On Soft Lattice Topological Spaces

Abstract

In this paper, we introduce Soft Lattice topological spaces which are defined over a soft lattice L with a fixed set of parameter P and it is also a generalisation of soft topological spaces. The notion of soft L-open sets, soft L-closed sets, soft L-closure, soft L-interior point and soft L-neighbourhood are introduced. Further some basic properties of Soft L-Topology are also investigated.

Keywords:

• Soft set
• soft L-set
• soft topology
• soft L-topology

1. Introduction

In 1999 Molodtsov [1] initiated the novel concept of soft set theory which is completely new approach for modelling, vagueness and uncertainties. Soft set theory has a rich potential for applications in several directions, few of which had been shown by Molodtsov in [1]. Also Maji et al. [2] studied soft sets initiated by Molodtsov [1] and defines equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples and basic operations are also defined. The algebraic structure of set theory dealing with uncertainties has also been studied by some authors [3,4,15–21,23,25,26]. The concept of soft set has been extended to soft lattices and soft fuzzy sets by Li  [5] in the year 2010. Shabir and Naz [6] introduced the concept of soft topological spaces in the year 2011 and studied some basic properties. In our work, we use the notion of soft set initiated by Molodtsov [7] and extend this idea to the field of soft lattices and obtain the topological properties of soft lattices.

Soft Lattice topological spaces (Soft L-topological spaces or Soft L-space) are defined over an initial universe X with a fixed set of parameters P. We discuss some basic properties of soft L-topological spaces and define soft L-open and soft L-closed sets. The soft L-closure of a soft lattice is defined which is, in fact, a generalisation of closure of a set in a broader sense. The newly introduced concept of parameters comes into play with the collection of parameterised topologies on the initial universe. Corresponding to each parameter, we get a topological space and this makes the involvement of parameters more significant. We can say that a soft topological space gives a parameterised family of topologies on the initial universe but the converse is not true i.e. we cannot construct a soft topological space if we are given some topologies for each parameter. Consequently we can say that the soft topological spaces are more generalised than the classical topological spaces.

During the process of theory development, we also see that the properties of parameterised topologies correspond to that of soft topological spaces in some particular situations. Similarly, Soft Lattice topological space is more generalised than Soft topological space.

2. Preliminaries and Basic Definitions

Let L be a complete lattice and its universal bounds denoted by ⊥ and ⊤. Assume that L is consistent i.e. ⊥ is distinct from ⊤. Thus $\mathrm{\perp }\le \alpha \le \mathrm{\top }$ for every $\alpha \in L.$ Also $\vee \varphi =\mathrm{\perp }$ and $\wedge \varphi =\mathrm{\top }$. The two point lattice $\{\mathrm{\perp },\mathrm{\top }\}$ is denoted by 2. A unary operation $\mathrm{\prime }:L⟶L$ is quasi complementation. It is an involution (i.e. ${\alpha }^{{}^{″}}=\alpha$ for all $\alpha \in L$) that inverts the ordering. (i.e. $\alpha \le \beta ⟹{\beta }^{\mathrm{\prime }}\le {\alpha }^{\mathrm{\prime }}$). De Morgan's laws also hold in $(L,\mathrm{\prime })$. (i.e. $(\vee A{)}^{\mathrm{\prime }}=\wedge \{{\alpha }^{\mathrm{\prime }}:\alpha \in A\}$ and $(\wedge A{)}^{\mathrm{\prime }}=\vee \{{\alpha }^{\mathrm{\prime }}:\alpha \in A\}$ for every $A\subset L$). Moreover, ${\mathrm{\perp }}^{\mathrm{\prime }}=\mathrm{\top }$ and ${\mathrm{\top }}^{\mathrm{\prime }}=\mathrm{\perp }$. Based on these facts, in this paper we use a completely distributive lattice $(L,\mathrm{\prime })$ as a complete lattice equipped with an order reserving involution [10].

Definition 2.1

[3,9,22]

Let X be an initial universe set and P be a set of parameters. The power set of X is denoted as $\mathrm{\wp }(X)$ and $B\subset P$. Then a pair $(P,B)$ is said to be a soft set over X, where the mapping P is given by $P:B\to \mathrm{\wp }(X)$.

i.e. a soft set over X is regarded as a parametrised family of subsets of the universe X. For $b\in B$, the set of approximate elements of the soft set $(P,B)$ denoted by $P(b)$.

Definition 2.2

[3,9,22]

Let $(P,A)$ and $(Q,B)$ be soft sets over a common universe X, then $(Q,B)$ is a soft subset of $(P,A)$ if

1. $B\subset A$, and

2. for all $b\in B$, $Q(b)$ and $P(b)$ are identical approximations.

We denote it by $(Q,B)\tilde{\subset }(P,A)$. If $(P,A)$ is a soft subset of $(Q,B)$ then $(Q,B)$ is a soft super set of $(P,A)$ and it is denoted by $(Q,B)\tilde{\supset }(P,A)$.

If $(P,A)$ is a soft subset of $(Q,B)$ and $(Q,B)$ is a soft subset of $(P,A)$, then these two soft sets $(P,A)$ and $(Q,B)$ over a common universe X are said to be soft equal.

Definition 2.3

[3,9,22]

Let $P=\{g_1,g_2,g_3,\dots ,g_i\}$ be a set of parameters. The NOT set of P denoted by $\mathrm{\neg }P$ is defined by $\mathrm{\neg }P=\{\mathrm{\neg }{g}_1,\mathrm{\neg }{g}_2,\mathrm{\neg }{g}_3,\dots ,\mathrm{\neg }{g}_i\}$, where $\mathrm{\neg }{g}_i=not{g}_n$ for all n.

Definition 2.4

[3,9,22]

Let $(Q,B)$ be a soft set, then $(Q,B{)}^c$ is complement or neg-complement of a soft set $(Q,B)$ and is defined by $Q^c=(Q^c,\mathrm{\neg }B)$, where $Q^c:\mathrm{\neg }B⟶\mathrm{\wp }(X)$ is a mapping given by $Q^c(\mathrm{\neg }\beta )=X-Q(\beta )$, for every $\mathrm{\neg }\beta \in \mathrm{\neg }B$. Also $(Q^c{)}^c=Q$ and $((Q,B{)}^c{)}^c=(Q,B)$.

Definition 2.5

[3,9,22]

Let $(Q,B)$ be a soft set over X, then $(Q,B)$ is said to be a NULL soft set if ∀ $b\in B$, $Q(b)=\varphi$,(null-set) and is denoted by $Q_{\varphi }$.

A soft set $(Q,B)$ over X is said to be absolute soft set denoted by $\overline{B}$, if ∀$b\in B$, $Q(b)=X$. Clearly ${\overline{B}}^c=Q_{\varphi }$, $Q_{\varphi }^c=\overline{B}$.

Definition 2.6

[5,8,24]

Let triplet $M=(f,X,L)$, where L is a complete lattice, $f:X⟶\mathrm{\wp }(L)$ is a mapping, X is a universe set, then M is called the soft lattice denoted by $f_P^L$.

i.e. for every $x\in X$, $f_P^L$ is a soft lattice over L, if $f(x)$ is a sub lattice of L.

