Implicative neutrosophic LI-ideals of lattice implication algebras

Abstract

Lattice implication algebra is a new logical algebraic system which is established by combining lattice and implication algebra. As a generalization of intuitionistic fuzzy set, neutrosophic set is introduced which deals with indeterminate membership in addition to degree of membership and degree of non-membership. In this article, the notion of neutrosophic set theory is applied to lattice implication algebras. The concept of implicative neutrosophic LI-ideals of a lattice implication algebra is introduced, and several properties are investigated. Relationship between a neutrosophic LI-ideal and an implicative neutrosophic LI-ideal is discussed, and conditions for a neutrosophic LI-ideal to be an implicative neutrosophic LI-ideal are provided. Characterizations of an implicative neutrosophic LI-ideal are considered, and the extension property of an implicative neutrosophic LI-ideal is studied.

Keywords

(implicative) LI-ideal (implicative) neutrosophic LI-ideal 06F35 03G25 08A72

1 Introduction

As a new logical algebraic system, lattice implication algebras are established by combining lattice and implication algebra (see [21]). The fuzzy set theory was introduced by Zadeh [22] in 1965, in order to explain the situations where data are imprecise or vague. On the other hand in 1986, a generalized form of fuzzy set namely, intuitionistic fuzzy set was introduced by Atanassaov [1] which deals with both the degree of membership (belongingness) and non-membership (non-belongingness) of an elements within a set. Konwar et al. studied intuitionistic fuzzy n-normed linear spaces, intuitionistic fuzzy n-Banach spaces, and intuitionistic fuzzy hypergroupoids (see [11 –14]). Smarandache [17] generalized the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS), and presented many examples. In neutrosophic sets, indeterminate membership is treated in addition to degree of membership and degree of non-membership. Neutrosophic set theory is applied to various part which is refered to the site

http://fs.gallup.unm.edu/neutrosophy.htm.

Jun and his colleagues applied the notion of neutrosophic set theory to BCK/BCI-algebras (see [7–10 , 19]). Borzooei et al. [3] studied commutative generalized neutrosophic ideals in BCK-algebras. Borzooei et al. [2] introduced the concept of neutrosophic LI-ideals and neutrosophic lattice ideals of a lattice implication algebra, and investigated several properties. They discussed relationship between a neutrosophic LI-ideal and a neutrosophic lattice ideal, and provided conditions for a neutrosophic lattice ideal to be a neutrosophic LI-ideal. Jun et al. studied LI-ideals in lattice implication algebras (see [4]), and Liu et al. studied implicative and prime LI-ideals in lattice implication algebras (see [6]).

In this paper, we apply the notion of neutrosophic set theory to lattice implication algebras. We introduce the concept of implicative neutrosophic LI-ideals of a lattice implication algebra, and investigate several properties. We discuss relationship between a neutrosophic LI-ideal and an implicative neutrosophic LI-ideal. We provide conditions for a neutrosophic LI-ideal to be an implicative neutrosophic LI-ideal. We consider characterizations of an implicative neutrosophic LI-ideal. We study the extension property of an implicative neutrosophic LI-ideal.

2 Preliminaries

By a lattice implication algebra we mean a bounded lattice (L, ∨ , ∧ , 0, 1) with order-reversing involution “ ′ ” and a binary operation “ → ” satisfying the following axioms:

y → (z → x) = z → (y → x) ,

y → y = 1,

y → z = z′ → y′,

y → z = z → y = 1 ⇒ x = z,

(y → z) → z = (z → y) → y,

(y ∨ z) → x = (y → x) ∧ (z → x) ,

(y ∧ z) → x = (y → x) ∨ (z → x) ,

for all y, z, x ∈ L . A lattice implication algebra L is called a lattice H-implication algebra if it satisfies:

(\forall y, z, x \in L) (y \lor z \lor ((y \land z) \to x) = 1) .

(2.1)

We can define a partial ordering ≤ on a lattice implication algebra L by y ≤ z if and only if y → z = 1.

In a lattice implication algebra L, the following hold (see [20]):

0 → y = 1, 1 → y = y and y → 1 =1.

y → z ≤ (z → x) → (y → x) .

y ≤ z implies z → x ≤ y → x and z → y ≤ z → z.

y′ = y → 0.

y ∨ z = (y → z) → z.

((z → y) → z′) ′ = y ∧ z = ((y → z) → y′) ′ .

y ≤ (y → z) → z.

