1 Introduction

Most of the problems in engineering, medical science, economics and social science etc. have vagueness and various uncertainties. To overcome these uncertainties, some kinds of theories were given which we can use as mathematical tools for dealing with uncertainties. However, these theories have their own difficulties. In 1999, Molodtsov [1] initiated a novel concept of soft set theory, which is a completely new approach for modeling vagueness and uncertainty. From then on, works on the soft set theory are progressing rapidly. After Molodtsov’s work, same different applications of soft sets were studied in [2, 3]. Furthermore Maji, Biswas and Roy worked on soft set theory in [4, 5]. Roy et al. presented some applications of this notion to decision making problems in [6]. The algebraic structures of soft sets have been studied by some authors [7–14]. Birkhoff’s work in 1930 started the general development of lattice theory [15]. The lattice theory has been applied to many kinds of fields. Recently, the work introducing the soft set theory to the lattice theory and the fuzzy set theory have been initiated. Fu [16] and Çağman et al. [17] presented the nation of the soft lattice and derived the properties of the soft lattice and discussed the relationship between the soft lattices. Karaaslan et al. [18] introduced the fuzzy soft lattice theory, some related properties on it. Florentine Smarandache for the first time introduced the concept of neutrosophy in 1995 [19, 20]. Maji [21] had combined the neutrosophic set with soft sets and introduced a new mathematical model neutrosophic soft set. Later Broumi and Smarandache defined the concepts interval-valued neutrosophic soft sets and intuitionistic neutrosophic soft sets in [22, 23]. Different algebraic structures and their applications were studied in the neutrosophic soft set context [24–31].

In this paper, we apply the notion of neutrosophic soft sets introduced by [21] to the lattice theory and present the notion of neutrosophic soft lattice, which is different from the one presented by [16–18]. The organization of this paper is as follows: in Sect. 2, some basic concepts and some related properties are introduced. In Sect. 3, the notion of neutrosophic soft lattice is presented and some related properties are defined, then the neutrosophic soft sublattice, complete neutrosophic soft lattice, distributive neutrosophic soft lattice, modular neutrosophic soft lattice, neutrosophic soft chain are presented and their related properties are studied. Section 4 concludes the paper.

2 Preliminaries

Definition 2.1

[19] Let U be a space of points (objects), with a generic element in U denoted by u. A neutrosophic set (N-set) Ain U is characterized by a truth-membership function \(T_A \), an indeterminacy-membership function \(I_A \)and a falsity-membership function \(F_A \). \(T_A (u)\); \(I_A (u)\) and \(F_A \)(u) are real standard or nonstandard subsets of [0, 1]. It can be written as

$$\begin{aligned} A=\{<u,(T_A (u),I_A (u),F_A (u))>:u\in U,T_A (u),I_A (u),F_A (u)\in [0,1]\}. \end{aligned}$$

There is no restriction on the sum of \(T_A \)(u); \(I_A \)(u) and \(F_A \)(u), so

0 \(\le \)sup \(T_A (u)\) + sup \(I_A (u)\) + sup \(F_A (u) \quad \le \) 3.

Definition 2.2

[21] Let U be an initial universe set and E be a set of parameters. Consider E. Let P(U) denote the set of all neutrosophic sets of U. The collection \(\left( {F,A} \right) \) is termed to be the soft neutrosophic set over U, where F is a mapping given by \(F:A\rightarrow P(U)\).

Definition 2.3

[32] Let P be a non-empty ordered set.

  1. (i)

    If \(x y\) and \(x\wedge y\) exist for all \(x,y\in P\), then P is called a lattice,

  2. (ii)

    If and \(\wedge S\) exist for all \(S\subseteq P\), then P is called a complete lattice.

Definition 2.4

[15] An algebra is called a lattice if L is a nonempty set, are binary operations on L, both \(\wedge \) and \(\vee \) are idempotent, commutative and associative, and they satisfy the two absorption identities. That is, for all \(a,b,c\in L\)

  1. 1.

    \(a\wedge a=a;a\vee a=a\),

  2. 2.

