Neutrosophic N-Structures Applied to BCK/BCI-Algebras

Abstract

Neutrosophic $\mathcal{N}$-structures with applications in $BCK/BCI$-algebras is discussed. The notions of a neutrosophic $\mathcal{N}$-subalgebra and a (closed) neutrosophic $\mathcal{N}$-ideal in a $BCK/BCI$-algebra are introduced, and several related properties are investigated. Characterizations of a neutrosophic $\mathcal{N}$-subalgebra and a neutrosophic $\mathcal{N}$-ideal are considered, and relations between a neutrosophic $\mathcal{N}$-subalgebra and a neutrosophic $\mathcal{N}$-ideal are stated. Conditions for a neutrosophic $\mathcal{N}$-ideal to be a closed neutrosophic $\mathcal{N}$-ideal are provided.

1. Introduction

$BCK$-algebras entered into mathematics in 1966 through the work of Imai and Iséki [1], and they have been applied to many branches of mathematics, such as group theory, functional analysis, probability theory and topology. Such algebras generalize Boolean rings as well as Boolean D-posets ($MV$-algebras). Additionally, Iséki introduced the notion of a $BCI$-algebra, which is a generalization of a $BCK$-algebra (see [2]).

A (crisp) set A in a universe X can be defined in the form of its characteristic function ${\mu }_A:X\to \{0,1\}$ yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set $A.$ So far, most of the generalizations of the crisp set have been conducted on the unit interval $[0,1]$, and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point $\left\{1\right\}$ into the interval $[0,1].$ Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply a mathematical tool. To attain such an object, Jun et al. [3] introduced a new function, called a negative-valued function, and constructed $\mathcal{N}$-structures. Zadeh [4] introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. As a generalization of fuzzy sets, Atanassov [5] introduced the degree of nonmembership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as an independent component in 1995 (published in 1998) and defined the neutrosophic set on three components:

$$\begin{array}{c}(\mathrm{t},\mathrm{i},\mathrm{f})=(\mathrm{truth},\mathrm{indeterminacy},falsehood)\hfill \end{array}$$

For more details, refer to the following site:

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

In this paper, we discuss a neutrosophic $\mathcal{N}$-structure with an application to $BCK/BCI$-algebras. We introduce the notions of a neutrosophic $\mathcal{N}$-subalgebra and a (closed) neutrosophic $\mathcal{N}$-ideal in a $BCK/BCI$-algebra, and investigate related properties. We consider characterizations of a neutrosophic $\mathcal{N}$-subalgebra and a neutrosophic $\mathcal{N}$-ideal. We discuss relations between a neutrosophic $\mathcal{N}$-subalgebra and a neutrosophic $\mathcal{N}$-ideal. We provide conditions for a neutrosophic $\mathcal{N}$-ideal to be a closed neutrosophic $\mathcal{N}$-ideal.

2. Preliminaries

We let $K\left(\tau \right)$ be the class of all algebras with type $\tau =(2,0)$. A BCI-algebra refers to a system $X:=(X,\ast ,\theta )\in K\left(\tau \right)$ in which the following axioms hold:

(I)

$\left(\right(x\ast y)\ast (x\ast z\left)\right)\ast (z\ast y)=\theta$,

(II)

$(x\ast (x\ast y\left)\right)\ast y=\theta$,

(III)

$x\ast x=\theta$,

(IV)

$x\ast y=y\ast x=\theta \Rightarrow x=y$.

for all $x,y,z\in X.$ If a BCI-algebra X satisfies $\theta \ast x=\theta$ for all $x\in X,$ then we say that X is a BCK-algebra. We can define a partial ordering ⪯ by

$$(\forall x,y\in X)(x⪯y\Rightarrow x\ast y=\theta )$$

In a BCK/BCI-algebra X, the following hold:

$$\begin{array}{c}(\forall x\in X)(x\ast \theta =x)\hfill \end{array}$$

(1)

$$\begin{array}{c}(\forall x,y,z\in X)\left(\right(x\ast y)\ast z=(x\ast z)\ast y)\hfill \end{array}$$

(2)

A non-empty subset S of a $BCK/BCI$-algebra X is called a subalgebra of X if $x\ast y\in S$ for all $x,y\in S.$

A subset I of a $BCK/BCI$-algebra X is called an ideal of X if it satisfies the following:

(I1)

$0\in I$,

(I2)

$(\forall x,y\in X)$$(x\ast y\in I,y\in I\Rightarrow x\in I)$.

We refer the reader to the books [6,7] for further information regarding BCK/BCI-algebras.

For any family $\{a_i\mid i\in \Lambda \}$ of real numbers, we define

$$\bigvee \{a_i\mid i\in \Lambda \}:=\left\{\begin{array}{cc}max\{a_i\mid i\in \Lambda \}\hfill & \mathrm{if}\Lambda \mathrm{is}\mathrm{finite}\hfill \\ sup\{a_i\mid i\in \Lambda \}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

$$\bigwedge \{a_i\mid i\in \Lambda \}:=\left\{\begin{array}{cc}min\{a_i\mid i\in \Lambda \}\hfill & \mathrm{if}\Lambda \mathrm{is}\mathrm{finite}\hfill \\ inf\{a_i\mid i\in \Lambda \}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

We denote by $\mathcal{F}(X,[-1,0\left]\right)$ the collection of functions from a set X to $[-1,0].$ We say that an element of $\mathcal{F}(X,[-1,0\left]\right)$ is a negative-valued function from X to $[-1,0]$ (briefly, $\mathcal{N}$-function on X). An $\mathcal{N}$-structure refers to an ordered pair $(X,f)$ of X and an $\mathcal{N}$-function f on X (see [3]). In what follows, we let X denote the nonempty universe of discourse unless otherwise specified.

A neutrosophic $\mathcal{N}$-structure over X (see [8]) is defined to be the structure:

$$X_{\mathbf{N}}:=\frac{X}{(T_N,I_N,F_N)}=\left\{\frac{x}{(T_N\left(x\right),I_N\left(x\right),F_N\left(x\right))}\mid x\in X\right\}$$

(3)

where $T_N$, $I_N$ and $F_N$ are $\mathcal{N}$-functions on X, which are called the negative truth membership function, the negative indeterminacy membership function and the negative falsity membership function, respectively, on X.