Definition 2.7

[5,8,24]

For two soft lattice $M=(f,X,L),N=(g,Y,L)$ over a common complete lattice, we say $M=(f,X,L)$ is a soft sub lattice of $N=(g,Y,L)$ if

1. $X\subset Y$ $\mathrm{\forall }ϵ\in X$.

2. $g(\epsilon )$ is extension of $f(\epsilon )$.

It is denoted by $(f,X,L)\subset (g,Y,L)$ ie, $f_X^L\subset f_Y^L$.

3. Soft Lattice Topology

Definition 3.1

The difference of two soft lattices $f_P^L$ and $g_P^L$ over L, denoted by $h_P^L=f_P^L\setminus g_P^L$, is defined as $h(p)=f(p)\setminus g(p)$ for all $p\in P$.

Example 3.1

Let $L=\{l_1,l_2,l_3\}$ be the lattice and $P=\{p_1,p_2\}$ be the parameter. Then $f_P^L$ and $g_P^L$ are two soft lattices over L which is defined as follows:

$f(p_1)=\{l_1,l_2\},g(p_1)=\{l_1\}$,

$f(p_2)=\{l_1,l_3\},g(p_2)=\{l_3\}$,

Let $h_P^L=f_P^L\setminus g_P^L$.

Then $h(p_1)=f(p_1)\setminus g(p_1)=\{l_1,l_2\}-\{l_1\}=\{l_2\}$ and

$h(p_2)=f(p_2)\setminus g(p_2)=\{l_1,l_3\}-\{l_3\}=\{l_1\}$.

Definition 3.2

Let $f_P^L$ be a soft lattice over L and $x\in L$. We say that $x\in f_P^L$ whenever $x\in f(a)$ for all $a\in P$.

Note: For any $x\in L$, $x\notin f_P^L$, if $x\notin f(a)$ for some $a\in P$.

Example 3.2

Let $L=\{l_1,l_2,l_3\}$ be the lattice, $P=\{p_1,p_2\}$ be the parameter and $f_P^L$ and $g_P^L$ are two soft lattices over L.

Let $l_1\in L$, then we say $l_1\in f_P^L$ whenever $l_1\in f(p_1)$ for all $p_1\in P$.

Similarly, let $l_3\in L$, then we say $l_3\in g_P^L$ whenever $l_3\in g(p_2)$ for all $p_2\in P$.

Definition 3.3

Let $f_P^L$ be a soft lattice over L and Y be a non-empty subset of L. Then the sub soft lattice of $f_P^L$ over Y denoted by $Y{f}_P^L$ is defined as $Yf(a)=Y\cap f(a)$, for every $a\in P$.

i.e. $Y{f}_P^L=Y\cap f_P^L$.

Example 3.3

Let $L=\{l_1,l_2,l_3,l_4\}$ be the lattice and $P=\{p_1,p_2\}$ be the parameter. Then $f_P^L$ be a soft lattices over L which is defined as $f(p_1)=\{l_1,l_2\},f(p_2)=\{l_3,l_4\}$.

Let $Y=\{l_1,l_2,l_3\}$ be the subset of L. Then sub soft lattice of $f_P^L$ over Y is defined by

$Yf(p_1)=Y\cap f(p_1)=\{l_1,l_2,l_3\}\cap \{l_1,l_2\}$

$Yf(p_1)=\{l_1,l_2\}$.

Also $Yf(p_2)=Y\cap f(p_2)=\{l_1,l_2,l_3\}\cap \{l_3,l_4\}$

$Yf(p_1)=\{l_3\}$.

Definition 3.4

The relative complement of a soft lattice $f_P^L$ is denoted by $(f_P^L{)}^{\mathrm{\prime }}$ and is defined as $(f_P^L{)}^{\mathrm{\prime }}=(f_P^{\mathrm{\prime }L})$ where $f^{\mathrm{\prime }}:P⟶\mathrm{\wp }(L)$ is a mapping given by $f^{\mathrm{\prime }}(a)=L-f(a)$ for all $a\in P$.

Example 3.4

Let $L=\{l_1,l_2\}$ be the lattice and $P=\{p_1,p_2\}$ be the parameter. Then $f_P^L$ be a soft lattices over L which is defined as $f(p_1)=\{l_1\},f(p_2)=\{l_2\}$

Now the relative complement of $f_P^L$ denoted by $(f_P^L{)}^{\mathrm{\prime }}$ is given by

$f^{\mathrm{\prime }}(p_1)=L-f(p_1)=\{l_1,l_2\}-\{l_1\}=\{l_2\}$

$f^{\mathrm{\prime }}(p_2)=L-f(p_2)=\{l_1,l_2\}-\{l_2\}=\{l_1\}$.

Definition 3.5

Let X be an initial universe set and P be the non-empty set of parameters.

Let τ be the collection of complete, uniquely complemented soft lattices over L, then τ is said to be a soft lattice topology on L if;

1. $\varphi ,L$ belongs to τ.

2. The union of any number of soft lattices in τ belongs to τ.

3. The intersection of any two soft lattices in τ belongs to τ.

Then the triplet $(L,\tau ,P)$ is called a Soft Lattice topological space (soft L-space or soft L-topological space) over L.

Example 3.5

Suppose $L=\{l_1,l_2,l_3\},P=\{p_1,p_2\}$ and $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L\}$, where $f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_2\},f_1(p_2)=\{l_1\}$,

$f_2(p_1)=\{l_2,l_3\},f_2(p_2)=\{l_1,l_2\}$,

$f_3(p_1)=\{l_1,l_2\},f_3(p_2)=\{L\}$,

$f_4(p_1)=\{l_1,l_2\},f_4(p_2)=\{l_1,l_3\}$

Therefore τ is a soft lattice topology.

So $(L,\tau ,P)$ is a soft L-topological space.

Further, ${\tau }_{p_1}=\{\varphi ,L,\{l_2\},\{l_2,l_3\},\{l_1,l_2\}\}$

and ${\tau }_{p_2}=\{\varphi ,L,\{l_1\},\{l_1,l_3\},\{l_1,l_2\}\}$ are topologies on L.

Hence these collections based on each parameter gives a soft lattice topology on L.

Example 3.6

Suppose $L=\{l_1,l_2,l_3,l_4,l_5\},P=\{p_1,p_2,p_3,p_4,p_5\}$ and $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L\}$, where $f_{1P}^L,f_{2P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_2,l_4\},f_2(p_1)=\{l_2\}$,

$f_1(p_2)=\{l_1,l_2\},f_2(p_2)=\{l_1\}$,

$f_1(p_3)=\{l_1,l_3,l_4,l_5\},f_2(p_3)=\{l_1,l_2,l_3\}$,

$f_1(p_4)=\{\mathrm{\varnothing }\},f_2(p_4)=\{L\}$,

$f_1(p_5)=\{l_1\},f_2(p_5)=\{l_1,l_4,l_3\}$.

Therefore τ is a soft lattice topology.

So $(L,\tau ,P)$ is a soft lattice topological space over L.

