrm{Lemma}1(\mathrm{v}))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(H_k^x(g_1)\Rightarrow {\mathcal{F}}_k^x(g_1))\odot \underset{x\in X}{\bigwedge }(H_k^x(g_2)\Rightarrow {\mathcal{F}}_k^x(g_2))\hfill \\ & ={({\mathcal{T}}_{\mathcal{F}})}_k(g_1)\odot {({\mathcal{T}}_{\mathcal{F}})}_k(g_2).\hfill \end{array}$$

(SO3)

For each $g_i\in {(L^X)}^E$, $i\in \Gamma$, we have

$$\begin{array}{cc}{({\mathcal{T}}_{\mathcal{F}})}_k(\underset{i\in \Gamma }{⨆}{g}_i)& =\underset{x\in X}{\bigwedge }(H_k^x(\underset{i\in \Gamma }{⨆}{g}_i)\Rightarrow {\mathcal{F}}_k^x(\underset{i\in \Gamma }{⨆}{g}_i))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(\underset{i\in \Gamma }{\bigvee }{H}_k^x(g_i)\Rightarrow \underset{i\in \Gamma }{\bigvee }{\mathcal{F}}_k^x(g_i))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(H_k^x(g_i)\Rightarrow {\mathcal{F}}_k^x(g_i)),(\mathrm{by}\mathrm{Lemma}1(\mathrm{ix}))\hfill \\ & \ge \underset{x\in X}{\bigwedge }\underset{i\in \Gamma }{\bigwedge }(H_k^x(g_i)\Rightarrow {\mathcal{F}}_k^x(g_i))\hfill \\ & \ge \underset{i\in \Gamma }{\bigwedge }\underset{x\in X}{\bigwedge }(H_k^x(g_i)\Rightarrow {\mathcal{F}}_k^x(g_i))\hfill \\ & =\underset{i\in \Gamma }{\bigwedge }{({\mathcal{T}}_{\mathcal{F}})}_k(g_i).\hfill \end{array}$$

Hence, ${\mathcal{T}}_{\mathcal{F}}$ is an $(L_{\odot },K_E)$-fs-topology.

(ii) Let ${\mathcal{F}}^x$ be a stratified $(L_{\odot },K_E)$-fs-filter and $H_k^x(\alpha \odot g)\le \alpha \odot H_k^x(g)$, for each $x\in X$, $k\in K$, $\alpha \in L$, $g\in {(L^X)}^E$. Then,

$$\begin{array}{cc}{({\mathcal{T}}_{\mathcal{F}})}_k(\alpha \odot g)& =\underset{x\in X}{\bigwedge }(H_k^x(\alpha \odot g)\Rightarrow {\mathcal{F}}_k^x(\alpha \odot g))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(\alpha \odot H_k^x(g)\Rightarrow \alpha \odot {\mathcal{F}}_k^x(g))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(H_k^x(g)\Rightarrow {\mathcal{F}}_k^x(g)),(\mathrm{by}\mathrm{Lemma}1(\mathrm{vi}))\hfill \\ & ={({\mathcal{T}}_{\mathcal{F}})}_k(g).\hfill \end{array}$$

□

Theorem 7.

Let $F=\{{\mathcal{F}}^x:x\in X\}$ be a family of $(L_{\odot },K_{1E_1})$-fs-filters for each $x\in X$ and $F=\{{\mathcal{F}}^y:y\in Y\}$ be a family of $(L_{\odot },K_{2E_2})$-fs-filters, $\forall y\in Y$. An operator ⊙ dominates *, which induces $u\Rightarrow v=\bigvee \{w\in L:u\ast w\le v\}$. For every $x\in X$, $H^x:K_1\to L^{{(L^X)}^{E_1}}$, where $H_k^x:=H^x(k):{(L^X)}^{E_1}\to L$ is a map for each $k\in K_1$ and for each $y\in Y$, $H^y:K_2\to L^{{(L^Y)}^{E_2}}$, where $H_k^x:=H^x(k):{(L^X)}^{E_2}\to L$ is a map for each $k\in K_2$, and $H^x,H^y$ satisfy the conditions (H1), (H2) and (H3) of Theorem 6. Let $\varphi :X\to Y$, $\psi :E_1\to E_2$ and $\eta :K_1\to K_2$ be maps. Then,

(i) 

$(H_k^x({\varphi }_{\psi }^{\leftarrow }(g))\Rightarrow H_{\eta (k)}^{\varphi (x)}(g))\ast ({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }((g)))\le {\bigwedge }_{x\in X}(H_{\eta (k)}^{\varphi (x)}(g)\to {\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to {\bigwedge }_{x\in X}(H_k^x{\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{F}}_k^x{\varphi }_{\psi }^{\leftarrow }((g))).$

(ii) 

If $H_k^x({\varphi }_{\psi }^{\leftarrow }(g))\le H_{\eta (k)}^{\varphi (x)}(g)$, then ${\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g)\to {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }(g))\le {({\mathcal{T}}_G)}_{\eta (k)}(g)\to {({\mathcal{T}}_F)}_k$

$({\varphi }_{\psi }^{\leftarrow }(g))$.