We note that every neutrosophic $\mathcal{N}$-structure $X_{\mathbf{N}}$ over X satisfies the condition:

$$(\forall x\in X)\left(-3\le T_N\left(x\right)+I_N\left(x\right)+F_N\left(x\right)\le 0\right)$$

3. Application in $\mathit{BCK}/\mathit{BCI}$-Algebras

In this section, we take a $BCK/BCI$-algebra X as the universe of discourse unless otherwise specified.

Definition 1.

A neutrosophic $\mathcal{N}$-structure $X_{\mathbf{N}}$ over X is called a neutrosophic $\mathcal{N}$-subalgebra of X if the following condition is valid:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{T}_N(x\ast y)\le \bigvee \{T_N\left(x\right),T_N\left(y\right)\}\hfill \\ I_N(x\ast y)\ge \bigwedge \{I_N\left(x\right),I_N\left(y\right)\}\hfill \\ F_N(x\ast y)\le \bigvee \{F_N\left(x\right),F_N\left(y\right)\}\hfill \end{array}\right)\end{array}$$

(4)

Example 1.

Consider a $BCK$-algebra $X=\{\theta ,a,b,c\}$ with the following Cayley table.

*	$\mathbf{\theta }$	$\mathit{a}$	$\mathit{b}$	$\mathit{c}$
θ	θ	θ	θ	θ
a	a	θ	θ	a
b	b	a	θ	b
c	c	c	c	θ

The neutrosophic $\mathcal{N}$-structure

$$\begin{array}{c}\hfill X_{\mathbf{N}}=\left\{\frac{\theta }{(-0.7,-0.2,-0.6)},\frac{a}{(-0.5,-0.3,-0.4)},\frac{b}{(-0.5,-0.3,-0.4)},\frac{c}{(-0.3,-0.8,-0.5)}\right\}\end{array}$$

over X is a neutrosophic $\mathcal{N}$-subalgebra of X.

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X and let $\alpha ,\beta ,\gamma \in [-1,0]$ be such that $-3\le \alpha +\beta +\gamma \le 0$. Consider the following sets:

$$\begin{array}{c}{T}_N^{\alpha }:=\{x\in X\mid T_N\left(x\right)\le \alpha \}\hfill \\ I_N^{\beta }:=\{x\in X\mid I_N\left(x\right)\ge \beta \}\hfill \\ F_N^{\gamma }:=\{x\in X\mid F_N\left(x\right)\le \gamma \}\hfill \end{array}$$

The set

$$\begin{array}{c}\hfill X_{\mathbf{N}}(\alpha ,\beta ,\gamma ):=\{x\in X\mid T_N\left(x\right)\le \alpha ,I_N\left(x\right)\ge \beta ,F_N\left(x\right)\le \gamma \}\end{array}$$

is called the $(\alpha ,\beta ,\gamma )$-level set of $X_{\mathbf{N}}$. Note that

$$X_{\mathbf{N}}(\alpha ,\beta ,\gamma )=T_N^{\alpha }\cap I_N^{\beta }\cap F_N^{\gamma }$$

Theorem 1.

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X and let $\alpha ,\beta ,\gamma \in [-1,0]$ be such that $-3\le \alpha +\beta +\gamma \le 0$. If $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-subalgebra of X, then the nonempty $(\alpha ,\beta ,\gamma )$-level set of $X_{\mathbf{N}}$ is a subalgebra of X.

Proof. 

Let $\alpha ,\beta ,\gamma \in [-1,0]$ be such that $-3\le \alpha +\beta +\gamma \le 0$ and $X_{\mathbf{N}}(\alpha ,\beta ,\gamma )\ne \mathsf{\varnothing }$. If $x,y\in X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$, then $T_N\left(x\right)\le \alpha$, $I_N\left(x\right)\ge \beta$, $F_N\left(x\right)\le \gamma$, $T_N\left(y\right)\le \alpha$, $I_N\left(y\right)\ge \beta$ and $F_N\left(y\right)\le \gamma$. It follows from Equation (4) that

$T_N(x\ast y)\le \bigvee \{T_N\left(x\right),T_N\left(y\right)\}\le \alpha$,

$I_N(x\ast y)\ge \bigwedge \{I_N\left(x\right),I_N\left(y\right)\}\ge \beta$, and

$F_N(x\ast y)\le \bigvee \{F_N\left(x\right),F_N\left(y\right)\}\le \gamma$.

Hence, $x\ast y\in X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$, and therefore $X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$ is a subalgebra of X. ☐

Theorem 2.

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X and assume that $T_N^{\alpha }$, $I_N^{\beta }$ and $F_N^{\gamma }$ are subalgebras of X for all $\alpha ,\beta ,\gamma \in [-1,0]$ with $-3\le \alpha +\beta +\gamma \le 0$. Then $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-subalgebra of X.

Proof. 

Assume that there exist $a,b\in X$ such that $T_N(a\ast b)>\bigvee \{T_N\left(a\right),T_N\left(b\right)\}$. Then $T_N(a\ast b)>t_{\alpha }\ge \bigvee \{T_N\left(a\right),T_N\left(b\right)\}$ for some $t_{\alpha }\in [-1,0)$. Hence $a,b\in T_N^{t_{\alpha }}$ but $a\ast b\notin T_N^{t_{\alpha }}$, which is a contradiction. Thus

$$T_N(x\ast y)\le \bigvee \{T_N\left(x\right),T_N\left(y\right)\}$$

for all $x,y\in X$. If $I_N(a\ast b)<\bigwedge \{I_N\left(a\right),I_N\left(b\right)\}$ for some $a,b\in X$, then

$$\begin{array}{c}\hfill I_N(a\ast b)<t_{\beta }<\bigwedge \{I_N\left(a\right),I_N\left(b\right)\}\end{array}$$

where $t_{\beta }:=\frac{1}{2}\left\{I_N(a\ast b)+\bigwedge \{I_N\left(a\right),I_N\left(b\right)\}\right\}$. Thus $a,b\in I_N^{t_{\beta }}$ and $a\ast b\notin I_N^{t_{\beta }}$, which is a contradiction. Therefore

$$I_N(x\ast y)\ge \bigwedge \{I_N\left(x\right),I_N\left(y\right)\}$$

for all $x,y\in X$. Now, suppose that there exist $a,b\in X$ and $t_{\gamma }\in [-1,0)$ such that

$$F_N(a\ast b)>t_{\gamma }\ge \bigvee \{F_N\left(a\right),F_N\left(b\right)\}$$

Then $a,b\in F_N^{t_{\gamma }}$ and $a\ast b\notin F_N^{t_{\gamma }}$, which is a contradiction. Hence

$$F_N(x\ast y)\le \bigvee \{F_N\left(x\right),F_N\left(y\right)\}$$

for all $x,y\in X$. Therefore $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-subalgebra of X. ☐

Because $[-1,0]$ is a completely distributive lattice with respect to the usual ordering, we have the following theorem.