    \(a\wedge b=b\wedge a;a\vee b=b\vee a\),

  3. 3.

    \((a\wedge b)\wedge c=a\wedge (b\wedge c);(a\vee b)\vee c=a\vee (b\vee c)\),

  4. 4.

    \(a\wedge \left( {a\vee b} \right) =a;a\vee \left( {a\wedge b} \right) =a\).

Definition 2.5

[21] Let (F, A) and (G, B) be two neutrosophic soft sets over the common universe U.(F, A) is said to be a neutrosophic soft subset of (G, B) if \(A\subset B,\) and\(T_F \)(e)(x) \(\le T_G (\hbox {e})(\hbox {x}),\hbox {}I_F (\hbox {e})(\hbox {x})\) \(\le I_G (\hbox {e})(\hbox {x}),F_F (\hbox {e})(\hbox {x})\hbox {}\) \(\ge F_G (\hbox {e})(\hbox {x}),\hbox {}\forall e\in A,x\in U.\)We denote it by \((F,A)\subseteq (G,B).\)

Definition 2.6

[21] The complement of a neutrosophic soft set (F, A) denoted by \((F,A)^{c}=(F^{c},\lnot A)\) where \(F^{c}:\lnot A\rightarrow P\left( U \right) \) is a mapping given by \(F^{c}\left( \alpha \right) =\)neutrosophic soft complement with \(T_{F^{c}} \left( x \right) =F_F \left( x \right) ,I_{F^{c}} \left( x \right) =I_F \left( x \right) \hbox {and} F_{F^{c}} \left( x \right) =T_F \left( x \right) .\)

Definition 2.7

[21] Let \(\left( {H,A} \right) \) and \(\left( {G,B} \right) \) be two NSSs over the common universe U. Then the union of \(\left( {H,A} \right) \) and \(\left( {G,B} \right) \) is denoted by “\(\left( {H,A} \right) \tilde{\cup } \left( {G,B} \right) \)” and is defined by\(\left( {H,A} \right) \tilde{\cup } \left( {G,B} \right) =(K,C)\), where \(C=A \cup B\) and the truth-membership, indeterminacy-membership and falsity-membership of (K, C) are as follows:

$$\begin{aligned} T_{K\left( e \right) } \left( m \right) =\left\{ {{\begin{array}{l} {T_{H\left( e \right) } \left( m \right) , \quad if\quad e\in A-B,} \\ {T_{G\left( e \right) } \left( m \right) ,\quad if\quad e\in B-A,} \\ {\max \left( {T_{H\left( e \right) } \left( m \right) ,T_{G\left( e \right) } \left( m \right) } \right) ,\quad if\quad e\in A \cap B.} \\ \end{array} }} \right. \end{aligned}$$
$$\begin{aligned} I_{K\left( e \right) } \left( m \right) =\left\{ {{\begin{array}{l} {I_{H\left( e \right) } \left( m \right) , \quad if\quad e\in A-B,} \\ {I_{G\left( e \right) } \left( m \right) ,\quad if\quad e\in B-A,} \\ {\frac{I_{H(e)} \left( m \right) +I_{G(e)} \left( m \right) }{2},\quad if\quad e\in A \cap B.} \\ \end{array} }} \right. \end{aligned}$$
$$\begin{aligned} F_{K\left( e \right) } \left( m \right) =\left\{ {{\begin{array}{l} {F_{H\left( e \right) } \left( m \right) , \quad if\quad e\in A-B,} \\ {F_{G\left( e \right) } \left( m \right) ,\quad if\quad e\in B-A,} \\ {\min \left( {F_{H\left( e \right) } \left( m \right) ,F_{G\left( e \right) } \left( m \right) } \right) , \quad if\quad e\in A \cap B.} \\ \end{array} }} \right. \end{aligned}$$