(iii) 

If $\varphi :(X,{\mathcal{F}}^x)\to (Y,{\mathcal{G}}^{\varphi (x)})$ is an L-fuzzy soft filter map, then $\varphi :(X,{\mathcal{T}}_F)\to (Y,{\mathcal{T}}_G)$ is an L-fuzzy soft continuous.

Proof. 

(i)

$$\begin{array}{cc}& (H_k^x({\varphi }_{\psi }^{\leftarrow }(g))\Rightarrow H_{\eta (k)}^{\varphi (x)}(g))\ast ({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }((g)))\hfill \\ & \le (H_{\eta (k)}^{\varphi (x)}(g)\to {\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to (H_k^x({\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }((g)))),(\mathrm{by}\mathrm{Lemma}1(\mathrm{x}))\hfill \\ & \le \underset{x\in X}{\bigwedge }(H_{\eta (k)}^{\varphi (x)}(g)\to {\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to \underset{x\in X}{\bigwedge }(H_k^x({\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }((g)))).\hfill \end{array}$$

(ii)

$$\begin{array}{cc}& {({\mathcal{T}}_G)}_{\eta (k)}(g)\to {({\mathcal{T}}_F)}_k{\varphi }_{\psi }^{\leftarrow }((g))\hfill \\ & =\underset{y\in Y}{\bigwedge }(H_{\eta (k)}^y(g)\Rightarrow {\mathcal{G}}_{\eta (k)}^y(g))\to \underset{x\in X}{\bigwedge }(H_k^x({\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }((g)))))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(H_{\eta (k)}^{\varphi (x)}(g)\Rightarrow {\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to \underset{x\in X}{\bigwedge }(H_k^x({\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }((g))))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(H_k^x{\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to \underset{x\in X}{\bigwedge }(H_k^x{\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }((g))))\hfill \\ & \ge (H_k^x{\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g))\to (H_k^x{\varphi }_{\psi }^{\leftarrow }((g))\Rightarrow {\mathcal{F}}_k^x{\varphi }_{\psi }^{\leftarrow }((g))),(\mathrm{by}\mathrm{Lemma}2(\mathrm{vii}))\hfill \\ & \ge {\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g)\to {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }(g)),(\mathrm{by}\mathrm{Lemma}2(\mathrm{vi})).\hfill \end{array}$$

(iii)

If $\varphi :(X,{\mathcal{F}}^x)\to (Y,{\mathcal{G}}^{\varphi (x)})$ is an L-fuzzy soft filter map, then ${\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g)\to {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }(g))=1_L$. Then from (ii), we have ${({\mathcal{T}}_G)}_{\eta (k)}(g)\to {({\mathcal{T}}_F)}_k{\varphi }_{\psi }^{\leftarrow }((g))=1_L$. From Lemma 1(xi), we have ${({\mathcal{T}}_G)}_{\eta (k)}(g)\le {({\mathcal{T}}_F)}_k{\varphi }_{\psi }^{\leftarrow }((g))$. Thus, $\varphi :(X,{\mathcal{T}}_F)\to (Y,{\mathcal{T}}_G)$ is an L-fuzzy soft continuous.

□

5. The Products of $({\mathit{L}}_{\odot },{\mathit{K}}_{\mathit{E}})$-fs-Filters

Theorem 8.

Let $(X,\mathcal{T})$ be an $(L_{\odot },K_E)$-fs-topology and $\{{\mathcal{F}}^x:x\in X\}$ be a family of $(L_{\odot },K_E)$-fs-filters for each $x\in X$. An operation * dominates ⊙. For each $x\in X$, we define ${\mathcal{N}}_{\mathcal{T}}^x:K\to L^{{(L^X)}^E}$, where ${({\mathcal{N}}_{\mathcal{T}}^x)}_k:={\mathcal{N}}_{\mathcal{T}}^x(k):{(L^X)}^E\to L$ is a map for each $k\in K$ defined by:

$${({\mathcal{N}}_{\mathcal{T}}^x)}_k(f)=\underset{g⊑f}{\bigvee }({\mathcal{F}}_k^x(g)\ast {\mathcal{T}}_k(g)).$$

Then:

(i) 

${\mathcal{N}}_{\mathcal{T}}^x$ is an $(L_{\odot },K_E)$-fs-filter on X.