Further, ${\tau }_{p_1}=\{\varphi ,L,\{l_2\},\{l_2,l_4\}\}$,

${\tau }_{p_2}=\{\varphi ,L,\{l_1\},\{l_1,l_3\}\}$,

${\tau }_{p_3}=\{\varphi ,L,\{l_1,l_2,l_3\},\{l_1,l_3,l_4,l_5\}\}$,

${\tau }_{p_4}=\{\varphi ,L\}$,

${\tau }_{p_5}=\{\varphi ,L,\{l_1\},\{l_1,l_3,l_4\}\}$

are topologies on L.

Hence these collections based on each parameter gives a soft lattice topology on L.

Example 3.7

Let $L=\{0,1,l_1,l_2,l_3\}$ be the lattice where $l_1,l_2,l_3$ represents the students of class 12, $P=\{p_1,p_2\}$ be the parameter in which $p_1$: brilliant and $p_2$: average.

Let us consider a collection $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L\}$, where $f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L$ are soft lattices over L in which $f_1,f_2,f_3,f_4$ represents subjects like Mathematics, Physics, Chemistry, Computer science respectively. It is defined as follows,

$f_1(p_1)=\{l_2\},f_1(p_2)=\{l_1\}$,

$f_2(p_1)=\{l_2,l_3\},f_2(p_2)=\{l_1,l_2\}$,

$f_3(p_1)=\{l_1,l_2\},f_3(p_2)=\{L\}$,

$f_4(p_1)=\{l_1\},f_4(p_2)=\{l_1,l_3\}$

Therefore τ is a soft lattice topology.

Hence $(L,\tau ,P)$ is a soft L-topological space.

Further, ${\tau }_{p_1}=\{\varphi ,L,\{l_1\},\{l_2\},\{l_2,l_3\},\{l_1,l_2\}\}$

and ${\tau }_{p_2}=\{\varphi ,L,\{l_1\},\{l_1,l_3\},\{l_1,l_2\}\}$ are topologies on L.

Hence these collections based on each parameter gives a soft lattice topology on L.

Definition 3.6

Let $(L,\tau ,P)$ be a soft lattice topological space over L, then the members of τ are said to be soft L-open sets in L.

Definition 3.7

Let $(L,\tau ,P)$ be a soft lattice topological space over L. A soft lattice $f_P^L$ over L is said to be a soft L-closed set in L, if its relative complement $(f_P^L{)}^{\mathrm{\prime }}$ belongs to τ.

Definition 3.8

Let L be a lattice, P be the set of parameters and $\tau =\{\varphi ,L\}$. Then τ is called the soft indiscrete lattice topology on L and $(L,\tau ,P)$ is said to be a soft indiscrete lattice topological space over L.

Definition 3.9

Let L be a lattice, P be the set of parameters and let τ be the collection of all soft lattices which can be defined over L. Then τ is called the soft discrete lattice topology on L and $(L,\tau ,P)$ is said to be a soft discrete lattice topological space over L.

Proposition 3.1

Let $f_P^L$ and $g_P^L$ be two soft lattices over L. Then $(f_P^L\cup g_P^L{)}^{\mathrm{\prime }}=f_P^{\mathrm{\prime }L}\cap g_P^{\mathrm{\prime }L}$.

Proof.

Let $f_P^L\cup g_P^L=h_P^L$, where, $h(p)=f(p)\cup g(p)$, for all $p\in P$.

Then $\begin{array}{rl}{h}^{\mathrm{\prime }}(p)& =(f(p)\cup g(p){)}^{\mathrm{\prime }}\\ & =(f(p){)}^{\mathrm{\prime }}\cap (g(p){)}^{\mathrm{\prime }}\\ & =(f^{\mathrm{\prime }}(p))\cap (g^{\mathrm{\prime }}(p))\end{array}$ for all $p\in P$.

Thus $(h_P^L{)}^{\mathrm{\prime }}=f_P^{\mathrm{\prime }L}\cap g_P^{\mathrm{\prime }L}$.

Hence $(f_P^L\cup g_P^L{)}^{\mathrm{\prime }}=f_P^{\mathrm{\prime }L}\cap g_P^{\mathrm{\prime }L}$.

Proposition 3.2

Let $f_P^L$ and $g_P^L$ be two soft lattices over L. Then $(f_P^L\cap g_P^L{)}^{\mathrm{\prime }}=f_P^{\mathrm{\prime }L}\cup g_P^{\mathrm{\prime }L}$.

Proof.

Let $f_P^L\cap g_P^L=h_P^L$, where, $h(p)=f(p)\cap g(p)$, for all $p\in P$.

Then $\begin{array}{rl}{h}^{\mathrm{\prime }}(p)& =(f(p)\cap g(p){)}^{\mathrm{\prime }}\\ & =(f(p){)}^{\mathrm{\prime }}\cup (g(p){)}^{\mathrm{\prime }}\\ & =(f^{\mathrm{\prime }}(p))\cup (g^{\mathrm{\prime }}(p))\end{array}$ for all $p\in P$.

Thus $(h_P^L{)}^{\mathrm{\prime }}=f_P^{\mathrm{\prime }L}\cup g_P^{\mathrm{\prime }L}$.

Hence $(f_P^L\cap g_P^L{)}^{\mathrm{\prime }}=f_P^{\mathrm{\prime }L}\cup g_P^{\mathrm{\prime }L}$.

Proposition 3.3

Let $(L,\tau ,P)$ be a soft L-space. Then

(1)	$\mathrm{\varnothing },L$ are closed soft lattices over L.
(2)	The intersection of infinite number of soft L-closed sets is a soft L-closed set over L.
(3)	The finite union of soft L-closed sets is a soft L-closed set over L.

Proof.

(1) is obvious.

(2) Consider two soft lattices, say, $(f,X,L)$, and $(g,X,L)$ over Y.

Then

$((f,X,L)\cup (g,X,L))=(h,X,L)$; where $\mathrm{\forall }a\in X,h(a)=f(a)\cup g(a)$; $\begin{array}{rl}{h}^{\mathrm{\prime }}(a)& =(f(a)\cup g(a){)}^{\mathrm{\prime }};\mathrm{\forall }a\in X;\\ & =(f(a){)}^{\mathrm{\prime }}\cap (g(a){)}^{\mathrm{\prime }}\\ & =f^{\mathrm{\prime }}(a)\cap g^{\mathrm{\prime }}(a).\end{array}$ Thus we have, $(h,X,L{)}^{\mathrm{\prime }}=((f,X,L)\cup (g,X,L){)}^{\mathrm{\prime }}$

$=((f,X,L){)}^{\mathrm{\prime }}\cap ((g,X,L){)}^{\mathrm{\prime }}$;

i.e.; $((f,X,L)\cup (g,X,L){)}^{\mathrm{\prime }}=((f,X,L){)}^{\mathrm{\prime }}\cap ((g,X,L){)}^{\mathrm{\prime }}$

and this we can extend to any no. of soft lattices.

Since, RHS leads intersection of closed soft lattices, and $(LHS{)}^{\mathrm{\prime }}$ also being in τ.

(3) On the other hand we want to prove, $((f,X,L)\cap (g,X,L){)}^{\mathrm{\prime }}=(f,X,L{)}^{\mathrm{\prime }}\cup (g,X,L{)}^{\mathrm{\prime }}$

Consider two soft lattices, say, $(f,X,L)$, and $(g,X,L)$ over Y.