Definition 2.8

[21] Let \(\left( {H,A} \right) \) and \(\left( {G,B} \right) \) be two NSSs over the common universe U. Then the intersection of \(\left( {H,A}\right) \) and \(\left( {G,B} \right) \) is denoted by “\(\left( {H,A} \right) \tilde{\cap } \left( {G,B} \right) \)” and is defined by\(\left( {H,A} \right) \tilde{\cap } \left( {G,B} \right) =(K,C)\), where \(C=A \cap B\) and the truth-membership, indeterminacy-membership and falsity-membership of (K, C) are as follows:

$$\begin{aligned} T_{K\left( e \right) } \left( m \right)= & {} \min \left( {T_{H\left( e \right) } \left( m \right) ,T_{G\left( e \right) } \left( m \right) } \right) \\ I_{K(e)} \left( m \right)= & {} \frac{I_{H(e)} \left( m \right) +I_{G(e)} \left( m \right) }{2}\\ F_{K(e)} \left( m \right)= & {} \max \left( {F_{H\left( e \right) } \left( m \right) ,F_{G\left( e \right) } \left( m \right) } \right) , \quad if\quad e\in A \cap B. \end{aligned}$$

3 Lattice structures of neutrosophic soft sets

In this section, the notion of neutrosophic soft lattice is defined and several related properties are investigated.

Definition 3.1

Let \(N^{L}\) be a neutrosophic soft set over \(U,{\tilde{\vee }}\) and \(\tilde{\wedge } \) be two binary operation on \( N^{L}\). If elements of \(N^{L}\) are equipped with two commutative and associative binary operations \(\tilde{\vee } \) and \(\tilde{\wedge } \) which are connected by the absorption law, then algebraic structure \(( {N^{L},\tilde{\vee } ,\tilde{\wedge }} )\) is called a neutrosophic soft lattice.

Example 3.2

Let \(U=\left\{ {u_1 ,u_2 ,u_3 ,u_4 } \right\} \) be a universe set and \(N^{L}=\left\{ {F_A ,F_B ,F_C ,F_D } \right\} \tilde{\subseteq } NS (U)\).

Suppose that

$$\begin{aligned} F_A= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.5,0.2,0.7}} \right) ,\left( {e_2 ,\frac{u_2 }{0.6,0.3,0.4},\frac{u_3 }{0.4,0.1,0.7}} \right) ,\left( {e_3 ,\frac{u_4 }{0.6,0.3,0.8}} \right) } \right\} \\ F_B= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.4,0.1,0.5},\frac{u_2 }{0.8,0.2,0.6}} \right) ,\left( {e_2 ,\frac{u_3 }{0.5,0.2,0.3}} \right) } \right\} \\ F_C= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6}} \right) ,\left( {e_2 ,\frac{u_2 }{0.5,0.4,0.6},\frac{u_3 }{0.4,0.1,0.7}} \right) } \right\} \\ F_D= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.5,0.1,0.4},\frac{u_2 }{0.6,0.3,0.2}} \right) } \right\} \\ \end{aligned}$$

Then,\(\left( {N^{L},\tilde{\cup } ,\tilde{\cap } } \right) \)is a neutrosophicsoft lattice. Here binary operations are neutrosophic union and neutrosophic intersection. Hasse diagram of \(N^{L}\)is shown in Fig. 1.

Fig. 1
figure 1

A neutrosophic soft lattice structure

Full size image

Theorem 3.3

\(( {N^{L},{\tilde{\vee }},\tilde{\wedge }} )\) be a neutrosophic soft lattice and \(F_A ,F_B \in NS (U)\). Then

$$\begin{aligned} F_A \tilde{\wedge } F_B =F_A \Leftrightarrow F_A \tilde{\vee } F_B =F_B . \end{aligned}$$

Proof

$$\begin{aligned} F_A \tilde{\wedge } F_B= & {} F_{A} \tilde{\wedge } \left( {F_A \tilde{\vee } F_B } \right) \\= & {} \left( {F_A \tilde{\wedge } F_A } \right) \tilde{\wedge } \left( {F_A \tilde{\wedge } F_B } \right) \\= & {} F_A \tilde{\vee } F_A\\= & {} F_A \end{aligned}$$

