Neutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCI-Algebras

Abstract

The concept of a $(\in ,$ $\in )$-neutrosophic ideal is introduced, and its characterizations are established. The notions of neutrosophic permeable values are introduced, and related properties are investigated. Conditions for the neutrosophic level sets to be energetic, right stable, and right vanished are discussed. Relations between neutrosophic permeable S- and I-values are considered.

1. Introduction

The notion of neutrosophic set (NS) theory developed by Smarandache (see [1,2]) is a more general platform that extends the concepts of classic and fuzzy sets, intuitionistic fuzzy sets, and interval-valued (intuitionistic) fuzzy sets and that is applied to various parts: pattern recognition, medical diagnosis, decision-making problems, and so on (see [3,4,5,6]). Smarandache [2] mentioned that a cloud is a NS because its borders are ambiguous and because each element (water drop) belongs with a neutrosophic probability to the set (e.g., there are types of separated water drops around a compact mass of water drops, such that we do not know how to consider them: in or out of the cloud). Additionally, we are not sure where the cloud ends nor where it begins, and neither whether some elements are or are not in the set. This is why the percentage of indeterminacy is required and the neutrosophic probability (using subsets—not numbers—as components) should be used for better modeling: it is a more organic, smooth, and particularly accurate estimation. Indeterminacy is the zone of ignorance of a proposition’s value, between truth and falsehood.

Algebraic structures play an important role in mathematics with wide-ranging applications in several disciplines such as coding theory, information sciences, computer sciences, control engineering, theoretical physics, and so on. NS theory is also applied to several algebraic structures. In particular, Jun et al. applied it to $BCK/BCI$-algebras (see [7,8,9,10,11,12]). Jun et al. [8] introduced the notions of energetic subsets, right vanished subsets, right stable subsets, and (anti-)permeable values in $BCK/BCI$-algebras and investigated relations between these sets.

In this paper, we introduce the notions of neutrosophic permeable S-values, neutrosophic permeable I-values, $(\in ,$ $\in )$-neutrosophic ideals, neutrosophic anti-permeable S-values, and neutrosophic anti-permeable I-values, which are motivated by the idea of subalgebras (i.e., S-values) and ideals (i.e., I-values), and investigate their properties. We consider characterizations of $(\in ,$ $\in )$-neutrosophic ideals. We discuss conditions for the lower (upper) neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets to be S- and I-energetic. We provide conditions for a triple $(\alpha ,\beta ,\gamma )$ of numbers to be a neutrosophic (anti-)permeable S- or I-value. We consider conditions for the upper (lower) neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets to be right stable (right vanished) subsets. We establish relations between neutrosophic (anti-)permeable S- and I-values.

2. Preliminaries

An algebra $(X;\ast ,0)$ of type $(2,0)$ is called a $BCI$-algebra if it satisfies the following conditions:

(I)

$(\forall x,y,z\in X)$$(((x\ast y)\ast (x\ast z))\ast (z\ast y)=0);$

(II)

$(\forall x,y\in X)$$((x\ast (x\ast y))\ast y=0);$

(III)

$(\forall x\in X)$$(x\ast x=0);$

(IV)

$(\forall x,y\in X)$$(x\ast y=0,y\ast x=0\Rightarrow x=y).$

If a $BCI$-algebra X satisfies the following identity:

(V)

$(\forall x\in X)$$(0\ast x=0),$

then X is called a $BCK$-algebra. Any $BCK/BCI$-algebra X satisfies the following conditions:

$$\begin{array}{c}(\forall x\in X)\left(x\ast 0=x\right),\hfill \end{array}$$

(1)

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

(2)

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

(3)

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

(4)

where $x\le y$ if and only if $x\ast y=0.$ A nonempty 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:

$$\begin{array}{c}0\in I,\hfill \end{array}$$

(5)

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

(6)

We refer the reader to the books [13] and [14] for further information regarding $BCK/BCI$-algebras.

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

$$\begin{array}{c}\hfill \bigvee \{a_i\mid i\in \mathsf{\Lambda }\}=sup\{a_i\mid i\in \mathsf{\Lambda }\}\end{array}$$

and

$$\begin{array}{c}\hfill \bigwedge \{a_i\mid i\in \mathsf{\Lambda }\}=inf\{a_i\mid i\in \mathsf{\Lambda }\}.\end{array}$$

If $\mathsf{\Lambda }=\{1,2\}$, we also use $a_1\vee a_2$ and $a_1\wedge a_2$ instead of $\bigvee \{a_i\mid i\in \{1,2\}\}$ and $\bigwedge \{a_i\mid i\in \{1,2\}\}$, respectively.