Then

$((f,X,L)\cap (g,X,L))=(h,X,L)$; where $\mathrm{\forall }a\in X,h(a)=f(a)\cap g(a)$; $\begin{array}{rl}{h}^{\mathrm{\prime }}(a)& =(f(a)\cap g(a){)}^{\mathrm{\prime }};\mathrm{\forall }a\in X;\\ & =(f(a){)}^{\mathrm{\prime }}\cup (g(a){)}^{\mathrm{\prime }}\\ & =f^{\mathrm{\prime }}(a)\cup g^{\mathrm{\prime }}(a).\end{array}$ Thus we have, $\begin{array}{rl}(h,X,L{)}^{\mathrm{\prime }}& =((f,X,L)\cap (g,X,L){)}^{\mathrm{\prime }}\\ & =((f,X,L){)}^{\mathrm{\prime }}\cup ((g,X,L){)}^{\mathrm{\prime }};\end{array}$ i.e.; $((f,X,L)\cap (g,X,L){)}^{\mathrm{\prime }}=((f,X,L){)}^{\mathrm{\prime }}\cup ((g,X,L){)}^{\mathrm{\prime }}$.

Definition 3.10

Let $(L,\tau ,P)$ be a soft L-topological space. Then the collection $P(\tau )=\{{\tau }_a:a\in P\}$ denotes the paramterised family of topologies induced by the soft L-topology τ.

Proposition 3.4

Let $(L,\tau ,P)$ be a soft L-space. Then the set ${\tau }_a=\{f(a)⏐f_P^L\in \tau \}$ for all $a\in P$ gives a topology on L.

Proof.

From definition 3.3, for any $a\in P$, we have ${\tau }_a=\{f(a)⏐f_P^L\in \tau \}$.Now,

$(i)$ $\varphi ,L\in \tau$ implies that $\varphi ,L\in {\tau }_a$.

$(ii)$ Let $\{f_i(a)⏐i\in I\}$ be a collection of sets in ${\tau }_a$.

Since $f_{iP}^L\in \tau ,\mathrm{\forall }i\in I$ so that ${\cup }_{i\in P}{f}_{iP}^L\in \tau$, thus ${\cup }_{i\in P}{f}_i(a)\in {\tau }_a$.

$(iii)$ Let $f(a),g(a)\in {\tau }_a$ for some soft lattices $f_P^L,g_P^L\in \tau$.

Since $f_P^L\cap g_P^L\in \tau$ ;

$⟹f_P^L\cap g_P^L\in {\tau }_a$.

which describes all sufficient conditions for a topology for ${\tau }_a$ on L for each $a\in P$.

The above proposition shows that based on each parameter $a\in P$, we have a topology on L. Therefore a soft L-topology gives a parameterised family of topologies on L.

Remark 3.1

The following example is to show that the converse of above proposition does not hold.

Example 3.8

Let $L=\{l_1,l_2,l_3\},P=\{p_1,p_2\}$ and $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L\}$, where $f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_2\},f_1(p_2)=\{l_1\}$,

$f_2(p_1)=\{l_1,l_2\},f_2(p_2)=\{l_1,l_2\}$,

$f_3(p_1)=\{l_2,l_3\},f_3(p_2)=\{l_1,l_2\}$,

$f_4(p_1)=\{l_2\},f_4(p_2)=\{l_1,l_3\}$,

Then τ is not a soft L-topology because $f_{2P}^L\cup f_{3P}^L=g_P^L$, where $g(p_1)=L$ and $g(p_2)=\{l_1,l_2\}$ and so $g_P^L\notin \tau$.

Also, ${\tau }_{p_1}=\{\varphi ,L,\{l_2\},\{l_2,l_3\},\{l_1,l_2\}\}$

and ${\tau }_{p_2}=\{\varphi ,L,\{l_1\},\{l_1,l_3\},\{l_1,l_2\}\}$ are topologies on L.

Hence this example shows that any collections of soft lattices need not to be a soft L-topology, even if the collection corresponding to each parameter defines a topology on L.

Remark 3.2

Using the Definition 3.10, the following proposition follows.

Proposition 3.5

Let $(L,\tau ,P)$ be a soft L-topological space over L with parameter space P. Then $|P(\tau )|\le |P|$ and $|{\tau }_a|\le |\tau |$ for every $a\in P$.

Proof.

Let τ be a soft L-topological space over L with parameter space P.

Define $\gamma :P⟶P(\tau )$ by $\gamma (a)={\tau }_a$.

Clearly, γ is onto but it need not be one-to-one.

This implies $|P(\tau )|\le |P|$.

Again define ${\lambda }_a:\tau ⟶{\tau }_a$ by ${\lambda }_a(f)=f(a)$.

Clearly, ${\lambda }_a$ is onto but it need not be one-to-one.

This implies $|{\tau }_a|\le |\tau |$.

Example 3.9

Let $L=\{l_1,l_2,l_3\},P=\{p_1,p_2\}$ and a collection $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,f_{5P}^L,f_{6P}^L,f_{7P}^L,f_{8P}^L,f_{9P}^L\}$, where $f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,f_{5P}^L,f_{6P}^L,f_{7P}^L,f_{8P}^L,f_{9P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_2\},f_1(p_2)=\{l_1\}$,

$f_2(p_1)=\{l_2,l_3\},f_2(p_2)=\{l_1,l_2\}$,

$f_3(p_1)=\{l_1,l_2\},f_3(p_2)=\{l_1,l_2\}$,

$f_4(p_1)=\{l_1,l_2\},f_4(p_2)=\{l_1,l_3\}$,

$f_5(p_1)=\{L\},f_5(p_2)=\{l_1,l_2\}$,

$f_6(p_1)=\{l_2\},f_6(p_2)=\{l_1,l_2\}$,

$f_7(p_1)=\{l_2,l_3\},f_7(p_2)=\{L\}$,

$f_8(p_1)=\{l_1,l_2\},f_8(p_2)=\{L\}$,

$f_9(p_1)=\{l_2\},f_9(p_2)=\{L\}$,

Then τ defines a soft L-topology on L and $(L,\tau ,P)$ is a soft L-topological space over L.

Also, ${\tau }_{p_1}=\{\varphi ,L,\{l_2\},\{l_2,l_3\},\{l_1,l_2\}\}$ and

${\tau }_{p_2}=\{\varphi ,L,\{l_1\},\{l_1,l_2\},\{l_1,l_3\}\}$ are topologies on L.

Define $\gamma :P⟶P(\tau )$ by $\gamma (p_1)={\tau }_{p_1}$ and $\gamma (p_2)={\tau }_{p_2}$.

Here $p_1\ne p_2$ and ${\tau }_{p_1}\ne {\tau }_{p_2}$.

Since $\gamma (p_1)\ne \gamma (p_2)$, γ is one-to-one.

Here $|P(\tau )|=2,|P|=2$ and hence $|P(\tau )|=|P|$.

Now let us define ${\lambda }_{p_1}:\tau ⟶{\tau }_{p_1}$ by ${\lambda }_{p_1}(f_P^L)=f(p_1)$.

Also $f_1(p_1)=f_6(p_1)$ but $f_{1P}^L\ne f_{6P}^L$.