Conversely,

$$\begin{aligned} F_A \tilde{\vee } F_B= & {} \left( {F_A \tilde{\wedge } F_B } \right) \widetilde{\vee } F_B\\= & {} \left( {F_A \tilde{\vee } F_B } \right) \tilde{\wedge } \left( {F_B \tilde{\vee } F_B } \right) \\= & {} F_B \tilde{\wedge } F_B\\= & {} F_{B} \end{aligned}$$

\(\square \)

Theorem 3.4

\(( {N^{L},\tilde{\vee } ,\tilde{\wedge } })\) be a neutrosophic soft lattice and \(F_A ,F_B \in NS (U)\). Then the relation \(\tilde{\le } \) which is defined by

$$\begin{aligned} F_A \tilde{\le }F_B \Leftrightarrow F_A \tilde{\wedge } F_B =F_A or F_A \tilde{\vee } F_B =F_B \end{aligned}$$

is an ordering relation on NS (U).

Proof

  1. (i)

    \(\forall F_A \in N^{L}, \tilde{\le } \) is reflexive, \(F_A \tilde{\le } F_A \Leftrightarrow F_A \tilde{\wedge } F_A =F_A \)

  2. (ii)

    \(\forall F_A ,F_B \in N^{L}, \tilde{\le } \) is antisymmetric. Let \(F_A \tilde{\le } F_B \) and \(F_B \tilde{\le } F_A .\) Then

    $$\begin{aligned} F_A= & {} F_A \tilde{\wedge } F_B\\= & {} F_B \tilde{\wedge } F_A\\= & {} F_{B.} \end{aligned}$$
  3. (iii)

    \(\forall F_A ,F_B and F_C \in N^{L},\tilde{\le } \) is transitive. If \(F_A \tilde{\le } F_B \) and \(F_B \tilde{\le } F_C \Rightarrow F_A \tilde{\le } F_C \). Indeed

    $$\begin{aligned} F_A \tilde{\wedge } F_C= & {} \left( {F_A \widetilde{\wedge } F_B } \right) \tilde{\wedge } F_C\\= & {} \left( {F_A } \right) \tilde{\wedge } \left( {F_B \tilde{\wedge } F_C } \right) \\= & {} F_A \tilde{\wedge } F_B\\= & {} F_A . \end{aligned}$$

\(\square \)

Theorem 3.5

\(( {N^{L},\tilde{\vee },\tilde{\wedge } })\) be a neutrosophic soft lattice and \(F_A ,F_B \in NS (U)\). Then \(F_A \tilde{\vee } F_B \) and \(F_A \tilde{\wedge } F_B \)are the least upper and the greatest lower bound of \(F_A \hbox {and}F_B \), respectively.

Proof

Suppose that \(F_A \tilde{\wedge } F_B \) is not the greatest lower bound of \(F_A \hbox {and}F_B \). Then there exists \(F_C \in NS (U)\) such that \(F_A \tilde{\wedge } F_B \tilde{\le } F_C \tilde{\le } F_A \)and \(F_A \tilde{\wedge } F_B \tilde{\le } F_C \tilde{\le } F_B \). Hence \(F_C \tilde{\wedge } F_C \tilde{\le } F_A \tilde{\wedge } F_B \). Thus \(F_C \tilde{\le } F_A \tilde{\wedge } F_B \). Therefore \(F_C =F_A \tilde{\wedge } F_{B} \).But this is a contradiction.\(F_A \tilde{\vee } F_B \) being the least upper bound of \(F_A \hbox {and}F_B \) can be shown similarly. \(\square \)

Lemma 3.6

Let \(N^{L}\in NS (U)\). Then neutrosophic soft lattice inclusion relation \(\tilde{\subseteq } \) that is defined by

$$\begin{aligned} F_A \tilde{\subseteq } F_B \Leftrightarrow F_A {\tilde{\cup } } F_B =F_B \hbox {or} F_A {\tilde{\cap } } F_B =F_A \end{aligned}$$

is an ordering relation on \(N^{L}\).