We let X be a nonempty set. A NS in X (see [1]) is a structure of the form

$$\begin{array}{c}\hfill A:=\{\langle x;A_T(x),A_I(x),A_F(x)\rangle \mid x\in X\},\end{array}$$

where $A_T:X\to [0,1]$ is a truth membership function, $A_I:X\to [0,1]$ is an indeterminate membership function, and $A_F:X\to [0,1]$ is a false membership function. For the sake of simplicity, we use the symbol $A=(A_T,$ $A_I,$ $A_F)$ for the NS

$$\begin{array}{c}\hfill A:=\{\langle x;A_T(x),A_I(x),A_F(x)\rangle \mid x\in X\}.\end{array}$$

A subset A of a $BCK/BCI$-algebra X is said to be S-energetic (see [8]) if it satisfies

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\ast y\in A\Rightarrow \{x,y\}\cap A\ne \varnothing \right).\end{array}$$

(7)

A subset A of a $BCK/BCI$-algebra X is said to be I-energetic (see [8]) if it satisfies

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(y\in A\Rightarrow \{x,y\ast x\}\cap A\ne \varnothing \right).\end{array}$$

(8)

A subset A of a $BCK/BCI$-algebra X is said to be right vanished (see [8]) if it satisfies

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

(9)

A subset A of a $BCK/BCI$-algebra X is said to be right stable (see [8]) if $A\ast X:=\{a\ast x\mid a\in A,x\in X\}\subseteq A$.

3. Neutrosophic Permeable Values

Given a NS $A=(A_T,$ $A_I,$ $A_F)$ in a set X, $\alpha ,$ $\beta \in (0,1]$ and $\gamma \in [0,1)$, we consider the following sets:

$$\begin{array}{c}\hfill U_T^{\in }(A;\alpha )=\{x\in X\mid A_T(x)\ge \alpha \},U_T^{\in }{(A;\alpha )}^{\ast }=\{x\in X\mid A_T(x)>\alpha \},\end{array}$$

$$\begin{array}{c}\hfill U_I^{\in }(A;\beta )=\{x\in X\mid A_I(x)\ge \beta \},U_I^{\in }{(A;\beta )}^{\ast }=\{x\in X\mid A_I(x)>\beta \},\end{array}$$

$$\begin{array}{c}\hfill U_F^{\in }(A;\gamma )=\{x\in X\mid A_F(x)\le \gamma \},U_F^{\in }{(A;\gamma )}^{\ast }=\{x\in X\mid A_F(x)<\gamma \},\end{array}$$

$$\begin{array}{c}\hfill L_T^{\in }(A;\alpha )=\{x\in X\mid A_T(x)\le \alpha \},L_T^{\in }{(A;\alpha )}^{\ast }=\{x\in X\mid A_T(x)<\alpha \},\end{array}$$

$$\begin{array}{c}\hfill L_I^{\in }(A;\beta )=\{x\in X\mid A_I(x)\le \beta \},L_I^{\in }{(A;\beta )}^{\ast }=\{x\in X\mid A_I(x)<\beta \},\end{array}$$

$$\begin{array}{c}\hfill L_F^{\in }(A;\gamma )=\{x\in X\mid A_F(x)\ge \gamma \},L_F^{\in }{(A;\gamma )}^{\ast }=\{x\in X\mid A_F(x)>\gamma \}.\end{array}$$

We say $U_T^{\in }(A;\alpha )$, $U_I^{\in }(A;\beta )$, and $U_F^{\in }(A;\gamma )$ are upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X, and $L_T^{\in }(A;\alpha )$, $L_I^{\in }(A;\beta )$, and $L_F^{\in }(A;\gamma )$ are lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X, where $\mathsf{\Phi }\in \{T,I,F\}$. We say $U_T^{\in }{(A;\alpha )}^{\ast }$, $U_I^{\in }{(A;\beta )}^{\ast }$, and $U_F^{\in }{(A;\gamma )}^{\ast }$ are strong upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X, and $L_T^{\in }{(A;\alpha )}^{\ast }$, $L_I^{\in }{(A;\beta )}^{\ast }$, and $L_F^{\in }{(A;\gamma )}^{\ast }$ are strong lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X, where $\mathsf{\Phi }\in \{T,I,F\}$.