Theorem 3.

If $\{X_{N_i}\mid i\in \mathbb{N}\}$ is a family of neutrosophic $\mathcal{N}$-subalgebras of X, then $\left(\{X_{N_i}\mid i\in \mathbb{N}\},\subseteq \right)$ forms a complete distributive lattice.

Proposition 1.

If a neutrosophic $\mathcal{N}$-structure $X_{\mathbf{N}}$ over X is a neutrosophic $\mathcal{N}$-subalgebra of X, then $T_N\left(\theta \right)\le T_N\left(x\right)$, $I_N\left(\theta \right)\ge I_N\left(x\right)$ and $F_N\left(\theta \right)\le F_N\left(x\right)$ for all $x\in X$.

Proof. 

Straightforward. ☐

Theorem 4.

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-subalgebra of X. If there exists a sequence $\left\{a_n\right\}$ in X such that ${\displaystyle \underset{n\to \infty }{lim}}{T}_N\left(a_n\right)=-1$, ${\displaystyle \underset{n\to \infty }{lim}}{I}_N\left(a_n\right)=0$ and ${\displaystyle \underset{n\to \infty }{lim}}{F}_N\left(a_n\right)=-1$, then $T_N\left(\theta \right)=-1$, $I_N\left(\theta \right)=0$ and $F_N\left(\theta \right)=-1$.

Proof. 

By Proposition 1, we have $T_N\left(\theta \right)\le T_N\left(x\right)$, $I_N\left(\theta \right)\ge I_N\left(x\right)$ and $F_N\left(\theta \right)\le F_N\left(x\right)$ for all $x\in X$. Hence $T_N\left(\theta \right)\le T_N\left(a_n\right)$, $I_N\left(a_n\right)\le I_N\left(\theta \right)$ and $F_N\left(\theta \right)\le F_N\left(a_n\right)$ for every positive integer n. It follows that

$$\begin{array}{c}-1\le T_N\left(\theta \right)\le \underset{n\to \infty }{lim}{T}_N\left(a_n\right)=-1\hfill \\ 0\ge I_N\left(\theta \right)\ge \underset{n\to \infty }{lim}{I}_N\left(a_n\right)=0\hfill \\ -1\le F_N\left(\theta \right)\le \underset{n\to \infty }{lim}{F}_N\left(a_n\right)=-1\hfill \end{array}$$

Hence $T_N\left(\theta \right)=-1$, $I_N\left(\theta \right)=0$ and $F_N\left(\theta \right)=-1$. ☐

Proposition 2.

If every neutrosophic $\mathcal{N}$-subalgebra $X_{\mathbf{N}}$ of X satisfies:

$$T_N(x\ast y)\le T_N\left(y\right),I_N(x\ast y)\ge I_N\left(y\right),F_N(x\ast y)\le F_N\left(y\right)$$

(5)

for all $x,y\in X$, then $X_{\mathbf{N}}$ is constant.

Proof. 

Using Equations (1) and (5), we have $T_N\left(x\right)=T_N(x\ast \theta )\le T_N\left(\theta \right)$, $I_N\left(x\right)=I_N(x\ast \theta )\ge I_N\left(\theta \right)$ and $F_N\left(x\right)=F_N(x\ast \theta )\le F_N\left(\theta \right)$ for all $x\in X$. It follows from Proposition 1 that $T_N\left(x\right)=T_N\left(\theta \right)$, $I_N\left(x\right)=I_N\left(\theta \right)$ and $F_N\left(x\right)=F_N\left(\theta \right)$ for all $x\in X$. Therefore $X_{\mathbf{N}}$ is constant. ☐

Definition 2.

A neutrosophic $\mathcal{N}$-structure $X_{\mathbf{N}}$ over X is called a neutrosophic $\mathcal{N}$-ideal of X if the following assertion is valid:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{T}_N\left(\theta \right)\le T_N\left(x\right)\le \bigvee \{T_N(x\ast y),T_N\left(y\right)\}\hfill \\ I_N\left(\theta \right)\ge I_N\left(x\right)\ge \bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\hfill \\ F_N\left(\theta \right)\le F_N\left(x\right)\le \bigvee \{F_N(x\ast y),F_N\left(y\right)\}\hfill \end{array}\right)\end{array}$$

(6)

Example 2.

The neutrosophic $\mathcal{N}$-structure $X_{\mathbf{N}}$ over X in Example 1 is a neutrosophic $\mathcal{N}$-ideal of X.

Example 3.

Consider a $BCI$-algebra $X:=Y\times \mathbb{Z}$ where $(Y,\ast ,\theta )$ is a $BCI$-algebra and $(\mathbb{Z},-,0)$ is the adjoint $BCI$-algebra of the additive group $(\mathbb{Z},+,0)$ of integers (see [6]). Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X given by

$$X_{\mathbf{N}}=\left\{\frac{x}{(\alpha ,0,\gamma )}\mid x\in Y\times (\mathbb{N}\cup \left\{0\right\})\right\}\cup \left\{\frac{x}{(0,\beta ,0)}\mid x\notin Y\times (\mathbb{N}\cup \left\{0\right\})\right\}$$

where $\alpha ,\gamma \in [-1,0)$ and $\beta \in (-1,0]$. Then $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-ideal of X.

Proposition 3.

Every neutrosophic $\mathcal{N}$-ideal $X_{\mathbf{N}}$ of X satisfies the following assertions:

$$\begin{array}{c}\hfill (x,y\in X)\left(x⪯y\Rightarrow T_N\left(x\right)\le T_N\left(y\right),I_N\left(x\right)\ge I_N\left(y\right),F_N\left(x\right)\le F_N\left(y\right)\right)\end{array}$$

(7)

Proof. 