(ii) 

If ${\mathcal{F}}^x$ is a stratified $(L_{\odot },K_E)$-fs-filter and $\mathcal{T}$ is an enriched $(L_{\odot },K_E)$-fs-topology, then ${\mathcal{N}}_{\mathcal{T}}^x$ is a stratified $(L_{\odot },K_E)$-fs-filter.

(iii) 

If ${\mathcal{F}}_k^x⊒{\mathcal{H}}_k^x$, then ${({\mathcal{T}}_{{\mathcal{N}}_{\mathcal{T}}^x})}_k⊒{\mathcal{T}}_k$, $k\in K$.

(iv) 

If ${\mathcal{F}}_k^x⊑{\mathcal{H}}_k^x$, then ${({\mathcal{N}}_{{\mathcal{T}}_{\mathcal{F}}}^x)}_k⊑{\mathcal{F}}_k^x$, $k\in K$.

Proof. 

(i)

(SF1) ${({\mathcal{N}}_{\mathcal{T}}^x)}_k(0_X)={\mathcal{F}}_k^x(0_X)\ast {\mathcal{T}}_k(0_X)=0_L$. and ${({\mathcal{N}}_{\mathcal{T}}^x)}_k(1_X)={\mathcal{F}}_k^x(1_X)\ast {\mathcal{T}}_k(1_X)=1_L$.

(SF2) For each $x\in X$, $k\in K$, $f,g\in {(L^X)}^E$, we have

$$\begin{array}{cc}& {({\mathcal{N}}_{\mathcal{T}}^x)}_k(f)\odot {({\mathcal{N}}_{\mathcal{T}}^x)}_k(g)\hfill \\ & =\underset{\nu ⊑f}{\bigvee }({\mathcal{F}}_k^x(\nu )\ast {\mathcal{T}}_k(\nu ))\odot \underset{\mu ⊑g}{\bigvee }({\mathcal{F}}_k^x(\mu )\ast {\mathcal{T}}_k(\mu ))\hfill \\ & =\underset{\nu ⊑f}{\bigvee }\underset{\mu ⊑g}{\bigvee }(({\mathcal{F}}_k^x(\nu )\ast {\mathcal{T}}_k(\nu ))\odot ({\mathcal{F}}_k^x(\mu )\ast {\mathcal{T}}_k(\mu )))\hfill \\ & \le \underset{\nu ⊑f}{\bigvee }\underset{\mu ⊑g}{\bigvee }(({\mathcal{F}}_k^x(\nu )\odot {\mathcal{F}}_k^x(\mu ))\ast ({\mathcal{T}}_k(\nu )\odot {\mathcal{T}}_k(\mu ))),(\sin \mathrm{ce}\ast \mathrm{dominates}\odot )\hfill \\ & \le \underset{\nu \odot \mu ⊑f\odot g}{\bigvee }(({\mathcal{F}}_k^x(\nu \odot \mu ))\ast ({\mathcal{T}}_k(\nu \odot \mu )))\hfill \\ & \le \underset{h⊑f\odot g}{\bigvee }({\mathcal{F}}_k^x(h)\ast {\mathcal{T}}_k(h))\hfill \\ & ={({\mathcal{N}}_{\mathcal{T}}^x)}_k(f\odot g).\hfill \end{array}$$

(SF3) It is easy.

(ii)

For each $\beta \in L$, $k\in K$, $g\in {(L^X)}^E$, we have

$$\begin{array}{cc}{({\mathcal{N}}_{\mathcal{T}}^x)}_k(\beta \odot g)& =\underset{\mu ⊑\beta \odot g}{\bigvee }({\mathcal{F}}_k^x(\mu )\ast {\mathcal{T}}_k(\mu ))\hfill \\ & \ge \underset{\beta \odot \nu ⊑\beta \odot g}{\bigvee }({\mathcal{F}}_k^x(\beta \odot \nu )\ast {\mathcal{T}}_k(\beta \odot \nu ))\hfill \\ & \ge \underset{\beta \odot \nu ⊑\beta \odot g}{\bigvee }((\beta \odot {\mathcal{F}}_k^x(\nu ))\ast {\mathcal{T}}_k(\nu ))\hfill \\ & \ge \beta \odot \underset{\nu ⊑g}{\bigvee }({\mathcal{F}}_k^x(\nu )\ast {\mathcal{T}}_k(\nu )),(\mathrm{by}\mathrm{Lemma}1(\mathrm{iv}))\hfill \\ & =\beta \odot {({\mathcal{N}}_{\mathcal{T}}^x)}_k(g).\hfill \end{array}$$

(iii)