Let G be a subset of a lattice implication algebra L with 0 ∈ G. Then G is called

an LI-ideal of L (see [5]) if it satisfies $(\forall y \in L) (\forall z \in G) ((y \to z)^{'} \in G \Rightarrow y \in G),$ (2.2)

an implicative LI-ideal (briefly, ILI-ideal) of L (see [6]) if it satisfies

$\begin{matrix} (\forall y, z, x \in L) ((((y \to z)^{'} \to z)^{'} \to x)^{'} \in G, \\ z \in G \Rightarrow (y \to z)^{'} \in G) . \end{matrix}$ (2.3)

Lemma 2.1.([5]). Every LI-ideal G of L satisfies the following implication: $(\forall y \in G) (\forall z \in L) (z \leq y \Rightarrow z \in G) .$

Let L be a non-empty set. A neutrosophic set (NS) in L (see [16]) is a structure of the form: $\begin{matrix} N_{\sim} : = {〈 y; N_{T} (y), N_{I} (y), N_{F} (y) 〉 ∣ y \in L} \end{matrix}$ where $N_{T} : L \to [0, 1]$ is a truth membership function, $N_{I} : L \to [0, 1]$ is an indeterminate membership function, and $N_{F} : L \to [0, 1]$ is a false membership function. For the sake of simplicity, we shall use the symbol $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ for the neutrosophic set $\begin{matrix} N_{\sim} : = {〈 y; N_{T} (y), N_{I} (y), N_{F} (y) 〉 ∣ y \in L} . \end{matrix}$

Given a neutrosophic set $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ in a lattice implication algebra L, we consider the following sets. $\begin{matrix} L (N_{T}; α) : = {y \in L ∣ N_{T} (y) \geq α}, \\ L (N_{I}; β) : = {y \in L ∣ N_{I} (y) \geq β}, \\ L (N_{F}; γ) : = {y \in L ∣ N_{F} (y) \leq γ}, \end{matrix}$ which are called neutrosophic level subsets of L.

We refer the reader to the books [21] for further information regarding lattice implication algebras, and to the site “http://fs.gallup.unm.edu/neutrosophy.htm” for further information regarding neutrosophic set theory.

3 Implicative neutrosophic LI-ideals

In what follows, let L denote a lattice implication algebra unless otherwise specified, and we consider the following sets.

$\begin{matrix} Γ_{N} & : = {(y, z) \in L \times L ∣ N_{T} (y) \geq N_{T} (z), N_{I} (y) \\ \geq N_{I} (z), N_{F} (y) \leq N_{F} (z)} \end{matrix}$ (3.1) and

$\begin{matrix} Γ_{N (min, max)} \\ : = {\frac{x}{(y, z)} \in \frac{L}{L \times L} | \begin{matrix} N_{T} (x) \geq min {N_{T} (y), N_{T} (z)} \\ N_{I} (x) \geq min {N_{I} (y), N_{I} (z)} \\ N_{F} (x) \leq max {N_{F} (y), N_{F} (z)} \end{matrix}} . \end{matrix}$ (3.2)

It is clear that $(\forall y, z \in L) ((y, z) \in Γ_{N} \Leftrightarrow \frac{y}{(z, z)} \in Γ_{N (min, max)}),$ (3.3) $\begin{matrix} (\forall y, z, x \in L) (\frac{x}{(y, z)} \in Γ_{N (min, max)} \\ \Rightarrow \frac{x}{(z, y)} \in Γ_{N (min, max)}), \end{matrix}$ (3.4) $\begin{matrix} (\forall y, z, x \in L) ((y, z) \in Γ_{N}, (z, x) \in Γ_{N} \\ \Rightarrow (y, x) \in Γ_{N}) . \end{matrix}$ (3.5)

Proposition 3.1. For any a, y, z, x ∈ L, we have $\begin{matrix} \frac{a}{(y, z)} & \in Γ_{N (min, max)}, (z, x) \in Γ_{N} \\ \Rightarrow \frac{a}{(y, x)} \in Γ_{N (min, max)}, \end{matrix}$ (3.6) $\begin{matrix} (a, y) & \in Γ_{N}, \frac{y}{(z, x)} \in Γ_{N (min, max)} \\ \Rightarrow \frac{a}{(z, x)} \in Γ_{N (min, max)} . \end{matrix}$ (3.7)

Proof. Let a, y, z, x ∈ L be such that $\frac{a}{(y, z)} \in Γ_{N (min, max)}$ and $(z, x) \in Γ_{N}$ . Then $\begin{matrix} N_{T} (a) \geq min {N_{T} (y), N_{T} (z)}, \\ N_{I} (a) \geq min {N_{I} (y), N_{I} (z)}, \\ N_{F} (a) \leq max {N_{F} (y), N_{F} (z)} \end{matrix}$ and $N_{T} (z) \geq N_{T} (x)$ , $N_{I} (z) \geq N_{I} (x)$ , $N_{F} (z) \leq N_{F} (x)$ . It follows that $\begin{matrix} N_{T} (a) \geq min {N_{T} (y), N_{T} (z)} \geq min {N_{T} (y), N_{T} (x)}, \\ N_{I} (a) \geq min {N_{I} (y), N_{I} (z)} \geq min {N_{I} (y), N_{I} (x)}, \\ N_{F} (a) \leq max {N_{F} (y), N_{F} (z)} \leq max {N_{F} (y), N_{F} (x)} . \end{matrix}$ Hence $\frac{a}{(y, x)} \in Γ_{N (min, max)}$ . Now assume that $(a, y) \in Γ_{N}$ and $\frac{y}{(z, x)} \in Γ_{N (min, max)}$ for all a, y, z, x ∈ L. Then

$N_{T} (a) \geq N_{T} (y), N_{I} (a) \geq N_{I} (y), N_{F} (a) \leq N_{F} (y),$ (3.8) and