Let $x,y\in X$ be such that $x⪯y$. Then $x\ast y=\theta$, and so

$T_N\left(x\right)\le \bigvee \{T_N(x\ast y),T_N\left(y\right)\}=\bigvee \{T_N\left(\theta \right),T_N\left(y\right)\}=T_N\left(y\right)$

$I_N\left(x\right)\ge \bigwedge \{I_N(x\ast y),I_N\left(y\right)\}=\bigwedge \{I_N\left(\theta \right),I_N\left(y\right)\}=I_N\left(y\right)$

$F_N\left(x\right)\le \bigvee \{F_N(x\ast y),F_N\left(y\right)\}=\bigvee \{F_N\left(\theta \right),F_N\left(y\right)\}=F_N\left(y\right)$

This completes the proof. ☐

Proposition 4.

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-ideal of X. Then

(1)

$T_N(x\ast y)\le T_N((x\ast y)\ast y)\iff T_N((x\ast z)\ast (y\ast z))\le T_N((x\ast y)\ast z)$

(2)

$I_N(x\ast y)\ge I_N((x\ast y)\ast y)\iff I_N((x\ast z)\ast (y\ast z))\ge I_N((x\ast y)\ast z)$

(3)

$F_N(x\ast y)\le F_N((x\ast y)\ast y)\iff F_N((x\ast z)\ast (y\ast z))\le F_N((x\ast y)\ast z)$

for all $x,y,z\in X$.

Proof. 

Note that

$$\left(\right(x\ast (y\ast z))\ast z)\ast z⪯(x\ast y)\ast z$$

(8)

for all $x,y,z\in X$. Assume that $T_N(x\ast y)\le T_N((x\ast y)\ast y)$, $I_N(x\ast y)\ge I_N((x\ast y)\ast y)$ and $F_N(x\ast y)\le F_N((x\ast y)\ast y)$ for all $x,y\in X$. It follows from Equation (2) and Proposition 3 that

$$\begin{array}{cc}\hfill T_N((x\ast z)\ast (y\ast z))& =T_N((x\ast (y\ast z))\ast z)\hfill \\ & \le T_N(((x\ast (y\ast z))\ast z)\ast z)\hfill \\ & \le T_N((x\ast y)\ast z)\hfill \end{array}$$

$$\begin{array}{cc}\hfill I_N((x\ast z)\ast (y\ast z))& =I_N((x\ast (y\ast z))\ast z)\hfill \\ & \ge I_N(((x\ast (y\ast z))\ast z)\ast z)\hfill \\ & \ge I_N((x\ast y)\ast z)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill F_N((x\ast z)\ast (y\ast z))& =F_N((x\ast (y\ast z))\ast z)\hfill \\ & \le F_N(((x\ast (y\ast z))\ast z)\ast z)\hfill \\ & \le F_N((x\ast y)\ast z)\hfill \end{array}$$

for all $x,y\in X$.

Conversely, suppose

$$\begin{array}{c}{T}_N((x\ast z)\ast (y\ast z))\le T_N((x\ast y)\ast z)\hfill \\ I_N((x\ast z)\ast (y\ast z))\ge I_N((x\ast y)\ast z)\hfill \\ F_N((x\ast z)\ast (y\ast z))\le F_N((x\ast y)\ast z)\hfill \end{array}$$

(9)

for all $x,y,z\in X$. If we substitute z for y in Equation (9), then

$$\begin{array}{c}{T}_N(x\ast z)=T_N((x\ast z)\ast \theta )=T_N((x\ast z)\ast (z\ast z))\le T_N((x\ast z)\ast z)\hfill \\ I_N(x\ast z)=I_N((x\ast z)\ast \theta )=I_N((x\ast z)\ast (z\ast z))\ge I_N((x\ast z)\ast z)\hfill \\ F_N(x\ast z)=F_N((x\ast z)\ast \theta )=F_N((x\ast z)\ast (z\ast z))\le F_N((x\ast z)\ast z)\hfill \end{array}$$

for all $x,z\in X$ by using (III) and Equation (1). ☐

Theorem 5.

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X and let $\alpha ,\beta ,\gamma \in [-1,0]$ be such that $-3\le \alpha +\beta +\gamma \le 0$. If $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-ideal of X, then the nonempty $(\alpha ,\beta ,\gamma )$-level set of $X_{\mathbf{N}}$ is an ideal of X.

Proof. 

Assume that $X_{\mathbf{N}}(\alpha ,\beta ,\gamma )\ne \mathsf{\varnothing }$ for $\alpha ,\beta ,\gamma \in [-1,0]$ with $-3\le \alpha +\beta +\gamma \le 0$. Clearly, $\theta \in X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$. Let $x,y\in X$ be such that $x\ast y\in X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$ and $y\in X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$. Then $T_N(x\ast y)\le \alpha$, $I_N(x\ast y)\ge \beta$, $F_N(x\ast y)\le \gamma$, $T_N\left(y\right)\le \alpha$, $I_N\left(y\right)\ge \beta$ and $F_N\left(y\right)\le \gamma$. It follows from Equation (6) that

$$\begin{array}{c}{T}_N\left(x\right)\le \bigvee \{T_N(x\ast y),T_N\left(y\right)\}\le \alpha \hfill \\ I_N\left(x\right)\ge \bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\ge \beta \hfill \\ F_N\left(x\right)\le \bigvee \{F_N(x\ast y),F_N\left(y\right)\}\le \gamma \hfill \end{array}$$

so that $x\in X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$. Therefore $X_{\mathbf{N}}(\alpha ,\beta ,\gamma )$ is an ideal of X. ☐

Theorem 6.

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X and assume that $T_N^{\alpha }$, $I_N^{\beta }$ and $F_N^{\gamma }$ are ideals of X for all $\alpha ,\beta ,\gamma \in [-1,0]$ with $-3\le \alpha +\beta +\gamma \le 0$. Then $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-ideal of X.

Proof. 