Definition 1

([7]). A NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK/BCI$-algebra X is called an $(\in ,$ $\in )$- neutrosophic subalgebra of X if the following assertions are valid:

$$\begin{array}{c}x\in U_T^{\in }(A;{\alpha }_x),y\in U_T^{\in }(A;{\alpha }_y)\Rightarrow x\ast y\in U_T^{\in }(A;{\alpha }_x\wedge {\alpha }_y),\hfill \\ x\in U_I^{\in }(A;{\beta }_x),y\in U_I^{\in }(A;{\beta }_y)\Rightarrow x\ast y\in U_I^{\in }(A;{\beta }_x\wedge {\beta }_y),\hfill \\ x\in U_F^{\in }(A;{\gamma }_x),y\in U_F^{\in }(A;{\gamma }_y)\Rightarrow x\ast y\in U_F^{\in }(A;{\gamma }_x\vee {\gamma }_y),\hfill \end{array}$$

(10)

for all $x,y\in X$, ${\alpha }_x,{\alpha }_y,$ ${\beta }_x,{\beta }_y\in (0,1]$ and ${\gamma }_x,{\gamma }_y\in [0,1)$.

Lemma 1 

([7]). A NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK/BCI$-algebra X is an $(\in ,$ $\in )$-neutrosophic subalgebra of X if and only if $A=(A_T,$ $A_I,$ $A_F)$ satisfies

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{A}_T(x\ast y)\ge A_T(x)\wedge A_T(y)\hfill \\ A_I(x\ast y)\ge A_I(x)\wedge A_I(y)\hfill \\ A_F(x\ast y)\le A_F(x)\vee A_F(y)\hfill \end{array}\right).\end{array}$$

(11)

Proposition 1.

Every $(\in ,$ $\in )$-neutrosophic subalgebra $A=(A_T,$ $A_I,$ $A_F)$ of a $BCK/BCI$-algebra X satisfies

$$\begin{array}{c}\hfill (\forall x\in X)\left(A_T(0)\ge A_T(x),A_I(0)\ge A_I(x),A_F(0)\le A_F(x)\right).\end{array}$$

(12)

Proof. 

Straightforward. ☐

Theorem 1.

If $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic subalgebra of a $BCK/BCI$-algebra X, then the lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are S-energetic subsets of X, where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Let $x,y\in X$ and $\alpha \in (0,1]$ be such that $x\ast y\in L_T^{\in }(A;\alpha )$. Then

$$\begin{array}{c}\hfill \alpha \ge A_T(x\ast y)\ge A_T(x)\wedge A_T(y),\end{array}$$

and thus $A_T(x)\le \alpha$ or $A_T(y)\le \alpha$; that is, $x\in L_T^{\in }(A;\alpha )$ or $y\in L_T^{\in }(A;\alpha )$. Thus $\{x,y\}\cap L_T^{\in }(A;\alpha )\ne \varnothing$. Therefore $L_T^{\in }(A;\alpha )$ is an S-energetic subset of X. Similarly, we can verify that $L_I^{\in }(A;\beta )$ is an S-energetic subset of X. We let $x,y\in X$ and $\gamma \in [0,1)$ be such that $x\ast y\in L_F^{\in }(A;\gamma )$. Then

$$\begin{array}{c}\hfill \gamma \le A_F(x\ast y)\le A_F(x)\vee A_F(y).\end{array}$$

It follows that $A_F(x)\ge \gamma$ or $A_F(y)\ge \gamma$; that is, $x\in L_F^{\in }(A;\gamma )$ or $y\in L_F^{\in }(A;\gamma )$. Hence $\{x,y\}\cap L_F^{\in }(A;\gamma )\ne \varnothing$, and therefore $L_F^{\in }(A;\gamma )$ is an S-energetic subset of X. ☐

Corollary 1.

If $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic subalgebra of a $BCK/BCI$-algebra X, then the strong lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are S-energetic subsets of X, where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Straightforward. ☐

The converse of Theorem 1 is not true, as seen in the following example.

Example 1.