$\begin{matrix} N_{T} (y) \geq min {N_{T} (z), N_{T} (x)}, \\ N_{I} (y) \geq min {N_{I} (z), N_{I} (x)}, \\ N_{F} (y) \leq max {N_{F} (z), N_{F} (x)} . \end{matrix}$ (3.9) Combining (3.8) and (3.9) induces $\begin{matrix} N_{T} (a) & \geq min {N_{T} (z), N_{T} (x)}, \\ N_{I} (a) & \geq min {N_{I} (z), N_{I} (x)}, \\ N_{F} (a) & \leq max {N_{F} (z), N_{F} (x)} . \end{matrix}$ This shows that $\frac{a}{(z, x)} \in Γ_{N (min, max)}$ .□

Definition 3.2.([2]). A neutrosophic set $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ in L is called a neutrosophic LI-ideal (briefly, NLI-ideal) of L if the following assertions are valid. $(\forall y \in L) ((0, y) \in Γ_{N}),$ (3.10) $(\forall y, z \in L) (\frac{y}{((y \to z)^{'}, z)} \in Γ_{N (min, max)}) .$ (3.11) Lemma 3.3.([2]). Every NLI-ideal $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ of L satisfies the following assertion. $(\forall y, z \in L) (y \leq z \Rightarrow (y, z) \in Γ_{N}) .$ (3.12)

Theorem 3.4. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be a neutrosophic set in L. Then the following are equivalent.

$N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L.

$N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ satisfies: $\begin{matrix} (\forall y, z, x \in L) ((x \to y)^{'} \leq z \\ \Rightarrow \frac{x}{(y, z)} \in Γ_{N (min, max)}) . \end{matrix}$ (3.13)

Proof. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of L. Then $\frac{x}{((x \to y)^{'}, y)} \in Γ_{N (min, max)}$ by (3.11). Assume that (x → y) ′ ≤ z for all y, z, x ∈ L. Then $((x \to y)^{'}, z) \in Γ_{N}$ by Lemma 3.3. It follows from (3.4) and (3.6) that $\frac{x}{(y, z)} \in Γ_{N (min, max)}$ .

Conversely, assume that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ satisfies the condition (3.13). Since (0 → y) ′ ≤ y for all y ∈ L, we have $\frac{0}{(y, y)} \in Γ_{N (min, max)}$ , that is, $(0, y) \in Γ_{N}$ . Since (y → z) ′ ≤ (y → z) ′ for all y, z ∈ L, we get $\frac{y}{(z, (y \to z)^{'})} \in Γ_{N (min, max)}$ and so $\frac{y}{((y \to z)^{'}, z)} \in Γ_{N (min, max)}$ by (3.4). Therefore $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L.□

Definition 3.5. A neutrosophic set $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ in L is called an implicative neutrosophic LI-ideal (briefly, INLI-ideal) of L if it satisfies (3.10) and

$\begin{matrix} (\forall y, z, x \in L) \\ (\frac{(y \to z)^{'}}{((((y \to z)^{'} \to z)^{'} \to x)^{'}, x)} \in Γ_{N (min, max)}) . \end{matrix}$ (3.14)

Example 3.6. Let L = {0, a, b, 1} be a set with Cayley tables as follows: $\begin{matrix} x & x^{'} \\ 0 & 1 \\ a & b \\ b & a \\ 1 & 0 \end{matrix} \begin{matrix} \to & 0 & a & b & 1 \\ 0 & 1 & 1 & 1 & 1 \\ a & b & 1 & b & 1 \\ b & a & a & 1 & 1 \\ 1 & 0 & a & b & 1 \end{matrix}$ Define ∨- and ∧-operations on L as follows: $y \lor z : = (y \to y) \to y, y \land z : = ((y^{'} \to y^{'}) \to y^{'})^{'}$ for all y, z ∈ L . Then L is a lattice implication algebra (see [21]). Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be a neutrosophic set in L defined by Table 1.

Table 1

Tabular representation of $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$

L	0	a	b	1
$N_{T} (y)$	0.6	0.3	0.3	0.3
$N_{I} (y)$	0.8	0.5	0.8	0.5
$N_{F} (y)$	0.2	0.2	0.7	0.7

It is routine to verify that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.

Example 3.7. Let L = {0, a, b, c, d, 1} be a set with Hasse diagram and Cayley tables as follows:

$\begin{matrix} x & x^{'} \\ 0 & 1 \\ a & c \\ b & d \\ c & a \\ d & b \\ 1 & 0 \end{matrix}$ $\begin{matrix} \to & 0 & a & b & c & d & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ a & c & 1 & b & c & b & 1 \\ b & d & a & 1 & b & a & 1 \\ c & a & a & 1 & 1 & a & 1 \\ d & b & 1 & 1 & b & 1 & 1 \\ 1 & 0 & a & b & c & d & 1 \end{matrix}$

Define ∨- and ∧-operations on L as follows: $y \lor z : = (y \to y) \to y, y \land z : = ((y^{'} \to y^{'}) \to y^{'})^{'}$ for all y, z ∈ L . Then L is a lattice implication algebra (see [5]). Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be a neutrosophic set in L defined by Table 2.