Since ${\lambda }_{p_1}(f_{1P}^L)={\lambda }_{p_1}(f_{6P}^L)=\{l_2\}$, ${\lambda }_{p_1}$ is not one-to-one.

Here $|{\tau }_{p_1}|=5,|\tau |=11$ and hence $|{\tau }_{p_1}|<|\tau |$.

Again let us define ${\lambda }_{p_2}:\tau ⟶{\tau }_{p_2}$ by ${\lambda }_{p_2}(f_P^L)=f(p_2)$.

Also $f_2(p_2)=f_3(p_2)$ but $f_{2P}^L\ne f_{3P}^L$.

Since ${\lambda }_{p_2}(f_{1P}^L)={\lambda }_{p_2}(f_{3P}^L)=\{l_1,l_2\}$, ${\lambda }_{p_2}$ is not one-to-one.

Here $|{\tau }_{p_2}|=5,|\tau |=11$ and hence $|{\tau }_{p_2}|<|\tau |$.

Example 3.10

Let $L=\{l_1,l_2\},P=\{p_1,p_2\}$ and a collection $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,f_{5P}^L,f_{6P}^L\}$, where $f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,f_{5P}^L,f_{6P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_2\},f_1(p_2)=\{l_2\}$,

$f_2(p_1)=\{L\},f_2(p_2)=\{l_2,l_3\}$,

$f_3(p_1)=\{l_2\},f_3(p_2)=\{L\}$,

$f_4(p_1)=\{l_2\},f_4(p_2)=\{l_2,l_3\}$,

$f_5(p_1)=\{l_2,l_3\},f_5(p_2)=\{L\}$,

$f_6(p_1)=\{l_2,l_3\},f_6(p_2)=\{l_2,l_3\}$,

Then τ defines a soft L-topology on L and $(L,\tau ,P)$ is a soft L-topological space over L.

Also, ${\tau }_{p_1}=\{\varphi ,L,\{l_2\},\{l_2,l_3\}\}$ and

${\tau }_{p_2}=\{\varphi ,L,\{l_2\},\{l_2,l_3\}\}$ are topologies on L.

Define $\gamma :P⟶P(\tau )$ by $\gamma (p_1)={\tau }_{p_1}$ and $\gamma (p_2)={\tau }_{p_2}$.

Here $p_1\ne p_2$ and ${\tau }_{p_1}={\tau }_{p_2}$.

Since $\gamma (p_1)=\gamma (p_2)$, γ is not one-to-one.

Here $|P(\tau )|=1,|P|=2$ and hence $|P(\tau )|<|P|$.

Now let us define ${\lambda }_{p_1}:\tau ⟶{\tau }_{p_1}$ by ${\lambda }_{p_1}(f_P^L)=f(p_1)$.

Also $f_1(p_1)=f_3(p_1)$ but $f_{1P}^L\ne f_{3P}^L$.

Since ${\lambda }_{p_1}(f_{1P}^L)={\lambda }_{p_1}(f_{3P}^L)=\{l_2\}$, ${\lambda }_{p_1}$ is not one-to-one.

Here $|{\tau }_{p_1}|=4,|\tau |=18$ and hence $|{\tau }_{p_1}|<|\tau |$.

Again let us define ${\lambda }_{p_2}:\tau ⟶{\tau }_{p_2}$ by ${\lambda }_{p_2}(f_P^L)=f(p_2)$.

Also $f_2(p_2)=f_4(p_2)$ but $f_{2P}^L\ne f_{4P}^L$.

Since ${\lambda }_{p_2}(f_{1P}^L)={\lambda }_{p_2}(f_{4P}^L)=\{l_2,l_3\}$, ${\lambda }_{p_2}$ is not one-to-one.

Here $|{\tau }_{p_2}|=4,|\tau |=8$ and hence $|{\tau }_{p_2}|<|\tau |$.

Example 3.11

Let $L=\{l_1,l_2\},P=\{p_1,p_2\}$ and a collection $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L\}$, where $f_{1P}^L,f_{2P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_2,l_3\},f_1(p_2)=\{l_2,l_3\}$,

$f_2(p_1)=\{l_2\},f_2(p_2)=\{l_2\}$,

Then τ defines a soft L-topology on L and $(L,\tau ,P)$ is a soft L-topological space over L.

Also, ${\tau }_{p_1}=\{\varphi ,L,\{l_2\},\{l_2,l_3\}\}$ and

${\tau }_{p_2}=\{\varphi ,L,\{l_2\},\{l_2,l_3\}\}$ are topologies on L.

Define $\gamma :P⟶P(\tau )$ by $\gamma (p_1)={\tau }_{p_1}$ and $\gamma (p_2)={\tau }_{p_2}$.

Here $p_1\ne p_2$ and ${\tau }_{p_1}={\tau }_{p_2}$.

Since $\gamma (p_1)=\gamma (p_2)$, γ is not one-to-one.

Here $|P(\tau )|=1,|P|=2$ and hence $|P(\tau )|<|P|$.

Now let us define ${\lambda }_{p_1}:\tau ⟶{\tau }_{p_1}$ by ${\lambda }_{p_1}(f_P^L)=f(p_1)$.

Also $f_1(p_1)\ne f_2(p_1)$ but $f_{1P}^L\ne f_{2P}^L$.

Since ${\lambda }_{p_1}(f_{1P}^L)={\lambda }_{p_1}(f_{2P}^L)=\{l_2,l_3\}$, ${\lambda }_{p_1}$ is one-to-one.

Here $|{\tau }_{p_1}|=4,|\tau |=4$ and hence $|{\tau }_{p_1}|=|\tau |$.

Again let us define ${\lambda }_{p_2}:\tau ⟶{\tau }_{p_2}$ by ${\lambda }_{p_2}(f_P^L)=f(p_2)$.

Also $f_1(p_2)\ne f_2(p_2)$ but $f_{1P}^L\ne f_{2P}^L$.

Since ${\lambda }_{p_2}(f_{1P}^L)={\lambda }_{p_2}(f_{2P}^L)=\{l_2\}$, ${\lambda }_{p_2}$ is one-to-one.

Here $|{\tau }_{p_2}|=4,|\tau |=4$ and hence $|{\tau }_{p_2}|=|\tau |$.

Proposition 3.6

Suppose $(L,{\tau }_1,P)$ and $(L,{\tau }_2,P)$ be two soft L-topological spaces over the same universe X, then $(L,{\tau }_1\cap {\tau }_2,P)$ is a soft lattice topological space over L.

Proof.

(i) $\varphi ,L$ belong to ${\tau }_1\cap {\tau }_2$.

(ii) Let $\{f_{iP}^L⏐i\in I\}$ be a family of soft sets in ${\tau }_1\cap {\tau }_2$.

Then $f_{iP}^L\in {\tau }_1$ and $f_{iP}^L\in {\tau }_2$, for all $i\in I$.

So ${\cup }_{i\in P}{f}_{iP}^L\in {\tau }_1$ and ${\cup }_{i\in P}{f}_{iP}^L\in {\tau }_2$.

Thus ${\cup }_{i\in P}{f}_{iP}^L\in {\tau }_1\cap {\tau }_2$.

$(iii)$ Let $f_P^L,g_P^L\in {\tau }_1\cap {\tau }_2$.Then $f_P^L,g_P^L\in {\tau }_1$ and ${\tau }_2$.