Since * dominates ⊙, by Lemma 1(iv), $u\to v\ge u\Rightarrow v$. Then,

$$\begin{array}{cc}{({\mathcal{T}}_{{\mathcal{N}}_{\mathcal{T}}^x})}_k(f)& =\underset{x\in X}{\bigwedge }({\mathcal{H}}_k^x(f)\to {({\mathcal{N}}_{\mathcal{T}}^x)}_k(f))\hfill \\ & =\underset{x\in X}{\bigwedge }({\mathcal{H}}_k^x(f)\to \underset{g⊑f}{\bigvee }({\mathcal{F}}_k^x(g)\ast {\mathcal{T}}_k(g)))\hfill \\ & \ge \underset{x\in X}{\bigwedge }({\mathcal{H}}_k^x(f)\to ({\mathcal{F}}_k^x(f)\ast {\mathcal{T}}_k(f)))\hfill \\ & \ge \underset{x\in X}{\bigwedge }({\mathcal{H}}_k^x(f)\Rightarrow ({\mathcal{F}}_k^x(f)\ast {\mathcal{T}}_k(f)))\hfill \\ & \ge {\mathcal{T}}_k(f).\hfill \end{array}$$

(iv)

$$\begin{array}{cc}{({\mathcal{N}}_{{\mathcal{T}}_{\mathcal{F}}}^x)}_k(g)& =\underset{\nu ⊑g}{\bigvee }({\mathcal{F}}_k^x(\nu )\ast {({\mathcal{T}}_{\mathcal{F}})}_k(\nu ))\hfill \\ & =\underset{\nu ⊑g}{\bigvee }({\mathcal{F}}_k^x(\nu )\ast \underset{x\in X}{\bigwedge }({\mathcal{H}}_k^x(\nu ))\to {\mathcal{F}}_k^x(\nu ))\hfill \\ & \le \underset{\nu ⊑g}{\bigvee }({\mathcal{F}}_k^x(\nu )\ast ({\mathcal{H}}_k^x(\nu ))\to {\mathcal{F}}_k^x(\nu ))\hfill \\ & \le \underset{\nu ⊑g}{\bigvee }({\mathcal{F}}_k^x(\nu )\ast ({\mathcal{F}}_k^x(\nu ))\to {\mathcal{F}}_k^x(\nu ))\hfill \\ & \le \underset{\nu ⊑g}{\bigvee }{\mathcal{F}}_k^x(\nu )\hfill \\ & ={\mathcal{F}}_k^x(g).\hfill \end{array}$$

□

Theorem 9.

Let $(X,{\mathcal{T}}^1)$ be an $(L_{\odot },K_{1E_1})$-fs-topology on X and $(X,{\mathcal{T}}^2)$ be an $(L_{\odot },K_{2E_2})$-fs-topology on Y. Let $F=\{{\mathcal{F}}^x:x\in X\}$ be a family of $(L_{\odot },K_{1E_1})$-fs-filters for each $x\in X$ and $G=\{{\mathcal{G}}^y:y\in Y\}$ be a family of $(L_{\odot },K_{2E_2})$-fs-filters for each $y\in Y$. Let $\varphi :X\to Y$, $\psi :E_1\to E_2$ and $\eta :K_1\to K_2$ be maps. If the operation * dominates ⊙, we have

(i) 

$({\mathcal{T}}_{\eta (k)}^2(h)\to {\mathcal{T}}_k^1({\varphi }_{\psi }^{\leftarrow }(h)))\ast ({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(h)\to {\mathcal{F}}_k^x{\varphi }_{\psi }^{\leftarrow }(h))\le {({\mathcal{N}}_{{\mathcal{T}}^2}^{\varphi (x)})}_{\eta (k)}(h)\to {({\mathcal{N}}_{{\mathcal{T}}^1}^x)}_k$

$({\varphi }_{\psi }^{\leftarrow }(h))$.

(ii) 

If $\varphi :(X,{\mathcal{T}}^1)\to (Y,{\mathcal{T}}^2)$ is an L- fuzzy soft continuous and $\varphi :(X,{\mathcal{F}}^x)\to (Y,{\mathcal{G}}^{\varphi (x)})$ is an L-fuzzy soft filter map, then $\varphi :(X,{\mathcal{N}}_{{\mathcal{T}}^1}^x)\to (Y,{\mathcal{N}}_{{\mathcal{T}}^2}^{\varphi (x)})$ is an L-fuzzy soft filter map.

Proof. 