Proof

For all \(F_A ,F_B and F_C \in N^{L}\),

  1. (i)

    \(F_A \in N^{L},\tilde{\subseteq } \) is reflexive, \(F_A \tilde{\subseteq } F_A \Leftrightarrow F_A {\tilde{\cap } } F_A =F_A \)

  2. (ii)

    \(F_A ,F_B \in N^{L},\tilde{\subseteq } \) is antisymmetric. Let \(F_A \tilde{\subseteq } F_B \) and \(F_B \tilde{\subseteq } F_A \Leftrightarrow F_A =F_B \).

  3. (iii)

    \(F_A ,F_B and F_C \in N^{L},\tilde{\subseteq } \) is transitive. If \(F_A \tilde{\subseteq } F_B \) and \(F_B \tilde{\subseteq } F_C \Rightarrow F_A \tilde{\subseteq } F_C \).

\(\square \)

Corollary 3.7

\(( {N^{L}, {\tilde{\cup } } \hbox {},\hbox {} {\tilde{\cap } },\tilde{\subseteq } })\) is a neutrosophic soft lattice.

Definition 3.8

Let \(( {N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice and let \(F_A \in N^{L}\). If \(F_A \tilde{\le } F_B \) for all \(F_B \in N^{L}\), then \(F_A \) is called the minimum element of \(N^{L}\). If \(F_B \tilde{\le } F_A \) for all \(F_B \in N^{L}\), then \(F_A \) is called the maximum element of \(N^{L}\).

Definition 3.9

Let \(({N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice and let \(F_A \in N^{L}\). If \(F_B \tilde{\le } F_A \hbox {or}F_A \tilde{\le } F_B \) for all \(F_A ,F_B \in N^{L}\), then \(N^{L}\) is called a neutrosophic soft chain.

Example 3.10

Consider the neutrosophic soft lattice in Example 3.2. A neutrosophic soft subset \(N^{S}=\{ {F_A ,F_B ,F_D }\}\tilde{\subseteq } NS (U)\) of \(N^{L}\) is a neutrosophic soft chain. But \(( {N^{L}, {\tilde{\cup } } \hbox {},\hbox {}\mathop {\tilde{\cap }}\ ,\tilde{\subseteq } })\) is not a neutrosophic soft chain since \(F_B \) and \(F_C \)can not be comparable.

Definition 3.11

Let \(({N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice. If every subset of \(N^{L}\) have both a greatest lowers bound and a least upper bound, then \(N^{L}\) is called a complete neutrosophic soft lattice.

Example 3.12

Let \(U=\{ {u_1 ,u_2 ,u_3 ,u_4 ,u_5 }\}\) be a universe set and \(N^{L}=\{ {F_A ,F_B ,F_C ,F_D } \}\tilde{\subseteq } NS (U)\).

$$\begin{aligned} F_A= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.5,0.2,0.7},\frac{u_5 }{0.8,0.3,0.4}} \right) } \right\} \\ F_B= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.4,0.1,0.5},\frac{u_4 }{0.8,0.2,0.6},\frac{u_5 }{0.5,0.2,0.3}} \right) ,\left( {e_2 ,\frac{u_3 }{0.5,0.2,0.7},\frac{u_4 }{0.8,0.3,0.4}} \right) } \right\} \\ F_C= & {} \left\{ \left( {e_1 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_4 }{0.7,0.2,0.3},\frac{u_5 }{0.6,0.1,0.4}} \right) ,\right. \\&\left. \left( {e_2 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_4 }{0.7,0.2,0.3}} \right) \right\} \\ F_D= & {} F_\emptyset . \end{aligned}$$

Then \(( {N^{L}, {\tilde{\cup } } \hbox {},\hbox {}\mathop {\tilde{\cap }} ,\tilde{\subseteq } })\) is a complete neutrosophic soft lattice

Definition 3.13

Let \(({N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice and \(N^{M}\tilde{\subseteq } N^{L}\). If \(N^{M}\) is a neutrosophic soft lattice with the operations of\(N^{L}\), then \(N^{M}\) is called a neutrosophic sublattice of\(\hbox {}N^{\hbox {L}}\).