If there exist $a,b,c\in X$ such that $T_N\left(\theta \right)>T_N\left(a\right)$, $I_N\left(\theta \right)<I_N\left(b\right)$ and $F_N\left(\theta \right)>F_N\left(c\right)$, respectively, then $T_N\left(\theta \right)>a_t\ge T_N\left(a\right)$, $I_N\left(\theta \right)<b_i\le I_N\left(b\right)$ and $F_N\left(\theta \right)>c_f\ge F_N\left(c\right)$ for some $a_t,c_f\in [-1,0)$ and $b_i\in (-1,0]$. Then $\theta \notin T_N^{a_t}$, $\theta \notin I_N^{b_i}$ and $\theta \notin F_N^{c_f}$. This is a contradiction. Hence, $T_N\left(\theta \right)\le T_N\left(x\right)$, $I_N\left(\theta \right)\ge I_N\left(x\right)$ and $F_N\left(\theta \right)\le F_N\left(x\right)$ for all $x\in X$. Assume that there exist $a_t,b_t,a_i,b_i,a_f,b_f\in X$ such that $T_N\left(a_t\right)>\bigvee \{T_N(a_t\ast b_t),T_N\left(b_t\right)\}$, $I_N\left(a_i\right)<\bigwedge \{I_N(a_i\ast b_i),I_N\left(b_i\right)\}$ and $F_N\left(a_f\right)>\bigvee \{F_N(a_f\ast b_f),F_N\left(b_f\right)\}$. Then there exist $s_t,s_f\in [-1,0)$ and $s_i\in (-1,0]$ such that

$$\begin{array}{c}{T}_N\left(a_t\right)>s_t\ge \bigvee \{T_N(a_t\ast b_t),T_N\left(b_t\right)\}\hfill \\ I_N\left(a_i\right)<s_i\le \bigwedge \{I_N(a_i\ast b_i),I_N\left(b_i\right)\}\hfill \\ F_N\left(a_f\right)>s_f\ge \bigvee \{F_N(a_f\ast b_f),F_N\left(b_f\right)\}\hfill \end{array}$$

It follows that $a_t\ast b_t\in T_N^{s_t}$, $b_t\in T_N^{s_t}$, $a_i\ast b_i\in I_N^{s_i}$, $b_i\in I_N^{s_i}$, $a_f\ast b_f\in F_N^{s_f}$ and $b_f\in F_N^{s_f}$. However, $a_t\notin T_N^{s_t}$, $a_i\notin I_N^{s_i}$ and $a_f\notin F_N^{s_f}$. This is a contradiction, and so

$$\begin{array}{c}{T}_N\left(x\right)\le \bigvee \{T_N(x\ast y),T_N\left(y\right)\}\hfill \\ I_N\left(x\right)\ge \bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\hfill \\ F_N\left(x\right)\le \bigvee \{F_N(x\ast y),F_N\left(y\right)\}\hfill \end{array}$$

for all $x,y\in X$. Therefore $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-ideal of X. ☐

Proposition 5.

For any neutrosophic $\mathcal{N}$-ideal $X_{\mathbf{N}}$ of X, we have

$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y⪯z\Rightarrow \left\{\begin{array}{c}{T}_N\left(x\right)\le \bigvee \{T_N\left(y\right),T_N\left(z\right)\}\hfill \\ I_N\left(x\right)\ge \bigwedge \{I_N\left(y\right),I_N\left(z\right)\}\hfill \\ F_N\left(x\right)\le \bigvee \{F_N\left(y\right),F_N\left(z\right)\}\hfill \end{array}\right.\hfill \end{array}\right)\end{array}$$

(10)

Proof. 

Let $x,y,z\in X$ be such that $x\ast y⪯z$. Then $(x\ast y)\ast z=\theta$, and so

$$\begin{array}{c}{T}_N(x\ast y)\le \bigvee \{T_N((x\ast y)\ast z),T_N\left(z\right)\}=\bigvee \{T_N\left(\theta \right),T_N\left(z\right)\}=T_N\left(z\right)\hfill \\ I_N(x\ast y)\ge \bigwedge \{I_N((x\ast y)\ast z),I_N\left(z\right)\}=\bigwedge \{I_N\left(\theta \right),I_N\left(z\right)\}=I_N\left(z\right)\hfill \\ F_N(x\ast y)\le \bigvee \{F_N((x\ast y)\ast z),F_N\left(z\right)\}=\bigvee \{F_N\left(\theta \right),F_N\left(z\right)\}=F_N\left(z\right)\hfill \end{array}$$

It follows that

$$\begin{array}{c}{T}_N\left(x\right)\le \bigvee \{T_N(x\ast y),T_N\left(y\right)\}\le \bigvee \{T_N\left(y\right),T_N\left(z\right)\}\hfill \\ I_N\left(x\right)\ge \bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\ge \bigwedge \{I_N\left(y\right),I_N\left(z\right)\}\hfill \\ F_N\left(x\right)\le \bigvee \{F_N(x\ast y),F_N\left(y\right)\}\le \bigvee \{F_N\left(y\right),F_N\left(z\right)\}\hfill \end{array}$$

This completes the proof. ☐

Theorem 7.

In a $BCK$-algebra, every neutrosophic $\mathcal{N}$-ideal is a neutrosophic $\mathcal{N}$-subalgebra.

Proof. 

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-ideal of a $BCK$-algebra X. For any $x,y\in X$, we have

$$\begin{array}{cc}\hfill T_N(x\ast y)& \le \bigvee \{T_N((x\ast y)\ast x),T_N\left(x\right)\}=\bigvee \{T_N((x\ast x)\ast y),T_N\left(x\right)\}\hfill \\ & =\bigvee \{T_N(\theta \ast y),T_N\left(x\right)\}=\bigvee \{T_N\left(\theta \right),T_N\left(x\right)\}\hfill \\ & \le \bigvee \{T_N\left(x\right),T_N\left(y\right)\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill I_N(x\ast y)& \ge \bigwedge \{I_N((x\ast y)\ast x),I_N\left(x\right)\}=\bigwedge \{I_N((x\ast x)\ast y),I_N\left(x\right)\}\hfill \\ & =\bigwedge \{I_N(\theta \ast y),I_N\left(x\right)\}=\bigwedge \{I_N\left(\theta \right),I_N\left(x\right)\}\hfill \\ & \ge \bigwedge \{I_N\left(y\right),I_N\left(x\right)\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill F_N(x\ast y)& \le \bigvee \{F_N((x\ast y)\ast x),F_N\left(x\right)\}=\bigvee \{F_N((x\ast x)\ast y),F_N\left(x\right)\}\hfill \\ & =\bigvee \{F_N(\theta \ast y),F_N\left(x\right)\}=\bigvee \{F_N\left(\theta \right),F_N\left(x\right)\}\hfill \\ & \le \bigvee \{F_N\left(x\right),F_N\left(y\right)\}\hfill \end{array}$$

Hence $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-subalgebra of a $BCK$-algebra X. ☐

The converse of Theorem 7 may not be true in general, as seen in the following example.