(i)

For each $k\in K_1$, $f\in {(L^Y)}^{E_2}$, we have

$$\begin{array}{cc}& {({\mathcal{N}}_{{\mathcal{T}}^2}^{\varphi (x)})}_{\eta (k)}(h)\to {({\mathcal{N}}_{{\mathcal{T}}^1}^x)}_k({\varphi }_{\psi }^{\leftarrow }(h))\hfill \\ & =\underset{g⊑h}{\bigvee }({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g)\ast {\mathcal{T}}_{\eta (k)}^2(g))\to \underset{f⊑({\varphi }_{\psi }^{\leftarrow }(h)}{\bigvee }({\mathcal{F}}_k^x(f)\ast {\mathcal{T}}_k^1(f))\hfill \\ & \ge \underset{g⊑h}{\bigvee }({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g)\ast {\mathcal{T}}_{\eta (k)}^2(g))\to \underset{{\varphi }_{\psi }^{\leftarrow }(h)⊑({\varphi }_{\psi }^{\leftarrow }(h)}{\bigvee }({\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }(g))\ast {\mathcal{T}}_k^1({\varphi }_{\psi }^{\leftarrow }(g)))\hfill \\ & \ge \underset{g⊑h}{\bigvee }({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(g)\ast {\mathcal{T}}_{\eta (k)}^2(g))\to \underset{g⊑h}{\bigvee }({\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }(g))\ast {\mathcal{T}}_k^1({\varphi }_{\psi }^{\leftarrow }(g)))\hfill \\ & \ge ({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(h)\ast {\mathcal{T}}_{\eta (k)}^2(h))\to ({\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }(h))\ast {\mathcal{T}}_k^1({\varphi }_{\psi }^{\leftarrow }(h))),(\mathrm{by}\mathrm{Lemma}1(\mathrm{ix}))\hfill \\ & \ge ({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(h)\to {\mathcal{F}}_k^x({\varphi }_{\psi }^{\leftarrow }(h)))\ast ({\mathcal{T}}_{\eta (k)}^2(h))\to {\mathcal{T}}_k^1({\varphi }_{\psi }^{\leftarrow }(h))),(\mathrm{by}\mathrm{Lemma}1(\mathrm{v})).\hfill \end{array}$$

(ii)

Since $\varphi :(X,{\mathcal{T}}^1)\to (Y,{\mathcal{T}}^2)$ is an L- fuzzy soft continuous, $\varphi :(X,{\mathcal{F}}^x)\to (Y,{\mathcal{G}}^{\varphi (x)})$ is an L-fuzzy soft filter map and by (i), we have

$$\begin{array}{cc}{({\mathcal{N}}_{{\mathcal{T}}^2}^{\varphi (x)})}_{\eta (k)}(h)\to {({\mathcal{N}}_{{\mathcal{T}}^1}^x)}_k({\varphi }_{\psi }^{\leftarrow }(h))& \ge ({\mathcal{T}}_{\eta (k)}^2(h)\to {\mathcal{T}}_k^1({\varphi }_{\psi }^{\leftarrow }(h)))\ast ({\mathcal{G}}_{\eta (k)}^{\varphi (x)}(h)\to {\mathcal{F}}_k^x{\varphi }_{\psi }^{\leftarrow }(h))\hfill \\ & =1_L\ast 1_L=1_L.\hfill \end{array}$$

Then, from Lemma 1(xi), we have ${({\mathcal{N}}_{{\mathcal{T}}^2}^{\varphi (x)})}_{\eta (k)}(h)\le {({\mathcal{N}}_{{\mathcal{T}}^1}^x)}_k({\varphi }_{\psi }^{\leftarrow }(h))$.

Thus, $\varphi :(X,{\mathcal{N}}_{{\mathcal{T}}^1}^x)\to (Y,{\mathcal{N}}_{{\mathcal{T}}^2}^{\varphi (x)})$ is an L-fuzzy soft filter map. □

Theorem 10.

Let $F=\{{\mathcal{F}}^x:x\in X\}$ and $G=\{{\mathcal{G}}^x:x\in X\}$ be two families of $(L_{\odot },K_E)$-fs-filters satisfying the condition ${\mathcal{F}}_k^x(f_1)\ast {\mathcal{G}}_k^x(f_2)=0_L$, for each $f_1\odot f_2=0_X$, where an operation * dominates ⊙. Define ${\mathcal{F}}^x\ast {\mathcal{G}}^x:K\to L^{{(L^X)}^E}$ as:

$${({\mathcal{F}}^x\ast {\mathcal{G}}^x)}_k(f)=\bigvee \{{\mathcal{F}}_k^x(f_1)\ast {\mathcal{G}}_k^x(f_2):f_1\odot f_2⊑f\}.$$

Let ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ be two $(L_{\odot },K_E)$-fs-topologies on X. We define ${\mathcal{T}}^1\ast {\mathcal{T}}^2:K\to L^{{(L^X)}^E}$ as follows:

$${({\mathcal{T}}^1\ast {\mathcal{T}}^2)}_k(f)=\bigvee \{{\mathcal{T}}_k^1(f_1)\ast {\mathcal{T}}_k^2(f_2):f_1\odot f_2=f\}.$$

Then, we have:

(i) 

${\mathcal{F}}^x\ast {\mathcal{G}}^x$ is an $(L_{\odot },K_E)$-fs-filter on X, and it is finer than ${\mathcal{F}}^x$ and ${\mathcal{G}}^x$. If $\ast =\odot$, then ${\mathcal{F}}^x\odot {\mathcal{G}}^x$ is the coarsest $(L_{\odot },K_E)$-fs-filter on X, and it is finer than ${\mathcal{F}}^x$ and ${\mathcal{G}}^x$. Besides, if $\ast =\odot$ and ${\mathcal{F}}^x={\mathcal{G}}^x$, then ${\mathcal{F}}^x\odot {\mathcal{F}}^x={\mathcal{F}}^x$.