Table 2

Tabular representation of $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$

L	0	a	b	c	d	1
$N_{T} (y)$	0.7	0.7	0.5	0.5	0.7	0.5
$N_{I} (y)$	0.5	0.5	0.3	0.3	0.5	0.3
$N_{F} (y)$	0.3	0.3	0.6	0.6	0.3	0.6

It is routine to verify that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.

We discuss relationship between NLI-ideals and INLI-ideals.

Theorem 3.8. Every INLI-ideal is an NLI-ideal.

Proof. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an INLI-ideal of L. For any y, z ∈ L, we have $\begin{matrix} \frac{y}{((y \to z)^{'}, z)} & = \frac{(y \to 0)^{'}}{((((y \to 0)^{'} \to 0)^{'} \to z)^{'}, z)} \\ \in Γ_{N (min, max)} . \end{matrix}$ Therefore $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L.□

The converse of Theorem 3.8 is not true in general as seen in the following example.

Example 3.9. Let L = {0, a, b, 1} be a set with Cayley tables as follows:

x	x ^′	⊳	0	a	b	1
0	1	0	1	1	1	1
a	b	a	b	1	1	1
b	a	b	a	b	1	1
1	0	1	0	a	b	1

Define ∨- and ∧-operations on L as follows: $y \lor z : = (y \to y) \to y, y \land z : = ((y^{'} \to y^{'}) \to y^{'})^{'}$ for all y, z ∈ L . Then L is a lattice implication algebra (see [21]).

Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be a neutrosophic set in L defined by Table 3.

Table 3

Tabular representation of $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$

L	0	a	b	1
$N_{T} (y)$	0.6	0.3	0.3	0.3
$N_{I} (y)$	0.7	0.5	0.5	0.5
$N_{F} (y)$	0.4	0.7	0.7	0.7

It is routine to check that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L. But, it is not an INLI-ideal of L since $\frac{(b \to a)^{'}}{((((b \to a)^{'} \to a)^{'} \to 0)^{'}, 0)} \notin Γ_{N (min, max)}$ .

We provide conditions for an NLI-ideal to be an INLI-ideal.

Theorem 3.10. In a lattice H-implication algebra, every NLI-ideal is an INLI-ideal.

Proof. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of a lattice H-implication algebra L. Since y → (y → z) = y → z for all y, z ∈ L, it follows from (3.11) that $\begin{matrix} \frac{(y \to z)^{'}}{((((y \to z)^{'} \to z)^{'} \to x)^{'}, x)} \\ = \frac{(z^{'} \to y^{'})^{'}}{((((y \to z)^{'} \to z)^{'} \to x)^{'}, x)} \\ = \frac{(z^{'} \to (z^{'} \to y^{'}))^{'}}{((((y \to z)^{'} \to z)^{'} \to x)^{'}, x)} \\ = \frac{((y \to z)^{'} \to z)^{'}}{((((y \to z)^{'} \to z)^{'} \to x)^{'}, x)} \in Γ_{N (min, max)} \end{matrix}$ for all y, z, x ∈ L. Therefore $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.□

Lemma 3.11. If a neutrosophic set $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ in L satisfies the condition (3.10), then

$(\forall y, z \in L) (\frac{y}{(z, 0)} \in Γ_{N (min, max)} \Leftrightarrow (y, z) \in Γ_{N}) .$ (3.15)

Proof. Straightforward.□

Theorem 3.12. A neutrosophic set $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ in L is an INLI-ideal of L if and only if $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L which satisfies: $(\forall y, z \in L) (((y \to z)^{'}, ((y \to z)^{'} \to z)^{'}) \in Γ_{N}) .$ (3.16)

Proof. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an INLI-ideal of L. Then it is an NLI-ideal of L by Theorem 3.8, and $\begin{matrix} \frac{(y \to z)^{'}}{(((y \to z)^{'} \to z)^{'}, 0)} & = \frac{(y \to z)^{'}}{((((y \to z)^{'} \to z)^{'} \to 0)^{'}, 0)} \\ \in Γ_{N (min, max)} \end{matrix}$ for all y, z ∈ L. Thus (3.16) is valid by Lemma 3.11.

Conversely, let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of L with the condition (3.16). Then $\begin{matrix} \frac{((y \to z)^{'} \to z)^{'}}{((((y \to z)^{'} \to z)^{'} \to x)^{'}, x)} \in Γ_{N (min, max)}, \end{matrix}$ by (3.11), and so $\frac{(y \to z)^{'}}{((((y \to z)^{'} \to z)^{'} \to x)^{'}, x)} \in Γ_{N (min, max)}$ for all y, z, x ∈ L by (3.7). Therefore $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.□

Theorem 3.13. If an NLI-ideal $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ of L satisfies:

$\begin{matrix} (\forall y, z, x, u \in L) \\ (\frac{((y \to x)^{'} \to (z \to x)^{'})^{'}}{((((y \to z)^{'} \to x)^{'} \to u)^{'}, u)} \in Γ_{N (min, max)}), \end{matrix}$ (3.17) then $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.