Since $f_P^L\cap g_P^L\in {\tau }_1$ and $f_P^L\cap g_P^L\in {\tau }_2$

$⟹f_P^L\cap g_P^L\in {\tau }_1\cap {\tau }_2$.

Thus ${\tau }_1\cap {\tau }_2$ gives a soft lattice topology on L and $(L,{\tau }_1\cap {\tau }_2,P)$ is a soft lattice topological space over L.

Remark 3.3

The finite union of soft lattice topologies on L may not be a soft lattice topology on L.

This remark can be explained through the following example.

Example 3.12

Let $L=\{l_1,l_2,l_3\},P=\{p_1,p_2\}$ and ${\tau }_1=\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L\}$, ${\tau }_2=\{\varphi ,L,g_{1P}^L,g_{2P}^L,g_{3P}^L,g_{4P}^L\}$ where $f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,g_{1P}^L,g_{2P}^L,g_{3P}^L,g_{4P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_2\},f_1(p_2)=\{l_1\}$,

$f_2(p_1)=\{l_2,l_3\},f_2(p_2)=\{l_1,l_2\}$,

$f_3(p_1)=\{l_1,l_2\},f_3(p_2)=\{L\}$,

$f_4(p_1)=\{l_1,l_2\},f_4(p_2)=\{l_1,l_3\}$

and

$g_1(p_1)=\{l_2\},g_1(p_2)=\{l_1\}$,

$g_2(p_1)=\{l_2,l_3\},g_2(p_2)=\{l_1,l_2\}$,

$g_3(p_1)=\{l_1,l_2\},g_3(p_2)=\{l_1,l_2\}$,

$g_4(p_1)=\{l_2\},g_4(p_2)=\{l_1,l_3\}$

Now, we define $\begin{array}{rl}\tau & ={\tau }_1\cup {\tau }_2\\ & =\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,g_{1P}^L,g_{2P}^L,g_{3P}^L,g_{4P}^L\}\end{array}$ If we take, $f_{2P}^L\cup g_{3P}^L=h_P^L$.

Then $\begin{array}{rl}h(p_1)& =f_2(p_1)\cup g_3(p_1)\\ & =\{l_2,l_3\}\cup \{l_1,l_2\}\\ & =L\end{array}$ and $\begin{array}{rl}h(p_2)& =f_2(p_2)\cup g_3(p_2)\\ & =\{l_1,l_2\}\cup \{l_1,l_2\}\\ & =\{l_1,l_2\}\end{array}$ But $h_P^L\notin \tau$, thus τ is not a soft lattice topology on L.

Definition 3.11

Let $(L,\tau ,P)$ be a soft lattice topological space over L and $f_P^L$ be a soft lattice over L. Then the soft lattice closure of $f_P^L$, denoted by ${\overline{f}}_P^L$, is the intersection of all soft L-closed super sets of $f_P^L$.

Note: It is clear that, ${\overline{f}}_P^L$ is the smallest soft L-closed set which contains $f_P^L$, by Proposition 3.3.

Theorem 3.1

Let $(L,\tau ,P)$ be a soft lattice topological space over L, $f_P^L$ and $g_P^L$ are two soft lattices over L. Then

(i)	$\overline{\mathrm{\varnothing }}=\mathrm{\varnothing }$ and $\overline{L}=L$.
(ii)	$f_P^L\tilde{\subset }\overline{f_P^L}$.
(iii)	$f_P^L$ is a soft L-closed set iff $f_P^L=\overline{f_P^L}$.
(iv)	$\overline{\overline{f_P^L}}=\overline{f_P^L}$.
(v)	$f_P^L\tilde{\subset }{g}_P^L⟹\overline{f_P^L}\tilde{\subset }\overline{g_P^L}$.
(vi)	$\overline{f_P^L\cup g_P^L}=\overline{f_P^L}\cup \overline{g_P^L}$.
(vii)	$\overline{f_P^L\cap g_P^L}\tilde{\subset }\overline{f_P^L}\cap \overline{g_P^L}$.

Proof.

(i) and (ii) are obvious.

$(iii)$ If $f_P^L$ is a soft L-closed set, then $\overline{f_P^L}$ is itself a soft L-closed set which contains $f_P^L$.

So $\overline{f_P^L}$ is the smallest soft L-closed set containing $f_P^L$ and $f_P^L=\overline{f_P^L}$.

Conversely, suppose that $f_P^L=\overline{f_P^L}$.

Since $\overline{f_P^L}$ is a soft L-closed set, then $f_P^L$ is a soft L-closed set.

$(iv)$ Since $\overline{f_P^L}$ is a soft L-closed set, by (iii) $\overline{\overline{f_P^L}}=\overline{f_P^L}$.

$(v)$ Suppose $f_P^L\tilde{\subset }{g}_P^L$.

Therefore all soft L-closed super set of $g_P^L$ will also contain $f_P^L$.

i.e. any soft L-closed super set of $g_P^L$ is also a soft L-closed super set of $f_P^L$.

Hence the intersection of soft L-closed super sets of $f_P^L$ is contained in the soft intersection of soft L-closed super sets of $g_P^L$.

Thus $\overline{f_P^L}\tilde{\subset }\overline{g_P^L}$.

$(vi)$ Since $f_P^L\tilde{\subset }{f}_P^L\cup g_P^L$ and $g_P^L\tilde{\subset }{f}_P^L\cup g_P^L$, by (v),

$\overline{f_P^L}\tilde{\subset }\overline{f_P^L\cup g_P^L}$ and $\overline{g_P^L}\tilde{\subset }\overline{f_P^L\cup g_P^L}$.

Thus $\overline{f_P^L}\cup \overline{g_P^L}\tilde{\subset }\overline{f_P^L\cup g_P^L}$.

Conversely, suppose $f_P^L\tilde{\subset }\overline{f_P^L}$ and $g_P^L\tilde{\subset }\overline{g_P^L}$.

So $f_P^L\cup g_P^L\tilde{\subset }\overline{f_P^L}\cup \overline{g_P^L}$.

Since $\overline{f_P^L}\cup \overline{g_P^L}$ is a soft L-closed set being the union of two soft L-closed sets.

Therefore $\overline{f_P^L\cup g_P^L}\tilde{\subset }\overline{f_P^L}\cup \overline{g_P^L}$.

Hence $\overline{f_P^L\cup g_P^L}=\overline{f_P^L}\cup \overline{g_P^L}$.

$(vii)$ Since $f_P^L\cap g_P^L\tilde{\subset }{f}_P^L$ and $f_P^L\cap g_P^L\tilde{\subset }{g}_P^L$, by (v),

$\overline{f_P^L\cap g_P^L}\tilde{\subset }\overline{f_P^L}$ and $\overline{f_P^L\cap g_P^L}\tilde{\subset }\overline{g_P^L}$.

Hence $\overline{f_P^L\cap g_P^L}\tilde{\subset }\overline{f_P^L}\cap \overline{g_P^L}$.