Example 4.

Consider a $BCK$-algebra $X=\{\theta ,1,2,3,4\}$ with the following Cayley table.

*	$\mathbf{\theta }$	1	2	3	4
θ	θ	θ	θ	θ	θ
1	1	θ	θ	θ	θ
2	2	1	θ	1	θ
3	3	3	3	θ	θ
4	4	4	4	3	θ

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X, which is given as follows:

$$\begin{array}{cc}\hfill X_{\mathbf{N}}& =\{\frac{\theta }{(-0.8,0,-1)},\frac{1}{(-0.8,-0.2,-0.9)},\hfill \\ & \frac{2}{(-0.2,-0.6,-0.5)},\frac{3}{(-0.7,-0.4,-0.7)},\frac{4}{(-0.4,-0.8,-0.3)}\}\hfill \end{array}$$

Then $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-subalgebra of X, but it is not a neutrosophic $\mathcal{N}$-ideal of X as $T_N\left(2\right)=-0.2>-0.7=\bigvee \{T_N(2\ast 3),T_N\left(3\right)\}$, $I_N\left(4\right)=-0.8<-0.4=\bigwedge \{I_N(4\ast 3),I_N\left(3\right)\}$, or $F_N\left(4\right)=-0.3>-0.7=\bigvee \{F_N(4\ast 3),F_N\left(3\right)\}$.

Theorem 7 is not valid in a $BCI$-algebra; that is, if X is a $BCI$-algebra, then there is a neutrosophic $\mathcal{N}$-ideal that is not a neutrosophic $\mathcal{N}$-subalgebra, as seen in the following example.

Example 5.

Consider the neutrosophic $\mathcal{N}$-ideal $X_{\mathbf{N}}$ of X in Example 3. If we take $x:=(\theta ,0)$ and $y:=(\theta ,1)$ in $Y\times (\mathbb{N}\cup \{0\left\}\right)$, then $x\ast y=(\theta ,0)\ast (\theta ,1)=(\theta ,-1)\notin Y\times (\mathbb{N}\cup \{0\left\}\right)$. Hence

$$\begin{array}{c}{T}_N(x\ast y)=0>\alpha =\bigvee \{T_N\left(x\right),T_N\left(y\right)\}\hfill \\ I_N(x\ast y)=\beta <0=\bigwedge \{I_N\left(x\right),I_N\left(y\right)\}\mathrm{or}\hfill \\ F_N(x\ast y)=0>\gamma =\bigvee \{F_N\left(x\right),F_N\left(y\right)\}\hfill \end{array}$$

Therefore $X_{\mathbf{N}}$ is not a neutrosophic $\mathcal{N}$-subalgebra of X.

For any elements ${\omega }_t$, ${\omega }_i$, ${\omega }_f\in X$, we consider sets:

$$\begin{array}{c}{X}_{\mathbf{N}}^{{\omega }_t}:=\left\{x\in X\mid T_N\left(x\right)\le T_N\left({\omega }_t\right)\right\}\hfill \\ X_{\mathbf{N}}^{{\omega }_i}:=\left\{x\in X\mid I_N\left(x\right)\ge I_N\left({\omega }_i\right)\right\}\hfill \\ X_{\mathbf{N}}^{{\omega }_f}:=\left\{x\in X\mid F_N\left(x\right)\le F_N\left({\omega }_f\right)\right\}\hfill \end{array}$$

Clearly, ${\omega }_t\in X_{\mathbf{N}}^{{\omega }_t}$, ${\omega }_i\in X_{\mathbf{N}}^{{\omega }_i}$ and ${\omega }_f\in X_{\mathbf{N}}^{{\omega }_f}$.

Theorem 8.

Let ${\omega }_t$, ${\omega }_i$ and ${\omega }_f$ be any elements of X. If $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-ideal of X, then $X_{\mathbf{N}}^{{\omega }_t}$, $X_{\mathbf{N}}^{{\omega }_i}$ and $X_{\mathbf{N}}^{{\omega }_f}$ are ideals of X.

Proof. 

Clearly, $\theta \in X_{\mathbf{N}}^{{\omega }_t}$, $\theta \in X_{\mathbf{N}}^{{\omega }_i}$ and $\theta \in X_{\mathbf{N}}^{{\omega }_f}$. Let $x,y\in X$ be such that $x\ast y\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$ and $y\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$. Then

$$\begin{array}{c}{T}_N(x\ast y)\le T_N\left({\omega }_t\right),T_N\left(y\right)\le T_N\left({\omega }_t\right)\hfill \\ I_N(x\ast y)\ge I_N\left({\omega }_i\right),I_N\left(y\right)\ge I_N\left({\omega }_i\right)\hfill \\ F_N(x\ast y)\le F_N\left({\omega }_f\right),F_N\left(y\right)\le F_N\left({\omega }_f\right)\hfill \end{array}$$

It follows from Equation (6) that

$$\begin{array}{c}{T}_N\left(x\right)\le \bigvee \{T_N(x\ast y),T_N\left(y\right)\}\le T_N\left({\omega }_t\right)\hfill \\ I_N\left(x\right)\ge \bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\ge I_N\left({\omega }_i\right)\hfill \\ F_N\left(x\right)\le \bigvee \{F_N(x\ast y),F_N\left(y\right)\}\le F_N\left({\omega }_f\right)\hfill \end{array}$$

Hence $x\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$, and therefore $X_{\mathbf{N}}^{{\omega }_t}$, $X_{\mathbf{N}}^{{\omega }_i}$ and $X_{\mathbf{N}}^{{\omega }_f}$ are ideals of X. ☐

Theorem 9.