Proof. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of L which satisfies the condition (3.10). If we put x = z in (3.10) and use (I2), (a1) and (a4), then $\begin{matrix} \frac{(y \to z)^{'}}{((((y \to z)^{'} \to z)^{'} \to u)^{'}, u)} \\ = \frac{((y \to x)^{'} \to (z \to x)^{'})^{'}}{((((y \to z)^{'} \to x)^{'} \to u)^{'}, u)} \in Γ_{N (min, max)} \end{matrix}$ for all y, z, x, u ∈ L. Hence $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.□

Theorem 3.14. If an NLI-ideal $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ of L satisfies: $\begin{matrix} (\forall y, z, x \in L) (((((y \to x)^{'} \to (z \to x)^{'})^{'}), \\ ((y \to z)^{'} \to x)^{'}) \in Γ_{N}), \end{matrix}$ (3.18) then $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.

Proof. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of L which satisfies the condition (3.17). Then $\frac{((y \to z)^{'} \to x)^{'}}{((((y \to z)^{'} \to x)^{'} \to u)^{'}, u)} \in Γ_{N (min, max)}$ for all y, z, x, u ∈ L by (3.11). It follows from (3.17) and (3.7) that $\begin{matrix} \frac{((y \to x)^{'} \to (z \to x)^{'})^{'}}{((((y \to z)^{'} \to x)^{'} \to u)^{'}, u)} \in Γ_{N (min, max)} \end{matrix}$ for all y, z, x, u ∈ L. Therefore $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L by Theorem 3.13.□

Proposition 3.15. Every INLI-ideal $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ of L satisfies:

$(\forall y, z \in L) ((y, (y \to (z \to y)^{'})^{'}) \in Γ_{N}) .$ (3.19)

Proof. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an INLI-ideal of L. Then it is an NLI-ideal of L by Theorem 3.8. Let y, z ∈ L. Then

$\begin{matrix} 1 & = (z \to y) \to (y^{'} \to z^{'}) = y^{'} \to ((z \to y) \to z^{'}) \\ = y^{'} \to (z \to (z \to y)^{'}) = (z \to (z \to y)^{'})^{'} \to y \\ \leq (y \to (z \to y)^{'}) \to ((z \to (z \to y)^{'})^{'} \to (z \to y)^{'}) \\ = ((z \to (z \to y)^{'})^{'} \to (z \to y)^{'})^{'} \to (y \to (z \to y)^{'})^{'} \end{matrix}$

and so ((z → (z → y) ′) ′ → (z → y) ′) ′ → (y → (z → y) ′) ′ = 1, that is, $((z \to (z \to y)^{'})^{'} \to (z \to y)^{'})^{'} \leq (y \to (z \to y)^{'})^{'} .$ Hence

$\begin{matrix} (((z \to (z \to y)^{'})^{'} \to (z \to y)^{'})^{'}, (y \to (z \to y)^{'})^{'}) \\ \in Γ_{N} \end{matrix}$ (3.20) by Lemma 3.3. Using Theorem 3.12, we have

$\begin{matrix} ((z \to (z \to y)^{'})^{'}, ((z \to (z \to y)^{'})^{'} \to (z \to y)^{'})^{'}) \\ \in Γ_{N} . \end{matrix}$ (3.21) It follows from (3.11) that

$((z \to (z \to y)^{'})^{'}, (y \to (z \to y)^{'})^{'}) \in Γ_{N} .$ (3.22) Note that $\begin{matrix} 1 & = z^{'} \to (z \to y) = (z \to y)^{'} \to z \\ \leq (y \to (z \to y)^{'}) \to (y \to z) \\ = (y \to z)^{'} \to (y \to (z \to y)^{'})^{'}, \end{matrix}$ and so (y → z) ′ → (y → (z → y) ′) ′ = 1, that is, (y → z) ′ ≤ (y → (z → y) ′) ′. Thus ((y → z) ′, $(y \to (z \to y)^{'})^{'}) \in Γ_{N}$ by Lemma 3.3. Since $\begin{matrix} (y \to z)^{'} & = (z^{'} \to y^{'})^{'} = (((z^{'} \to y^{'}) \to y^{'}) \to y^{'})^{'} \\ = (y \to ((z^{'} \to y^{'}) \to y^{'})^{'})^{'} \\ = (y \to ((y^{'} \to z^{'}) \to z^{'})^{'})^{'} \\ = (y \to (z \to (z \to y)^{'})^{'})^{'}, \end{matrix}$ we get

$((y \to (z \to (z \to y)^{'})^{'})^{'}, (y \to (z \to y)^{'})^{'}) \in Γ_{N} .$ (3.23) Since $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L, we have

$\begin{matrix} \frac{y}{((y \to (z \to (z \to y)^{'})^{'})^{'}, (z \to (z \to y)^{'})^{'})} \\ \in Γ_{N (min, max)} . \end{matrix}$ (3.24) Hence $\frac{y}{((y \to (z \to y)^{'})^{'}, (z \to (z \to y)^{'})^{'})} \in Γ_{N (min, max)}$ (3.25) by (3.23), (3.24) and (3.6). It follows from (3.22) and (3.6) that $\begin{matrix} \frac{y}{((y \to (z \to y)^{'})^{'}, (y \to (z \to y)^{'})^{'})} \in Γ_{N (min, max)}, \end{matrix}$ that is, $(y, (y \to (z \to y)^{'})^{'}) \in Γ_{N}$ .□

Proposition 3.16. If an NLI-ideal $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ of L satisfies (3.19), then it satisfies the following assertion.