Definition 3.12

Let $(L,\tau ,P)$ be a soft lattice topological space over L and $f_P^L$ be a soft lattice over L. Then we associate with $f_P^L$, a soft lattice L, denoted by ${\overline{f}}_P^L$ and defined as $\overline{f}(\alpha )=\overline{f(\alpha )}$, where $\overline{f(\alpha )}$ is the soft L-closure of $f(\alpha )$ in ${\tau }_{\alpha }$ for each $\alpha \in P$.

Proposition 3.7

Let $(L,\tau ,P)$ be a soft lattice topological space over L and $f_P^L$ be a soft lattice over L. Then ${\overline{f}}_P^L\tilde{\subset }\overline{f_P^L}$.

Proof.

For any $a\in P$, $\overline{f(a)}$ is the smallest soft L-closed set in $(L,{\tau }_a)$ which contains $f(a)$.

Moreover if $f_P^L=h_P^L$, then $h(a)$ is also a soft L-closed set in $(L,{\tau }_a)$ containing $f(a)$.

This implies that ${\overline{f}}_P^L=\overline{f_P^L}\subseteq h(a)$.

Thus ${\overline{f}}_P^L\tilde{\subset }\overline{f_P^L}$.

Corollary 3.1

Let $(L,\tau ,P)$ be a soft lattice topological space over L, and $f_P^L$ be a soft lattice over L. Then ${\overline{f}}_P^L=\overline{f_P^L}$ if and only if $({\overline{f}}_P^L{)}^{\mathrm{\prime }}\in \tau$.

Proof.

If ${\overline{f}}_P^L=\overline{f_P^L}$,then ${\overline{f}}_P^L$ is a soft L-closed set and so $(f_P^L{)}^{\mathrm{\prime }}\in \tau$.

Conversely if $({\overline{f}}_P^L{)}^{\mathrm{\prime }}\in \tau$, then $f_P^L$ is a soft L-closed set containing $f_P^L$.

By above Proposition 3.7, ${\overline{f}}_P^L\tilde{\subset }\overline{f_P^L}$ and by the definition of soft L-closure of $f_P^L$, any soft L-closed set over L which contains $f_P^L$ will contain $\overline{f_P^L}$.

Thus ${\overline{f}}_P^L=\overline{f_P^L}$.

Example 3.13

Let $L=\{l_1,l_2,l_3\},P=\{p_1,p_2\}$ and $\tau =\{\varphi ,L,f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,f_{5P}^L,f_{6P}^L,f_{7P}^L\}$, where $f_{1P}^L,f_{2P}^L,f_{3P}^L,f_{4P}^L,f_{5P}^L,f_{6P}^L,f_{7P}^L$ are soft lattices over L, defined as follows,

$f_1(p_1)=\{l_1,l_2\},f_1(p_2)=\{l_1,l_2\}$,

$f_2(p_1)=\{l_2\},f_2(p_2)=\{l_1,l_3\}$,

$f_3(p_1)=\{l_2,l_3\},f_3(p_2)=\{l_1\}$,

$f_4(p_1)=\{l_2\},f_4(p_2)=\{l_1\}$

$f_5(p_1)=\{l_1,l_2\},f_5(p_2)=\{L\}$,

$f_6(p_1)=\{L\},f_6(p_2)=\{l_1,l_2\}$,

$f_7(p_1)=\{l_2,l_3\},f_7(p_2)=\{l_1,l_3\}$,

Then $(L,\tau ,P)$ is a soft L-topological space.

Let $f_P^L$ and $g_P^L$ are defined as follows:

$f(p_1)=\{l_1,l_3\},f(p_2)=\varphi ,$

$g(p_1)=\{l_2,l_3\},g(p_2)=\{l_1,l_2\}$.

Then $f_P^L\cap g_P^L=(f\cap g{)}_P^L$ is given by,

$(f\cap g)(p_1)=\{l_3\},(f\cap g)(p_2)=\varphi$.

Now, ${\overline{f}}_P^L=L\cap f_{2P}^{\mathrm{\prime }L}\cap f_{4P}^{\mathrm{\prime }L}=f_{2P}^{\mathrm{\prime }L}$ and ${\overline{g}}_P^L=L$.

Therefore ${\overline{f}}_P^L\cap {\overline{g}}_P^L={\overline{f}}_P^L$.

Also $\overline{f_P^L\cap g_P^L}=\cap \{L,f_{1P}^{\mathrm{\prime }L},f_{2P}^{\mathrm{\prime }L},f_{3P}^{\mathrm{\prime }L},f_{4P}^{\mathrm{\prime }L}\}=f_{5P}^{\mathrm{\prime }L}$.

So $\overline{f_P^L\cap g_P^L}\subset {\overline{f}}_P^L\cap {\overline{g}}_P^L$ but $\overline{f_P^L\cap g_P^L}\ne {\overline{f}}_P^L\cap {\overline{g}}_P^L$.

Next we see that

${\tau }_{p_1}=\{\varphi ,L,\{l_2\},\{l_2,l_3\},\{l_1,l_2\}\}$

and

${\tau }_{p_2}=\{\varphi ,L,\{l_1\},\{l_1,l_3\},\{l_1,l_2\}\}$.

Here $f_P^L$ is given by $\overline{f}(p_1)=\{l_1,l_3\},\overline{f}(p_2)=\varphi$.

Clearly $\overline{f_P^L}\tilde{\subset }{\overline{f}}_P^L$ but $\overline{f_P^L}\ne {\overline{f}}_P^L$.

Definition 3.13

Let $(L,\tau ,P)$ be a soft lattice topological space over L, $g_P^L$ be a soft lattice over L and $x\in L$. Then x is said to be a soft L-interior point of $g_P^L$ if there exists a soft L-open set $f_P^L$ such that $x\in f_P^L\subset g_P^L$.

Definition 3.14

Let $(L,\tau ,P)$ be a soft lattice topological space over L, $g_P^L$ be a soft lattice over L and $x\in L$. Then $g_P^L$ is said to be a soft lattice neighbourhood of x if there exists a soft L-open set $f_P^L$ such that $x\in f_P^L\subset g_P^L$.

Proposition 3.8

Let $(L,\tau ,P)$ be a soft lattice topological space over L, $g_P^L$ be a soft lattice over L and $x\in L$. If x is a soft L-interior point of $g_P^L$, then x is an interior point of $g(a)$ in $(L,{\tau }_a)$, for each $a\in P$.

Proof.

For any $a\in P$, $g(a)\subseteq L$.

If $x\in L$ is a soft L-interior point of $g_P^L$, then there exists $f_P^L\in \tau$ such that $x\in f_P^L\subset g_P^L$.

This means that, $x\in f(a)\subseteq g(a)$.

As $f(a)\in {\tau }_a$, so $f(a)$ is an soft L-open set in ${\tau }_a$ and $x\in {\tau }_a$.

This implies that x is an interior point of $g(a)$ in ${\tau }_a$.

Proposition 3.9

Let $(L,\tau ,P)$ be a soft lattice topological space over L. Then

(1)	each $x\in L$ has a soft lattice neighbourhood.
(2)	if $f_P^L$ and $g_P^L$ are soft lattice neighbourhoods of some $x\in L$, then $f_P^L\cap g_P^L$ is also a soft lattice neighbourhood of x.
(3)	if $f_P^L$ is a soft lattice neighbourhoods of $x\in L$ and $f_P^L\tilde{\subset }{g}_P^L$, then $g_P^L$ is a soft lattice neighbourhoods of $x\in L$.