Theorem 3.14

Let \(({N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice and \(N^{M}\tilde{\subseteq } N^{L}\). If\(F_A \tilde{\vee } F_B \in N^{M}\) and \(F_A \tilde{\wedge } F_B \in N^{M}\) for all \(F_A , F_B \in N^{M}\), then \(N^{M}\) is a neutrosophic soft lattice.

Proof

It is obvious from Definition 3.13. \(\square \)

Example 3.15

Let \(U= \{ {u_1 ,u_2 ,u_3 ,u_4 ,u_5 }\}\) be a universe set and \(N^{L}= \{ {F_A ,F_B ,F_C ,F_D }\} \tilde{\subseteq } NS (U)\).

$$\begin{aligned} F_A= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.5,0.2,0.7},\frac{u_5 }{0.8,0.3,0.4}} \right) } \right\} \\ F_B= & {} \left\{ {\left( {\frac{u_1 }{0.4,0.1,0.5},\frac{u_4 }{0.8,0.2,0.6},\frac{u_5 }{0.5,0.2,0.3}} \right) ,\left( {e_2 ,\frac{u_3 }{0.5,0.2,0.7},\frac{u_4 }{0.8,0.3,0.4}} \right) } \right\} \\ F_C= & {} \left\{ \left( {e_1 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_4 }{0.7,0.2,0.3},\frac{u_5 }{0.6,0.1,0.4}} \right) ,\right. \\&\left. \left( {e_2 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_4 }{0.7,0.2,0.3}} \right) \right\} \\ F_D= & {} F_\emptyset \end{aligned}$$

Then, if \(N^{M}=\{ {F_A ,F_B ,F_D }\}\tilde{\subseteq } NS (U)\), then \(N^{M}\)is a neutrosophic soft sublattice.

Definition 3.16

Let \(( {N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice and \(F_A ,F_B \hbox {and}F_C \in N^{L}\). If

$$\begin{aligned} \left( {F_A \tilde{\wedge } F_B } \right) \tilde{\vee } \left( {F_A \tilde{\wedge } F_C } \right) \tilde{\le } F_A \tilde{\wedge } \left( {F_B \tilde{\vee } F_C } \right) \end{aligned}$$

or

$$\begin{aligned} F_A \tilde{\wedge } \left( {F_B \tilde{\vee } F_C } \right) \tilde{\le } \left( {F_A \tilde{\wedge } F_B } \right) \tilde{\vee } \left( {F_A \tilde{\wedge } F_C } \right) \end{aligned}$$

then \(N^{L}\) is called a one-sided distributive neutrosophic soft lattice.

Theorem 3.17

Every neutrosophic soft lattice is a one-sided distributive neutrosophic soft lattice.

Proof

Let \(F_A ,F_B \hbox {and} F_C \in N^{L}.\) \(\square \)

Since \(F_A \tilde{\wedge } F_B \tilde{\le } F_A \) and \(F_A \tilde{\wedge } F_B \tilde{\le } F_B \tilde{\le } F_B \tilde{\vee } F_C \),

\(F_A \tilde{\wedge } F_B \tilde{\le } F_A \) and \(F_A \tilde{\wedge } F_B \tilde{\le } F_B \tilde{\wedge } F_C \) .

Therefore,

$$\begin{aligned} F_A \tilde{\wedge } F_B =\left( {F_A \tilde{\wedge } F_B } \right) \tilde{\wedge } \left( {F_A \tilde{\wedge } F_B } \right) \tilde{\le } F_A \tilde{\wedge } \left( {F_B \tilde{\vee } F_C } \right) \quad \left( a \right) \end{aligned}$$