Consider a $BCK$-algebra $X=\{0,1,2,3,4\}$ with the binary operation * that is given in

Table 1. Cayley table for the binary operation “*”.

*	0	1	2	3	4
0	0	0	0	0	0
1	1	0	0	0	0
2	2	1	0	0	1
3	3	2	1	0	2
4	4	1	1	1	0

(see [14]).

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in X that is given in

Table 2. Tabulation representation of $A=(A_T,$ $A_I,$ $A_F)$.

X	${\mathit{A}}_{\mathit{T}}(\mathit{x})$	${\mathit{A}}_{\mathit{I}}(\mathit{x})$	${\mathit{A}}_{\mathit{F}}(\mathit{x})$
0	$0.6$	$0.8$	$0.2$
1	$0.4$	$0.5$	$0.7$
2	$0.4$	$0.5$	$0.6$
3	$0.4$	$0.5$	$0.5$
4	$0.7$	$0.8$	$0.2$

.

If $\alpha \in [0.4,0.6)$, $\beta \in [0.5,0.8)$, and $\gamma \in (0.2,0.5]$, then $L_T^{\in }(A;\alpha )=\{1,2,3\}$, $L_I^{\in }(A;\beta )=\{1,2,3\}$, and $L_F^{\in }(A;\gamma )=\{1,2,3\}$ are S-energetic subsets of X. Because

$$\begin{array}{c}\hfill A_T(4\ast 4)=A_T(0)=0.6\ngeqq 0.7=A_T(4)\wedge A_T(4)\end{array}$$

and/or

$$\begin{array}{c}\hfill A_F(3\ast 2)=A_F(1)=0.7\nleqq 0.6=A_F(3)\vee A_F(2),\end{array}$$

it follows from Lemma 1 that $A=(A_T,$ $A_I,$ $A_F)$ is not an $(\in ,$ $\in )$-neutrosophic subalgebra of X.

Definition 2.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. Then $(\alpha ,\beta ,\gamma )$ is called a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$ if the following assertion is valid:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}x\ast y\in U_T^{\in }(A;\alpha )\Rightarrow A_T(x)\vee A_T(y)\ge \alpha ,\hfill \\ x\ast y\in U_I^{\in }(A;\beta )\Rightarrow A_I(x)\vee A_I(y)\ge \beta ,\hfill \\ x\ast y\in U_F^{\in }(A;\gamma )\Rightarrow A_F(x)\wedge A_F(y)\le \gamma \hfill \end{array}\right)\end{array}$$

(13)

Example 2.

Let $X=\{0,1,2,3,4\}$ be a set with the binary operation * that is given in

Table 3. Cayley table for the binary operation “*”.

*	0	1	2	3	4
0	0	0	0	0	0
1	1	0	1	1	0
2	2	2	0	2	0
3	3	3	3	0	3
4	4	4	4	4	0

.

Then $(X,\ast ,0)$ is a $BCK$-algebra (see [14]). Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in X that is given in

Table 4. Tabulation representation of $A=(A_T,$ $A_I,$ $A_F)$.

X	${\mathit{A}}_{\mathit{T}}(\mathit{x})$	${\mathit{A}}_{\mathit{I}}(\mathit{x})$	${\mathit{A}}_{\mathit{F}}(\mathit{x})$
0	$0.2$	$0.3$	$0.7$
1	$0.6$	$0.4$	$0.6$
2	$0.5$	$0.3$	$0.4$
3	$0.4$	$0.8$	$0.5$
4	$0.7$	$0.6$	$0.2$

.

It is routine to verify that $(\alpha ,\beta ,\gamma )\in (0,2,1]\times (0.3,1]\times [0,0.7)$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$.

Theorem 2.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ satisfies the following condition:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{A}_T(x\ast y)\le A_T(x)\vee A_T(y)\hfill \\ A_I(x\ast y)\le A_I(x)\vee A_I(y)\hfill \\ A_F(x\ast y)\ge A_F(x)\wedge A_F(y)\hfill \end{array}\right),\end{array}$$

(14)

then $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$.

Proof. 