(ii) 

If ${\mathcal{F}}^x$ or ${\mathcal{G}}^x$ is a stratified $(L_{\odot },K_E)$-fs-filter, then ${\mathcal{F}}^x\ast {\mathcal{G}}^x$ is a stratified $(L_{\odot },K_E)$-fs-filter on X.

(iii) 

${\mathcal{T}}^1\ast {\mathcal{T}}^2$ is an $(L_{\odot },K_E)$-fs-topology on X and it is finer than ${\mathcal{T}}^1$ and ${\mathcal{T}}^2$. If $\ast =\odot$, then ${\mathcal{T}}^1\odot {\mathcal{T}}^2$ is the coarsest $(L_{\odot },K_E)$-fs-topology on X, and it is finer than ${\mathcal{T}}^1$ and ${\mathcal{T}}^2$.

(iv) 

If ${\mathcal{T}}^1$ or ${\mathcal{T}}^2$ is an enriched $(L_{\odot },K_E)$-fs-topology, then ${\mathcal{T}}^1\ast {\mathcal{T}}^2$ is an enriched $(L_{\odot },K_E)$-fs-topology on X.

(v) 

${\mathcal{T}}_{F\ast G}⊒{\mathcal{T}}_F\ast {\mathcal{T}}_G$, where $F\ast G=\{{\mathcal{F}}^x\ast {\mathcal{G}}^x:x\in X\}$.

(vi) 

${\mathcal{N}}_{{\mathcal{T}}^1\odot {\mathcal{T}}^2}^x⊒{\mathcal{N}}_{{\mathcal{T}}^1}^x\odot {\mathcal{N}}_{{\mathcal{T}}^2}^x$.

Proof. 

(i)

(SF1) and (SF3) are clear.

(SF2) For each $f,g\in {(L^X)}^E$ and $k\in K$,

$$\begin{array}{l}{({\mathcal{F}}^x\ast {\mathcal{G}}^x)}_k(f)\odot {({\mathcal{F}}^x\ast {\mathcal{G}}^x)}_k(g)\\ =\bigvee \{{\mathcal{F}}_k^x(f_1)\ast {\mathcal{G}}_k^x(f_2):f_1\odot f_2⊑f\}\odot \bigvee \{{\mathcal{F}}_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2):g_1\odot g_2⊑g\}\\ =\bigvee \{({\mathcal{F}}_k^x(f_1)\ast {\mathcal{G}}_k^x(f_2))\odot ({\mathcal{F}}_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2)):f_1\odot f_2⊑f,g_1\odot g_2⊑g\}\\ \le \bigvee \{({\mathcal{F}}_k^x(f_1)\odot {\mathcal{F}}_k^x(g_1))\ast ({\mathcal{G}}_k^x(f_2)\odot {\mathcal{G}}_k^x(g_2)):f_1\odot f_2⊑f,g_1\odot g_2⊑g\},\\ (\mathrm{since}\ast \mathrm{dominates}\odot )\\ \le \bigvee \{{\mathcal{F}}_k^x(f_1\odot g_1)\ast {\mathcal{G}}_k^x(f_2\odot g_2):(f_1\odot f_2)\odot (g_1\odot g_2)⊑f\odot g\}\\ \le {({\mathcal{F}}^x\ast {\mathcal{G}}^x)}_k(f\odot g).\end{array}$$

If $\ast =\odot$, then for $f=1_X\odot f$, ${({\mathcal{F}}^x\odot {\mathcal{G}}^x)}_k(f)\ge {\mathcal{F}}_k^x(1_X)\odot {\mathcal{G}}_k^x(f)=1_L\odot {\mathcal{G}}_k^x(f)={\mathcal{G}}_k^x(f)$ and ${({\mathcal{F}}^x\odot {\mathcal{G}}^x)}_k(f)\ge {\mathcal{F}}_k^x(f)\odot {\mathcal{G}}_k^x(1_X)={\mathcal{F}}_k^x(f)\odot 1_L={\mathcal{F}}_k^x(f)$. If ${\mathcal{H}}^x$ is $(L_{\odot },K_E)$-fs-filter on X with ${\mathcal{F}}^x⊑{\mathcal{H}}^x$ and ${\mathcal{G}}^x⊑{\mathcal{H}}^x$, we have ${\mathcal{F}}_k^x\odot {\mathcal{G}}_k^x\le {\mathcal{H}}_k^x,\forall k\in K$. Then ${({\mathcal{F}}_k^x\odot {\mathcal{G}}^x)}_k\le {\mathcal{H}}_k^x,\forall k\in K$.