$\begin{matrix} (\forall y, z, x \in L) \\ (\frac{y}{(((y \to (z \to y)^{'})^{'} \to x)^{'}, x)} \in Γ_{N (min, max)}) . \end{matrix}$ (3.26)

Proof. Assume that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L satisfying (3.19). Then $\begin{matrix} \frac{(y \to (z \to y)^{'})^{'}}{(((y \to (z \to y)^{'})^{'} \to x)^{'}, x)} \in Γ_{N (min, max)} \end{matrix}$ for all y, z, x ∈ L by (3.11). It follows from (3.19) and (3.7) that $\frac{y}{(((y \to (z \to y)^{'})^{'} \to x)^{'}, x)} \in Γ_{N (min, max)}$ for all y, z, x ∈ L.□

Theorem 3.17. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of L which satisfies the condition (3.26). Then $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.

Proof. Assume that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal of L satisfying the condition (3.26). Taking x = 0 in (3.26) induces $\frac{y}{((y \to (z \to y)^{'})^{'}, 0)} \in Γ_{N (min, max)}$ , and so

$(y, (y \to (z \to y)^{'})^{'}) \in Γ_{N}$ (3.27) for all y, z ∈ L. Since $\begin{matrix} ((y \to z)^{'} \to z)^{'} \\ = (z^{'} \to (z^{'} \to y^{'}))^{'} \\ = (z^{'} \to ((y^{'} \to (z^{'} \to y^{'})) \to (z^{'} \to y^{'})))^{'} \\ = (z^{'} \to ((y^{'} \to (y \to z)) \to (y \to z)))^{'} \\ = (z^{'} \to (((y \to z) \to y^{'}) \to y^{'}))^{'} \\ = (((y \to z) \to y^{'}) \to (z^{'} \to y^{'}))^{'} \\ = ((y \to (y \to z)^{'}) \to (y \to z))^{'} \\ = ((y \to z)^{'} \to (y \to (y \to z)^{'})^{'})^{'}, \end{matrix}$ we have ((y → z) ′, ((y → z) ′ → z) ′) = ((y → z) ′, $((y \to z)^{'} \to (y \to (y \to z)^{'})^{'})^{'}) \in Γ_{N}$ . for all y, z ∈L. It follows from Theorem 3.12 that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.□

Corollary 3.18. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of L which satisfies the condition (3.19). Then $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.

Theorem 3.19. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be a neutrosophic set in L. Given ɛ_T, ɛ_I, ɛ_F ∈ [0, 1], consider the sets: $\begin{matrix} L (N_{\sim}; ɛ_{T}) : = {y \in L ∣ N_{T} (y) \geq ɛ_{T}}, \\ L (N_{\sim}; ɛ_{I}) : = {y \in L ∣ N_{T} (y) \geq ɛ_{I}}, \\ L (N_{\sim}; ɛ_{F}) : = {y \in L ∣ N_{T} (y) \leq ɛ_{F}} . \end{matrix}$ Then $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an NLI-ideal (resp. INLI-ideal) of L if and only if $L (N_{\sim}; ɛ_{T})$ , $L (N_{\sim}; ɛ_{I})$ and $L (N_{\sim}; ɛ_{F})$ are LI-ideals (resp. ILI-ideals) of L for all ɛ_T, ɛ_I, ɛ_F ∈ [0, 1].

Proof. The proof is routine and so it is omitted.□

Lemma 3.20.([6]). An LI-ideal I of L is an ILI-ideal of L if and only if the following assertion is valid.

$(\forall y, z \in L) (((y \to z)^{'} \to z)^{'} \in I \Rightarrow (y \to z)^{'} \in I) .$ (3.28)

Theorem 3.21. Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ be an NLI-ideal of L. Then the following are equvalent.

$N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L.

The set $\begin{matrix} I_{b} & : = {y \in L ∣ (y \to b)^{'} \in L (N_{\sim}; ɛ_{T}) \\ \cap L (N_{\sim}; ɛ_{I}) \cap L (N_{\sim}; ɛ_{F})} \end{matrix}$ is an LI-ideal of L for all b ∈ L and ɛ_T, ɛ_I, ɛ_F ∈ [0, 1].

Proof. Assume that I_b is an LI-ideal of L for any b ∈ L and ɛ_T, ɛ_I, ɛ_F ∈ [0, 1]. Let y, z ∈ L be such that $((y \to z)^{'} \to z)^{'} \in L (N_{\sim}; ɛ_{T})$ $\cap L (N_{\sim}; ɛ_{I}) \cap L (N_{\sim}; ɛ_{F})$ . Then (y → z) ′ ∈ I_z. Since $(z \to z)^{'} = 1^{'} = 0 \in L (N_{\sim}; ɛ_{T}) \cap L (N_{\sim}; ɛ_{I}) \cap L (N_{\sim}; ɛ_{F})$ , we have z ∈ I_z. If follows from (2.2) that y ∈ I_z. Thus $(y \to z)^{'} \in L (N_{\sim}; ɛ_{T})$ $\cap L (N_{\sim}; ɛ_{I}) \cap L (N_{\sim}; ɛ_{F})$ , and thus $L (N_{\sim}; ɛ_{T})$ , $L (N_{\sim}; ɛ_{I})$ and $L (N_{\sim}; ɛ_{F})$ are ILI-ideals of L by Lemma 3.20. Therefore $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L by Theorem 3.19.