Let $x,y\in X$ be such that $x\ast y\in U_T^{\in }(A;\alpha )$. Then

$$\begin{array}{c}\hfill \alpha \le A_T(x\ast y)\le A_T(x)\vee A_T(y).\end{array}$$

Similarly, if $x\ast y\in U_I^{\in }(A;\beta )$ for $x,y\in X$, then $A_I(x)\vee A_I(y)\ge \beta$. Now, let $a,b\in X$ be such that $a\ast b\in U_F^{\in }(A;\gamma )$. Then

$$\begin{array}{c}\hfill \gamma \ge A_F(a\ast b)\ge A_F(a)\wedge A_F(b).\end{array}$$

Therefore $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$. ☐

Theorem 3.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ satisfies the following conditions:

$$\begin{array}{c}\hfill (\forall x\in X)\left(A_T(0)\le A_T(x),A_I(0)\le A_I(x),A_F(0)\ge A_F(x)\right)\end{array}$$

(15)

and

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{A}_T(x)\le A_T(x\ast y)\vee A_T(y)\hfill \\ A_I(x)\le A_I(x\ast y)\vee A_I(y)\hfill \\ A_F(x)\ge A_F(x\ast y)\wedge A_F(y)\hfill \end{array}\right),\end{array}$$

(16)

then $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$.

Proof. 

Let $x,y,a,b,u,v\in X$ be such that $x\ast y\in U_T^{\in }(A;\alpha )$, $a\ast b\in U_I^{\in }(A;\beta )$, and $u\ast v\in U_F^{\in }(A;\gamma )$. Then

$$\begin{array}{cc}\hfill \alpha & \le A_T(x\ast y)\le A_T((x\ast y)\ast x)\vee A_T(x)\hfill \\ & =A_T((x\ast x)\ast y)\vee A_T(x)=A_T(0\ast y)\vee A_T(x)\hfill \\ & =A_T(0)\vee A_T(x)=A_T(x),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \beta & \le A_I(a\ast b)\le A_I((a\ast b)\ast a)\vee A_I(a)\hfill \\ & =A_I((a\ast a)\ast b)\vee A_I(a)=A_I(0\ast b)\vee A_I(a)\hfill \\ & =A_I(0)\vee A_I(a)=A_I(a),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \gamma & \ge A_F(u\ast v)\ge A_F((u\ast v)\ast u)\wedge A_F(u)\hfill \\ & =A_F((u\ast u)\ast v)\wedge A_F(u)=A_F(0\ast v)\wedge A_F(v)\hfill \\ & =A_F(0)\wedge A_F(v)=A_F(v)\hfill \end{array}$$

by Equations (3), (V), (15), and (16). It follows that

$$\begin{array}{c}{A}_T(x)\vee A_T(y)\ge A_T(x)\ge \alpha ,\hfill \\ A_I(a)\vee A_I(b)\ge A_I(a)\ge \beta ,\hfill \\ A_F(u)\wedge A_F(v)\le A_F(u)\le \gamma .\hfill \end{array}$$

Therefore $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$. ☐

Theorem 4.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$, then upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are S-energetic where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Let $x,y,a,b,u,v\in X$ be such that $x\ast y\in U_T^{\in }(A;\alpha )$, $a\ast b\in U_I^{\in }(A;\beta )$, and $u\ast v\in U_F^{\in }(A;\gamma )$. Using Equation (13), we have $A_T(x)\vee A_T(y)\ge \alpha$, $A_I(a)\vee A_I(b)\ge \beta$, and $A_F(u)\wedge A_F(v)\le \gamma$. It follows that

$$\begin{array}{c}\hfill A_T(x)\ge \alpha \mathrm{or}{A}_T(y)\ge \alpha ,\mathrm{that}\mathrm{is},x\in U_T^{\in }(A;\alpha )\mathrm{or}y\in U_T^{\in }(A;\alpha );\end{array}$$

$$\begin{array}{c}\hfill A_I(a)\ge \beta \mathrm{or}{A}_I(b)\ge \beta ,\mathrm{that}\mathrm{is},a\in U_I^{\in }(A;\beta )\mathrm{or}b\in U_I^{\in }(A;\beta );\end{array}$$

and

$$\begin{array}{c}\hfill A_F(u)\le \gamma \mathrm{or}{A}_F(v)\le \gamma ,\mathrm{that}\mathrm{is},u\in U_F^{\in }(A;\gamma )\mathrm{or}v\in U_F^{\in }(A;\gamma ).\end{array}$$