Moreover, if $\ast =\odot$ and ${\mathcal{F}}^x={\mathcal{G}}^x$, then

$$\begin{array}{cc}{({\mathcal{F}}^x\odot {\mathcal{F}}^x)}_k(h)& =\bigvee \{{\mathcal{F}}_k^x(f)\odot {\mathcal{F}}_k^x(g):f\odot g⊑h\}\hfill \\ & \le \bigvee \{{\mathcal{F}}_k^x(f\odot g):f\odot g⊑h\}\hfill \\ & \le {\mathcal{F}}_k^x(h).\hfill \end{array}$$

Thus, ${\mathcal{F}}^x\odot {\mathcal{F}}^x={\mathcal{F}}^x$.

(ii)

Let ${\mathcal{F}}^x$ be a stratified $(L_{\odot },K_E)$-fs-filter, then for each $k\in K$ we have

$$\begin{array}{cc}\beta \odot {({\mathcal{F}}^x\ast {\mathcal{G}}^x)}_k(g)& =\beta \odot \bigvee \{{\mathcal{F}}_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2):g_1\odot g_2⊑g\}\hfill \\ & =\bigvee \{\beta \odot ({\mathcal{F}}_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2)):g_1\odot g_2⊑g\}\hfill \\ & \le \bigvee \{(\beta \odot {\mathcal{F}}_k^x(g_1))\ast {\mathcal{G}}_k^x(g_2):g_1\odot g_2⊑g\}(\mathrm{by}\mathrm{Lemma}1(\mathrm{iv}))\hfill \\ & \le \bigvee \{{\mathcal{F}}_k^x(\beta \odot g_1)\ast {\mathcal{G}}_k^x(g_2):\beta \odot g_1\odot g_2⊑\beta \odot g\}\hfill \\ & \le {({\mathcal{F}}^x\ast {\mathcal{G}}^x)}_k(\beta \odot g).\hfill \end{array}$$

Then, ${\mathcal{F}}^x\ast {\mathcal{G}}^x$ is a stratified $(L_{\odot },K_E)$-fs-filter on X. Similarly if ${\mathcal{G}}^x$ is a stratified $(L_{\odot },K_E)$-fs-filter, then ${\mathcal{F}}^x\ast {\mathcal{G}}^x$ is a stratified $(L_{\odot },K_E)$-fs-filter on X.

(iii)

It can be proved by the same manner as in (i).

(iv)

Suppose that ${\mathcal{T}}^1$ is an enriched $(L_{\odot },K_E)$-fs-topology on X. Then, $\forall k\in K$ we have

$$\begin{array}{cc}{({\mathcal{T}}^1\ast {\mathcal{T}}^2)}_k(\beta \odot g)& =\bigvee \{{\mathcal{T}}_k^1(g_1)\ast {\mathcal{T}}_k^2(g_2):g_1\odot g_2=\beta \odot g\}\hfill \\ & \ge \bigvee \{{\mathcal{T}}_k^1(\beta \odot g_1)\ast {\mathcal{T}}_k^2(g_2):\beta \odot g_1\odot g_2=\beta \odot g\}\hfill \\ & \ge \bigvee \{{\mathcal{T}}_k^1(g_1)\ast {\mathcal{T}}_k^2(g_2):g_1\odot g_2=g\}\hfill \\ & \le {({\mathcal{T}}^1\ast {\mathcal{T}}^2)}_k(g).\hfill \end{array}$$

(v)

Since $H_k^x(g_1\odot g_2)\le H_k^x(g_1)\odot H_k^x(g_2)\le H_k^x(g_1)\ast H_k^x(g_2)$ and $u\to v=\bigvee \{w:u\odot w\le v\}$, by Theorem 6, we have:

$$\begin{array}{cc}{({\mathcal{T}}_{F\ast G})}_k(g)& =\underset{x\in X}{\bigwedge }(H_k^x(g)\to {({\mathcal{F}}^x\ast {\mathcal{G}}^x)}_k(g))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(\underset{g_1\odot g_2⊑g}{\bigvee }{H}_k^x(g_1\odot g_2)\to \underset{g_1\odot g_2⊑g}{\bigvee }({\mathcal{F}}_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2)))\hfill \\ & \ge \underset{x\in X}{\bigwedge }(\underset{g_1\odot g_2⊑g}{\bigvee }(H_k^x(g_1)\odot H_k^x(g_2))\to \underset{g_1\odot g_2⊑g}{\bigvee }({\mathcal{F}}_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2)))\hfill \\ & \ge \underset{x\in X}{\bigwedge }\underset{g_1\odot g_2⊑g}{\bigvee }(H_k^x(g_1)\ast {\mathcal{F}}_k^x(g_1))\to \underset{g_1\odot g_2⊑g}{\bigvee }({\mathcal{F}}_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2)))\hfill \\ & \ge \underset{x\in X}{\bigwedge }\underset{g_1\odot g_2⊑g}{\bigvee }(H_k^x(g_1)\to {\mathcal{F}}_k^x(g_1))\ast ({\mathcal{H}}_k^x(g_2)\to {\mathcal{G}}_k^x(g_2)))\hfill \\ & (\mathrm{by}\mathrm{Lemma}1(\mathrm{v}))\hfill \\ & \ge \underset{g_1\odot g_2⊑g}{\bigvee }(\underset{x\in X}{\bigwedge }(H_k^x(g_1)\to {\mathcal{F}}_k^x(g_1))\ast (\underset{x\in X}{\bigwedge }(H_k^x(g_1)\ast {\mathcal{G}}_k^x(g_2))))\hfill \\ & =\underset{g_1\odot g_2⊑g}{\bigvee }({({\mathcal{T}}_F)}_k(g_1)\ast {({\mathcal{T}}_G)}_k(g_2))\hfill \\ & ={({\mathcal{T}}_F\ast {\mathcal{T}}_G)}_k(g).\hfill \end{array}$$

(vi)

$$\begin{array}{cc}& {({\mathcal{N}}_{{\mathcal{T}}^1}^x\odot {\mathcal{N}}_{{\mathcal{T}}^2}^x)}_k(g)\hfill \\ & =\bigvee \{{({\mathcal{N}}_{{\mathcal{T}}^1}^x)}_k(g_1)\odot {({\mathcal{N}}_{{\mathcal{T}}^2}^x)}_k(g_2):g_1\odot g_2⊑g\}\hfill \\ & =\bigvee \{(\underset{\nu ⊑g_1}{\bigvee }({\mathcal{F}}_k^x(\nu )\ast {\mathcal{T}}_k^1(\nu ))\odot (\underset{\mu ⊑g_2}{\bigvee }({\mathcal{F}}_k^x(\mu )\ast {\mathcal{T}}_k^2(\mu )):g_1\odot g_2⊑g\}\hfill \\ & \le \bigvee \{(\underset{\nu \odot \mu ⊑g_1\odot g_2}{\bigvee }({\mathcal{F}}_k^x(\nu )\odot {\mathcal{F}}_k^x(\mu )\ast ({\mathcal{T}}_k^1(\nu ))\odot {\mathcal{T}}_k^2(\mu ))):g_1\odot g_2⊑g\}\hfill \\ & \le \bigvee \{\underset{\nu \odot \mu ⊑g_1\odot g_2}{\bigvee }({\mathcal{F}}_k^x(\nu \odot \mu ))\ast ({\mathcal{T}}_k^1(\nu )\odot {\mathcal{T}}_k^2(\mu )):g_1\odot g_2⊑g\}\hfill \\ & \le \bigvee \{{\mathcal{F}}_k^x(g_1\odot g_2)\ast {({\mathcal{T}}^1\odot {\mathcal{T}}^2)}_k(g_1\odot g_2)):g_1\odot g_2⊑g\}\hfill \\ & \le {({\mathcal{N}}_{{\mathcal{T}}^1\odot {\mathcal{T}}^2}^x)}_k(g).\hfill \end{array}$$

□

6. Conclusions

In this paper, we introduced the concept of a $(L_{\odot },K_E)$-fs-filter and investigated many of its properties. By using certain operations, we induced $(L_{\odot },K_E)$-fs-topologies from $(L_{\odot },K_E)$-fs-filters. In addition, we introduced the products of $(L_{\odot },K_E)$-fs-filters and the products of $(L_{\odot },K_E)$-fs-topologies and inspected their properties. We illustrated that the product of two $(L_{\odot },K_E)$-fs-filters (where either of them is stratified) is stratified, and the product of two $(L_{\odot },K_E)$-fs-topologies (where either of them is enriched) is enriched.

Since the $(L_{\odot },K_E)$-fs-filter is a useful tool for studying $(L_{\odot },K_E)$-fs-topology, we plan to study more properties about it. Our future work in this direction will be to present and study some type of convergence of an $(L_{\odot },K_E)$-fs-filter in an $(L_{\odot },K_E)$-fs-topological space.