And also we have \(F_A \tilde{\wedge } F_C \tilde{\le } F_A \) and \(F_A \tilde{\wedge } F_C \tilde{\le } F_C \tilde{\le } F_B \tilde{\vee } F_C \) . Since \(F_A \tilde{\wedge } F_C \tilde{\le } F_A \) and \(F_A \tilde{\wedge } F_C \tilde{\le } F_B \tilde{\vee } F_C \), then

$$\begin{aligned} F_A \tilde{\wedge } F_C =\left( {F_A \tilde{\wedge } F_C } \right) \tilde{\wedge } \left( {F_A \tilde{\wedge } F_C } \right) \tilde{\le } F_A \tilde{\wedge } \left( {F_B \tilde{\vee } F_C } \right) \quad \left( b \right) \end{aligned}$$

from \((a) \hbox {and} (b)\), we get the desired result,

$$\begin{aligned} \left( {F_A \tilde{\wedge } F_B } \right) \tilde{\vee } \left( {F_A \tilde{\wedge } F_C } \right) \tilde{\le } F_A \tilde{\wedge } \left( {F_B \tilde{\vee } F_C } \right) . \end{aligned}$$

Definition 3.18

\(( {N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice. If \(N^{L}\) satisfies the following axioms, it is called a distributive neutrosophic soft lattice;

  1. (i)

    \(F_A \tilde{\vee } \left( {F_B \tilde{\wedge } F_C } \right) =\left( {F_A \tilde{\vee } F_B } \right) \tilde{\wedge } \left( {F_A \tilde{\vee } F_C } \right) \)

  2. (ii)

    \(F_A \tilde{\wedge } \left( {F_B \tilde{\vee } F_C } \right) =\left( {F_A \tilde{\wedge } F_B } \right) \tilde{\vee } \left( {F_A \tilde{\wedge } F_C } \right) \)

for all \(F_A ,F_B \hbox {and}F_C \in N^{L}\).

Example 3.19

Let \(U=\{ {u_1 ,u_2 ,u_3 ,u_4 ,u_5 }\}\) be a universe set and \(N^{L}=\{ F_\emptyset ,F_A ,F_B ,F_C ,F_D ,F_E \} \tilde{\subseteq } NS (U).\) Then \(N^{L}\tilde{\subseteq } NS(U)\) is a neutrosophic soft lattice with the operations \({\tilde{\cup } } \hbox {and} {\tilde{\cap }} \). Suppose that

$$\begin{aligned} F_A= & {} \left\{ {\left( {e_1 ,\frac{u_5 }{0.5,0.2,0.7}} \right) ,\left( {e_2 ,\frac{u_1 }{0.6,0.2,0.7},\frac{u_2 }{0.5,0.1,0.6}} \right) } \right\} \\ F_B= & {} \left\{ {{\begin{array}{l} {\left( {e_1 ,\frac{u_1 }{0.4,0.1,0.5},\frac{u_3 }{0.8,0.2,0.6},\frac{u_5 }{0.5,0.2,0.3}} \right) ,\left( {e_2 ,\frac{u_2 }{0.4,0.2,0.3},\frac{u_4 }{0.8,0.3,0.4}} \right) ,} \\ {\left( {e_3 ,\frac{u_3 }{0.7,0.4,0.3},\frac{u_4 }{0.5,0.1,0.4}} \right) } \\ \end{array} }} \right\} \\ F_C= & {} \left\{ {{\begin{array}{l} {\left( {e_1 ,\frac{u_3 }{0.4,0.2,0.5},\frac{u_4 }{0.7,0.2,0.3},\frac{u_5 }{0.6,0.1,0.4}} \right) ,\left( {e_2 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_4 }{0.7,0.2,0.3}} \right) ,} \\ {\left( {e_3 ,\frac{u_3 }{0.7,0.4,0.3},\frac{u_4 }{0.5,0.1,0.4}} \right) } \\ \end{array} }} \right\} \\ F_D= & {} \left\{ \left( {e_1 ,\frac{u_5 }{0.6,0.1,0.4}} \right) ,\left( {e_2 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_3 }{0.7,0.2,0.3}} \right) ,\right. \\&\left. \left( {e_3 ,\frac{u_3 }{0.7,0.4,0.3},\frac{u_4 }{0.5,0.1,0.4}} \right) \right\} \\ F_E= & {} \left\{ {{\begin{array}{l} {\left( {e_1 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_3 }{0.7,0.2,0.3},\frac{u_5 }{0.6,0.1,0.4}} \right) ,\left( {e_2 ,\frac{u_1 }{0.4,0.2,0.5},\frac{u_2 }{0.5,0.4,0.6},\frac{u_3 }{0.7,0.2,0.3}} \right) ,} \\ {\left( {e_3 ,\frac{u_3 }{0.7,0.4,0.3},\frac{u_4 }{0.5,0.1,0.4},\frac{u_5 }{0.5,0.2,0.3}} \right) } \end{array} }} \right\} \\ F_\emptyset= & {} \emptyset \end{aligned}$$