Let ${\omega }_t$, ${\omega }_i$, ${\omega }_f\in X$ and let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X. Then

(1)

If $X_{\mathbf{N}}^{{\omega }_t}$, $X_{\mathbf{N}}^{{\omega }_i}$ and $X_{\mathbf{N}}^{{\omega }_f}$ are ideals of X, then the following assertion is valid:

$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}{T}_N\left(x\right)\ge \bigvee \{T_N(y\ast z),T_N\left(z\right)\}\Rightarrow T_N\left(x\right)\ge T_N\left(y\right)\hfill \\ I_N\left(x\right)\le \bigwedge \{I_N(y\ast z),I_N\left(z\right)\}\Rightarrow I_N\left(x\right)\le I_N\left(y\right)\hfill \\ F_N\left(x\right)\ge \bigvee \{F_N(y\ast z),F_N\left(z\right)\}\Rightarrow F_N\left(x\right)\ge F_N\left(y\right)\hfill \end{array}\right)\end{array}$$

(11)

(2)

If $X_{\mathbf{N}}$ satisfies Equation (11) and

$$\begin{array}{c}\hfill (\forall x\in X)\left(T_N\left(\theta \right)\le T_N\left(x\right),I_N\left(\theta \right)\ge I_N\left(x\right),F_N\left(\theta \right)\le F_N\left(x\right)\right)\end{array}$$

(12)

then $X_{\mathbf{N}}^{{\omega }_t}$, $X_{\mathbf{N}}^{{\omega }_i}$ and $X_{\mathbf{N}}^{{\omega }_f}$ are ideals of X for all ${\omega }_t\in \mathrm{Im}\left(T_N\right)$, ${\omega }_i\in \mathrm{Im}\left(I_N\right)$ and ${\omega }_f\in \mathrm{Im}\left(F_N\right)$.

Proof. 

(1) Assume that $X_{\mathbf{N}}^{{\omega }_t}$, $X_{\mathbf{N}}^{{\omega }_i}$ and $X_{\mathbf{N}}^{{\omega }_f}$ are ideals of X for ${\omega }_t$, ${\omega }_i$, ${\omega }_f\in X$. Let $x,y,z\in X$ be such that $T_N\left(x\right)\ge \bigvee \{T_N(y\ast z),T_N\left(z\right)\}$, $I_N\left(x\right)\le \bigwedge \{I_N(y\ast z),I_N\left(z\right)\}$ and $F_N\left(x\right)\ge \bigvee \{F_N(y\ast z),F_N\left(z\right)\}$. Then $y\ast z\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$ and $z\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$, where ${\omega }_t={\omega }_i={\omega }_f=x$. It follows from (I2) that $y\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$ for ${\omega }_t={\omega }_i={\omega }_f=x$. Hence $T_N\left(y\right)\le T_N\left({\omega }_t\right)=T_N\left(x\right)$, $I_N\left(y\right)\ge I_N\left({\omega }_i\right)=I_N\left(x\right)$ and $F_N\left(y\right)\le F_N\left({\omega }_f\right)=F_N\left(x\right)$.

(2) Let ${\omega }_t\in \mathrm{Im}\left(T_N\right)$, ${\omega }_i\in \mathrm{Im}\left(I_N\right)$ and ${\omega }_f\in \mathrm{Im}\left(F_N\right)$ and suppose that $X_{\mathbf{N}}$ satisfies Equations (11) and (12). Clearly, $\theta \in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$ by Equation (12). Let $x,y\in X$ be such that $x\ast y\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$ and $y\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$. Then

$$\begin{array}{c}{T}_N(x\ast y)\le T_N\left({\omega }_t\right),T_N\left(y\right)\le T_N\left({\omega }_t\right)\hfill \\ I_N(x\ast y)\ge I_N\left({\omega }_i\right),I_N\left(y\right)\ge I_N\left({\omega }_i\right)\hfill \\ F_N(x\ast y)\le F_N\left({\omega }_f\right),F_N\left(y\right)\le F_N\left({\omega }_f\right)\hfill \end{array}$$

which implies that $\bigvee \{T_N(x\ast y),T_N\left(y\right)\}\le T_N\left({\omega }_t\right)$, $\bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\ge I_N\left({\omega }_i\right)$, and $\bigvee \{F_N(x\ast y),F_N\left(y\right)\}\le F_N\left({\omega }_f\right)$. It follows from Equation (11) that $T_N\left({\omega }_t\right)\ge T_N\left(x\right)$, $I_N\left({\omega }_i\right)\le I_N\left(x\right)$ and $F_N\left({\omega }_f\right)\ge F_N\left(x\right)$. Thus, $x\in X_{\mathbf{N}}^{{\omega }_t}\cap X_{\mathbf{N}}^{{\omega }_i}\cap X_{\mathbf{N}}^{{\omega }_f}$, and therefore $X_{\mathbf{N}}^{{\omega }_t}$, $X_{\mathbf{N}}^{{\omega }_i}$ and $X_{\mathbf{N}}^{{\omega }_f}$ are ideals of X.☐

Definition 3.

A neutrosophic $\mathcal{N}$-ideal $X_{\mathbf{N}}$ of X is said to be closed if it is a neutrosophic $\mathcal{N}$-subalgebra of X.

Example 6.

Consider a $BCI$-algebra $X=\{\theta ,1,a,b,c\}$ with the following Cayley table.

*	$\mathbf{\theta }$	1	$\mathit{a}$	$\mathit{b}$	$\mathit{c}$
θ	θ	θ	a	b	c
1	1	θ	a	b	c
a	a	a	θ	c	b
b	b	b	c	θ	a
c	c	c	b	a	θ

Let $X_{\mathbf{N}}$ be a neutrosophic $\mathcal{N}$-structure over X which is given as follows:

$$\begin{array}{cc}\hfill X_{\mathbf{N}}& =\{\frac{\theta }{(-0.9,-0.3,-0.8)},\frac{1}{(-0.7,-0.4,-0.7)},\frac{a}{(-0.6,-0.8,-0.3)},\hfill \\ & \frac{b}{(-0.2,-0.6,-0.3)},\frac{c}{(-0.2,-0.8,-0.5)}\}\hfill \end{array}$$

Then $X_{\mathbf{N}}$ is a closed neutrosophic $\mathcal{N}$-ideal of X.