Proof.

(1) For any $x\in L$, $L\in \tau$, so $x\tilde{\subset }L$.

(2) Let $f_P^L$ and $g_P^L$ be the soft neighbourhoods of $x\in L$, then there exist $f_{1P}^L,f_{2P}^L\in \tau$ such that $x\in f_{1P}^L\tilde{\subset }{f}_P^L$ and $x\in f_{2P}^L\tilde{\subset }{f}_P^L$.

Now $x\in f_{1P}^L$ and $x\in f_{2P}^L$ implies that $x\in f_{1P}^L\cap f_{2P}^L$ and $f_{1P}^L\cap f_{2P}^L\in \tau$.

So we have $x\in f_{1P}^L\cap f_{2P}^L\tilde{\subset }{f}_P^L\cap g_P^L$.

Thus $f_P^L\cap g_P^L$ is also a soft lattice neighbourhood of x.

(3) Let $f_P^L$ is a soft lattice neighbourhoods of $x\in L$ and $f_P^L\tilde{\subset }{g}_P^L$.

Then by Definition 3.14, ∃ a soft open L-set $f_{1P}^L$ such that $f_{1P}^L\tilde{\subset }{f}_P^L\tilde{\subset }{g}_P^L$.

Thus $x\in f_{1P}^L\tilde{\subset }{g}_P^L$.

Hence $g_P^L$ is a soft lattice neighbourhoods of $x\in L$.

Proposition 3.10

Let $(L,\tau ,P)$ be a soft lattice topological space over L. For any soft L-open set $f_P^L$ over L, $f_P^L$ is a soft L-neighbourhood of each point of ${\cap }_{a\in P}f(a)$.

Proof.

Let $f_P^L\in \tau$. For any $x\in {\cap }_{a\in P}f(a)$, we have $x\in f(a)$ for each $a\in P$. Thus

$x\in f_P^L\subset f_P^L$ and so $f_P^L$ is a soft L-neighbourhood of x.

Definition 3.15

Let $(L,\tau ,P)$ be a soft lattice topological space over L and Z be a non-empty subset of L. Then ${\tau }_Z=\{Z{f}_P^L\mid f_P^L\in \tau \}$ is said to be the soft relative lattice topology on Z and $(Z,{\tau }_Z,P)$ is called a soft L-subspace of $f_P^L$.

Example 3.14

A soft discrete L-topological space is any soft L-subspace of a soft discrete L- topological space.

Example 3.15

A soft indiscrete L-topological space is any soft L-subspace of a soft indiscrete L-topological space.

Proposition 3.11

Let $(L,\tau ,P)$ be a soft lattice topological space over L and Z be a non-empty subset of L. Then $(Z,{\tau }_{aZ})$ is a subspace of $(L,{\tau }_a,P)$ for each $a\in P$.

Proof.

Since $(Z,{\tau }_{aZ})$ is a soft lattice topological space over L, so from the Definition 3.15, for any $a\in E$, $\begin{array}{rl}{\tau }_{aZ}& =[Z_{F(a)}⏐f_P^L\in \tau ].\\ & =[Z\cap F(a)⏐f_P^L\in \tau ].\\ & =[Z\cap F(a)⏐F(a)\in {\tau }_Z]\end{array}$ Hence $(Z,{\tau }_{aZ})$ is a subspace of $(L,{\tau }_a,P)$.

Proposition 3.12

Let $(L,{\tau }_Y,P)$ be a soft L-subspace of a soft L-topological space $(L,\tau ,P)$ and $f_P^L$ be a soft L-open set in Z. If $Z\in \tau$, then $f_P^L\in \tau$.

Proof.

Let $f_P^L$ be a soft L-open set in Z, then there exists a soft open set $g_P^L$ in Z such that, $f_P^L=Z\cap g_P^L$.

Now, if $Z\in \tau$, then $Z\cap g_P^L\in \tau$ by the third axiom of the definition of a soft L-topological space and hence $f_P^L\in \tau$.

Theorem 3.2

Let $(L,{\tau }_Z,P)$ be a soft L-subspace of a soft L-topological space $(L,\tau ,P)$ and $f_P^L$ be a soft L-open set in Z. Then

(i)	$f_P^L$ is soft L-open in Z iff $f_P^L=\tilde{Z}\cap g_P^L$.
(ii)	$f_P^L$ is soft L-closed in Z iff $f_P^L=\tilde{Z}\cap g_P^L$ for some soft L-closed set $g_P^L$ in L.

Proof.

$(i)$ This result follows from the definition of a soft L-subspace.

$(ii)$ If $f_P^L$ is a soft L-closed in Z, then $f_P^L=\tilde{Z}\setminus g_P^L$ for some $g_P^L\in {\tau }_Z$.

Now $g_P^L=\tilde{Z}\cap h_P^L$ for some $h_P^L\in L$.

For any $a\in P$, $\begin{array}{rl}f(a)& =Z(a)\setminus g(a)\\ & =Z\setminus g(a)\\ & =Z\setminus (Z(a)\cap h(a))\\ & =Z\setminus (Z\cap h(\alpha ))\\ & =Z\setminus h(a)\\ & =Z\cap (L\setminus h(a))\\ & =Z\cap (h(a){)}^{\mathrm{\prime }}\\ & =Z(a)\cap (h(a){)}^{\mathrm{\prime }}\end{array}$ Thus $f_P^L=\tilde{Z}\setminus h_P^{L^{\mathrm{\prime }}}$,

where $h_P^{L^{\mathrm{\prime }}}$ is soft L-closed set.

Conversely, assume that $f_P^L=\tilde{Z}\setminus g_P^L$ for some soft L-closed set $g_P^L$.

i.e. $g_P^{L^{\mathrm{\prime }}}\in \tau$.

Now if $g_P^L=(L,\tau ,P)\setminus h_P^L$, where $h_P^L\in \tau$, then for any $a\in P$, $\begin{array}{rl}f(a)& =Z(a)\cap g(a)\\ & =Z\cap g(a)\\ & =Z\cap (L(a)\setminus h(a))\\ & =Z\cap (L\setminus h(a))\\ & =Z\setminus h(a)\\ & =Z\setminus (Z\cap h(a))\\ & =Z(a)\setminus (Z(a))\cap h(a))\end{array}$ Thus $f_P^L=\tilde{Z}\setminus \tilde{Z}\cap h(a)$.

Since $f_P^L\in \tau$, so $\tilde{Z}\cap h(a)\in {\tau }_Z$ and

hence $f_P^L$ is soft L-closed in Z.

Acknowledgments

The authors are very much indebted to Dr. Sunil Jacob John, Department of Mathematics, National Institute of Technology, Calicut, Kerala, India for his constant encouragement throughout the preparation of this paper.

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No potential conflict of interest was reported by the author(s).

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Notes on contributors

Sandhya S. Pai

Sandhya S. Pai currently working as an assistant professor in the department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, India. Area of interest is Soft sets and Soft topological spaces.

T. Baiju

T. Baiju currently working as an associate professor, senior scale in the department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, Karnataka, India. Area of interest is Fuzzy Mathematics and Soft topological spaces.