\(\left( {N^{L}, {\tilde{\cup } } \hbox {},\hbox {} {\tilde{\cap }} ,\tilde{\subseteq } } \right) \) is a distributive neutrosophic soft lattice. The Hasse diagram of it is shown in Fig. 2.

Fig. 2
figure 2

A distributive neutrosophic soft lattice

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Definition 3.20

\(( {N^{L},\tilde{\vee } \hbox {},\hbox {}\tilde{\wedge } \hbox {},\tilde{\le } })\) be a neutrosophic soft lattice. Then \(N^{L}\)is called a neutrosophic soft modular lattice, if it is satisfies the following property:

$$\begin{aligned} F_C \tilde{\le } F_A \Rightarrow F_A \tilde{\wedge } \left( {F_B \tilde{\vee } F_C } \right) =\left( {F_{A} \tilde{\wedge } F_B } \right) \tilde{\vee } F_C \end{aligned}$$

for all \(F_A ,F_B \hbox {and}F_C \in N^{L}\).

Example 3.21

Let \(U=\{ {u_1 ,u_2 ,u_3 ,u_4 ,u_5 ,u_6 }\}\) be a universe set and \(N^{L}= \{ F_\emptyset ,F_A ,F_B ,F_C ,F_D \}\,\tilde{\subseteq }\,NS (U).\) Then \(N^{L}\,\tilde{\subseteq }\,NS(U)\) is a neutrosophic soft lattice with the operations \({\tilde{\cup } } \ \hbox {and} \ {\tilde{\cap }} \). Suppose that

$$\begin{aligned} F_A= & {} \left\{ {\left( {e_5 ,\frac{u_5 }{0.5,0.2,0.7}} \right) } \right\} \\ F_B= & {} \left\{ {\left( {e_1 ,\frac{u_1 }{0.4,0.1,0.5},\frac{u_2 }{0.8,0.2,0.6}} \right) ,\left( {e_2 ,\frac{u_4 }{0.8,0.3,0.4}} \right) } \right\} \\ F_C= & {} \left\{ {\left( {e_3 ,\frac{u_2 }{0.7,0.4,0.3}} \right) } \right\} \\ F_D= & {} \left\{ {{\begin{array}{l} {\left( {e_1 ,\frac{u_1 }{0.4,0.1,0.5},\frac{u_2 }{0.8,0.2,0.6},\frac{u_3 }{0.6,0.1,0.4}} \right) ,\left( {e_2 ,\frac{u_2 }{0.5,0.4,0.6},\frac{u_4 }{0.7,0.2,0.3}} \right) ,} \\ {\left( {e_3 ,\frac{u_2 }{0.7,0.4,0.3},\frac{u_5 }{0.5,0.1,0.4}} \right) ,\left( {e_5 ,\frac{u_1 }{0.7,0.4,0.3},\frac{u_5 }{0.5,0.1,0.4}} \right) } \end{array} }} \right\} \\ F_\emptyset= & {} \emptyset \end{aligned}$$

\(\left( {N^{L}, {\tilde{\cup } }\hbox {},\hbox {} {\tilde{\cap } } ,\tilde{\subseteq } } \right) \)is a neutrosophic soft modular lattice. The Hasse diagram of it is shown in Fig. 3

Fig. 3
figure 3

A neutrosophic soft modular lattice

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4 Conclusion

In this paper, we defined the concept of neutrosophic soft lattice as an algebraic structure and showed that these definitions are equivalent. We then investigated some related properties and some characterization theorems. To extend this work one can study the properties of neutrosophic soft set in other algebraic structures and fields. In addition, based on these results, we can further probe the applications of neutrosophic soft lattice.