Theorem 10.

Let X be a $BCI$-algebra, For any ${\alpha }_1,{\alpha }_2,{\gamma }_1,{\gamma }_2\in [-1,0)$ and ${\beta }_1,{\beta }_2\in (-1,0]$ with ${\alpha }_1<{\alpha }_2$, ${\gamma }_1<{\gamma }_2$ and ${\beta }_1>{\beta }_2$, let $X_{\mathbf{N}}:=\frac{X}{(T_N,I_N,F_N)}$ be a neutrosophic $\mathcal{N}$-structure over X given as follows:

$$\begin{array}{c}{T}_N:X\to [-1,0],x\mapsto \left\{\begin{array}{cc}{\alpha }_1\hfill & \mathrm{if}x\in X_{+}\hfill \\ {\alpha }_2\hfill & \mathrm{otherwise}\hfill \end{array}\right.\hfill \\ I_N:X\to [-1,0],x\mapsto \left\{\begin{array}{cc}{\beta }_1\hfill & \mathrm{if}x\in X_{+}\hfill \\ {\beta }_2\hfill & \mathrm{otherwise}\hfill \end{array}\right.\hfill \\ F_N:X\to [-1,0],x\mapsto \left\{\begin{array}{cc}{\gamma }_1\hfill & \mathrm{if}x\in X_{+}\hfill \\ {\gamma }_2\hfill & \mathrm{otherwise}\hfill \end{array}\right.\hfill \end{array}$$

where $X_{+}=\{x\in X\mid \theta ⪯x\}$. Then $X_{\mathbf{N}}$ is a closed neutrosophic $\mathcal{N}$-ideal of X.

Proof. 

Because $\theta \in X_{+}$, we have $T_N\left(\theta \right)={\alpha }_1\le T_N\left(x\right)$, $I_N\left(\theta \right)={\beta }_1\ge I_N\left(x\right)$ and $F_N\left(\theta \right)={\gamma }_1\le F_N\left(x\right)$ for all $x\in X$. Let $x,y\in X$. If $x\in X_{+}$, then

$$\begin{array}{c}{T}_N\left(x\right)={\alpha }_1\le \bigvee \{T_N(x\ast y),T_N\left(y\right)\}\hfill \\ I_N\left(x\right)={\beta }_1\ge \bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\hfill \\ F_N\left(x\right)={\gamma }_1\le \bigvee \{F_N(x\ast y),F_N\left(y\right)\}\hfill \end{array}$$

Suppose that $x\notin X_{+}$. If $x\ast y\in X_{+}$ then $y\notin X_{+}$, and if $y\in X_{+}$ then $x\ast y\notin X_{+}$. In either case, we have

$$\begin{array}{c}{T}_N\left(x\right)={\alpha }_2=\bigvee \{T_N(x\ast y),T_N\left(y\right)\}\hfill \\ I_N\left(x\right)={\beta }_2=\bigwedge \{I_N(x\ast y),I_N\left(y\right)\}\hfill \\ F_N\left(x\right)={\gamma }_2=\bigvee \{F_N(x\ast y),F_N\left(y\right)\}\hfill \end{array}$$

For any $x,y\in X$, if any one of x and y does not belong to $X_{+}$, then

$$\begin{array}{c}{T}_N(x\ast y)\le {\alpha }_2=\bigvee \{T_N\left(x\right),T_N\left(y\right)\}\hfill \\ I_N(x\ast y)\ge {\beta }_2=\bigwedge \{I_N\left(x\right),I_N\left(y\right)\}\hfill \\ F_N(x\ast y)\le {\gamma }_2=\bigvee \{F_N\left(x\right),F_N\left(y\right)\}\hfill \end{array}$$

If $x,y\in X_{+}$, then $x\ast y\in X_{+}$. Hence

$$\begin{array}{c}{T}_N(x\ast y)={\alpha }_1=\bigvee \{T_N\left(x\right),T_N\left(y\right)\}\hfill \\ I_N(x\ast y)={\beta }_1=\bigwedge \{I_N\left(x\right),I_N\left(y\right)\}\hfill \\ F_N(x\ast y)={\gamma }_1=\bigvee \{F_N\left(x\right),F_N\left(y\right)\}\hfill \end{array}$$

Therefore $X_{\mathbf{N}}$ is a closed neutrosophic $\mathcal{N}$-ideal of X.  ☐

Proposition 6.

Every closed neutrosophic $\mathcal{N}$-ideal $X_{\mathbf{N}}$ of a $BCI$-algebra X satisfies the following condition:

$$\begin{array}{c}\hfill (\forall x\in X)\left(T_N(\theta \ast x)\le T_N\left(x\right),I_N(\theta \ast x)\ge I_N\left(x\right),F_N(\theta \ast x)\le F_N\left(x\right)\right)\end{array}$$

(13)

Proof. 

Straightforward. ☐

We provide conditions for a neutrosophic $\mathcal{N}$-ideal to be closed.

Theorem 11.

Let X be a $BCI$-algebra. If $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-ideal of X that satisfies the condition of Equation (13), then $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-subalgebra and hence is a closed neutrosophic $\mathcal{N}$-ideal of X.

Proof. 

Note that $(x\ast y)\ast x⪯\theta \ast y$ for all $x,y\in X$. Using Equations (10) and (13), we have

$$\begin{array}{c}{T}_N(x\ast y)\le \bigvee \{T_N\left(x\right),T_N(\theta \ast y)\}\le \bigvee \{T_N\left(x\right),T_N\left(y\right)\}\hfill \\ I_N(x\ast y)\ge \bigwedge \{I_N\left(x\right),I_N(\theta \ast y)\}\ge \bigwedge \{I_N\left(x\right),I_N\left(y\right)\}\hfill \\ F_N(x\ast y)\le \bigvee \{F_N\left(x\right),F_N(\theta \ast y)\}\le \bigvee \{F_N\left(x\right),F_N\left(y\right)\}\hfill \end{array}$$

Hence $X_{\mathbf{N}}$ is a neutrosophic $\mathcal{N}$-subalgebra and is therefore a closed neutrosophic $\mathcal{N}$-ideal of X. ☐