Conversely, suppose that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L. Obviously 0 ∈ I_b for all b ∈ L. Let y, z ∈ L be such that (y → z) ′ ∈ I_b and z ∈ I_b. Then $((y \to z)^{'} \to b)^{'} \in L (N_{\sim}; ɛ_{T}) \cap L (N_{\sim}; ɛ_{I}) \cap L (N_{\sim}; ɛ_{F})$ and $(z \to b)^{'} \in L (N_{\sim}; ɛ_{T}) \cap L (N_{\sim}; ɛ_{I}) \cap L (N_{\sim}; ɛ_{F})$ . Since $\begin{matrix} z \to b & \leq (y \to z) \to (y \to b) = (y \to b)^{'} \to (y \to z)^{'} \\ \leq ((y \to z)^{'} \to b) \to ((y \to b)^{'} \to b) \\ = ((y \to b)^{'} \to b)^{'} \to ((y \to z)^{'} \to b)^{'}, \end{matrix}$ we get (((y → b) ′ → b) ′ → ((y → z) ′ → b) ′) ′ ≤ (z → b) ′. Hence $\begin{matrix} ((y \to b)^{'} \to b)^{'} & \in L (N_{\sim}; ɛ_{T}) \cap L (N_{\sim}; ɛ_{I}) \\ \cap L (N_{\sim}; ɛ_{F}), \end{matrix}$ and thus $(y \to b)^{'} \in L (N_{\sim}; ɛ_{T}) \cap L (N_{\sim}; ɛ_{I}) \cap L (N_{\sim}; ɛ_{F})$ by Theorem 3.19 and Lemma 3.20. Hence y ∈ I_b, and I_b is an LI-ideal of L.□

Theorem 3.22. (Extension property) Let $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ and $M_{\sim} = (M_{T},$ $M_{I},$ $M_{F})$ be NLI-ideals in L such that $N_{\sim} (0) = M_{\sim} (0)$ and $N_{\sim} \subseteq M_{\sim}$ , that is, $N_{T} \leq M_{T}$ , $N_{I} \leq M_{I}$ and $N_{F} \geq M_{F}$ . If $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L, then so is $M_{\sim} = (M_{T},$ $M_{I},$ $M_{F})$ .

Proof. Assume that $N_{\sim} = (N_{T},$ $N_{I},$ $N_{F})$ is an INLI-ideal of L. Since $N_{\sim} (0) = M_{\sim} (0)$ and $N_{\sim} \subseteq M_{\sim}$ , we have

$(\forall y \in L) ((y, 0) \in Γ_{N} \Rightarrow (y, 0) \in Γ_{M}) .$ (3.29) Let y, z ∈ L. If we take w = ((y → z) ′ → z) ′, then $\begin{matrix} (((y \to w)^{'} \to z)^{'} \to z)^{'} \\ = (z^{'} \to (z^{'} \to (w^{'} \to y^{'})))^{'} \\ = (w^{'} \to (z^{'} \to (z^{'} \to y^{'})))^{'} \\ = (w^{'} \to ((y \to z)^{'} \to z))^{'} \\ = (((y \to z)^{'} \to z)^{'} \to w)^{'} \\ = 1^{'} = 0 . \end{matrix}$ Hence

$\begin{matrix} (((y \to w)^{'} \to z)^{'}, 0) \\ = (((y \to w)^{'} \to z)^{'}, (((y \to w)^{'} \to z)^{'} \to z)^{'}) \in Γ_{N} \end{matrix}$

by Theorem 3.12, and so $(((y \to z)^{'} \to w)^{'}, 0) = (((y \to w)^{'} \to z)^{'}, 0) \in Γ_{M}$ by (3.29). Using (3.11) implies that $\frac{(y \to z)^{'}}{(((y \to z)^{'} \to w)^{'}, w)} \in Γ_{M (min, max)}$ . It follows from (3.6) that $\begin{matrix} \frac{(y \to z)^{'}}{(0, ((y \to z)^{'} \to z)^{'})} = \frac{(y \to z)^{'}}{(0, w)} \in Γ_{M (min, max)}, \end{matrix}$ that is, $((y \to z)^{'}, ((y \to z)^{'} \to z)^{'}) \in Γ_{M}$ . Therefore $M_{\sim} = (M_{T},$ $M_{I},$ $M_{F})$ is an INLI-ideal of L by Theorem 3.12.□

4 Conclusions

To establish an alternative logic for knowledge representation and reasoning, Xu has proposed a logical algebra—lattice implication algebra in 1993 by combining algebraic lattice and implication algebra in [Y. Xu, Lattice implication algebra. J. Southwest Jiaotong Univ, 28 (1993), no. 1, 20–27]. Using real standard or nonstandard subsets T, I and F of ] ^-0, 1⁺], Smarandache has introduced the notion of neutrosophic sets, which is a generalization of intuitionistic fuzzy set, in which indeterminate membership is dealt with in addition to degree of membership and degree of non-membership in intuitionistic fuzzy set. In this manuscript, we have applied the notion of neutrosophic set theory to lattice implication algebras. We have introduced the concept of implicative neutrosophic LI-ideals of a lattice implication algebra, and have investigated several properties. We have discussed relationship between a neutrosophic LI-ideal and an implicative neutrosophic LI-ideal, and have provided conditions for a neutrosophic LI-ideal to be an implicative neutrosophic LI-ideal. We have considered characterizations of an implicative neutrosophic LI-ideal, and have studied the extension property of an implicative neutrosophic LI-ideal.

Footnotes

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable suggestions. The second author is supported by a grant of National Natural Science Foundation of China (11971384).

References

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Borzooei

R.A.

, Sabetkish

, Neutrosophic LI-ideals in lattice implication algebras, Neutrosophic Sets and Systems (submitted).

Borzooei

R.A.

, Zhang

X.H.

, Smarandache

and Jun

Y.B.

, Commutative generalized neutrosophic ideals in BCK-algebras, Symmetry 10 (2018), 350. doi: 10.3390/sym10080350

Jun

Y.B.

, On LI-ideals and prime LI-ideals of lattice implication algebras, J. Korean Math. Soc. 36(2) (1999), 369–380.

Jun

Y.B.

, Roh

E.H.

and Xu

, LI-ideals in lattice implication algebras, Bull. Korean Math. Soc. 35(1) (1998), 13–24.

Liu

Y.L.

, Liu

S.Y.

, Xu

and Qin

K.Y.

, ILI-ideals and prime LI-ideals in lattice implication algebras, Inform. Sci. 155 (2003), 157–175.

Jun

Y.B.

, Neutrosophic subalgebras of several types in BCK/BCI-algebras, Ann. Fuzzy Math. Inform. 14(1) (2017), 75–86.

Jun

Y.B.

, Kim

S.J.

and Smarandache

, Interval neutrosophic sets with applications in BCK/BCI-algebra, Axioms 7 (2018), 23. doi: 10.3390/axioms7020023

Jun

Y.B.

, Smarandache

and Bordbar

, Neutrosophic

N

-structures applied to BCK/BCI-algebras, Information 8 (2017), 128. doi: 10.3390/info8040128

10.

Jun

Y.B.

, Smarandache

, Song

S.Z.

and Khan

, Neutrosophic positive implicative

N

-ideals in BCK/BCI-algebras, Axioms 7 (2018), 3. doi: 10.3390/axioms7010003

11.

Konwar

and Debnath

, Continuity and banach contraction principle in intuitionistic fuzzy n-normed linear spaces, J. Intell. Fuzzy Systems 33(4) (2017), 2363–2373. DOI: 10.3233/JIFS-17500

12.

Konwar

and Debnath

, Some new contractive conditions and related fixed point theorems in intuitionistic fuzzy n-Banach spaces, J. Intell. Fuzzy Systems 34(1) (2018), 361–372. DOI: 10.3233/JIFS-171372

13.

Konwar

, Davvaz

and Debnath

, Approximation of new bounded operators in intuitionistic fuzzy n-Banach spaces, J. Intell. Fuzzy Systems 35(6) (2018), 6301–6312. DOI: 10.3233/JIFS-181094

14.

Konwar

, Davvaz

and Debnath

, Results on generalized intuitionistic fuzzy hypergroupoids, J. Intell. Fuzzy Systems 36(3) (2019), 2571–2580. DOI: 10.3233/JIFS-181522

15.

Öztürk

M.A.

and Jun

Y.B.

, Neutrosophic ideals in BCK/BCI-algebras based on neutrosophic points, J. Inter. Math. Virtual Inst. 8 (2018), 1–17.

16.

Smarandache

, A unifying field in logics. Neutrosophy: Neutrosophic probability, set and logic, Rehoboth: American Research Press (1999).

17.

Smarandache

, Neutrosophic set, a generalization of intuitionistic fuzzy sets, International Journal of Pure and Applied Mathematics 24(5) (2005), 287–297.

18.

Song

S.Z.

, Khan

, Smarandache

and Jun

Y.B.

, A novel extension of neutrosophic sets and its application in BCK/BI-algebras, New Trends in Neutrosophic Theory and Applications (Volume II), Pons Editions, Brussels, Belium, EU 2018, 308–326, by Florentin Smarandache and Surapati Pramanik (Editors).

19.

Song

S.Z.

, Smarandache

and Jun

Y.B.

, Neutrosophic commutative

N

-ideals in BCK-algebras, Information 8 (2017), 130. doi: 10.3390/info8040130

20.

, Lattice implication algebras, J. Southwest Jiaotong Univ. 1 (1993), 20–27.

21.

, Ruan

, Qin

K.Y.

and Liu

, Lattice-valued logic, Studies in Fuzzyness and Soft Computing, Vol. 132, Springer-Verlag, Berlin, Heidelberg, New York, 2003.

22.

Zadeh

L.A.

, Fuzzy sets, Inform Control 8 (1965), 338–353.