Hence $\{x,y\}\cap U_T^{\in }(A;\alpha )\ne \varnothing$, $\{a,b\}\cap U_I^{\in }(A;\beta )\ne \varnothing$, and $\{u,v\}\cap U_F^{\in }(A;\gamma )\ne \varnothing$. Therefore $U_T^{\in }(A;\alpha )$, $U_I^{\in }(A;\beta )$, and $U_F^{\in }(A;\gamma )$ are S-energetic subsets of X. ☐

Definition 3.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. Then $(\alpha ,\beta ,\gamma )$ is called a neutrosophic anti-permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$ if the following assertion is valid:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}x\ast y\in L_T^{\in }(A;\alpha )\Rightarrow A_T(x)\wedge A_T(y)\le \alpha ,\hfill \\ x\ast y\in L_I^{\in }(A;\beta )\Rightarrow A_I(x)\wedge A_I(y)\le \beta ,\hfill \\ x\ast y\in L_F^{\in }(A;\gamma )\Rightarrow A_F(x)\vee A_F(y)\ge \gamma \hfill \end{array}\right).\end{array}$$

(17)

Example 3.

Let $X=\{0,1,2,3,4\}$ be a set with the binary operation * that is given in

Table 5. Cayley table for the binary operation “*”.

*	0	1	2	3	4
0	0	0	0	0	0
1	1	0	0	1	0
2	2	1	0	2	0
3	3	3	3	0	3
4	4	4	4	4	0

.

Then $(X,\ast ,0)$ is a $BCK$-algebra (see [14]). Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in X that is given in

Table 6. Tabulation representation of $A=(A_T,$ $A_I,$ $A_F)$.

X	${\mathit{A}}_{\mathit{T}}(\mathit{x})$	${\mathit{A}}_{\mathit{I}}(\mathit{x})$	${\mathit{A}}_{\mathit{F}}(\mathit{x})$
0	$0.7$	$0.6$	$0.4$
1	$0.4$	$0.5$	$0.6$
2	$0.4$	$0.5$	$0.6$
3	$0.5$	$0.2$	$0.7$
4	$0.3$	$0.3$	$0.9$

.

It is routine to verify that $(\alpha ,\beta ,\gamma )\in (0.3,1]\times (0.2,1]\times [0,0.9)$ is a neutrosophic anti-permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$.

Theorem 5.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic subalgebra of X, then $(\alpha ,\beta ,\gamma )$ is a neutrosophic anti-permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$.

Proof. 

Let $x,y,a,b,u,v\in X$ be such that $x\ast y\in L_T^{\in }(A;\alpha )$, $a\ast b\in L_I^{\in }(A;\beta )$, and $u\ast v\in L_F^{\in }(A;\gamma )$. Using Lemma 1, we have

$$\begin{array}{c}{A}_T(x)\wedge A_T(y)\le A_T(x\ast y)\le \alpha ,\hfill \\ A_I(a)\wedge A_I(b)\le A_I(a\ast b)\le \beta ,\hfill \\ A_F(u)\vee A_F(v)\ge A_F(u\ast v)\ge \gamma ,\hfill \end{array}$$

and thus $(\alpha ,\beta ,\gamma )$ is a neutrosophic anti-permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$. ☐

Theorem 6.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $(\alpha ,\beta ,\gamma )$ is a neutrosophic anti-permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$, then lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are S-energetic where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Let $x,y,a,b,u,v\in X$ be such that $x\ast y\in L_T^{\in }(A;\alpha )$, $a\ast b\in L_I^{\in }(A;\beta )$, and $u\ast v\in L_F^{\in }(A;\gamma )$. Using Equation (17), we have $A_T(x)\wedge A_T(y)\le \alpha$, $A_I(a)\wedge A_I(b)\le \beta$, and $A_F(u)\vee A_F(v)\ge \gamma$, which imply that

$$\begin{array}{c}\hfill A_T(x)\le \alpha \mathrm{or}{A}_T(y)\le \alpha ,\mathrm{that}\mathrm{is},x\in L_T^{\in }(A;\alpha )\mathrm{or}y\in L_T^{\in }(A;\alpha );\end{array}$$

$$\begin{array}{c}\hfill A_I(a)\le \beta \mathrm{or}{A}_I(b)\le \beta ,\mathrm{that}\mathrm{is},a\in L_I^{\in }(A;\beta )\mathrm{or}b\in L_I^{\in }(A;\beta );\end{array}$$

and

$$\begin{array}{c}\hfill A_F(u)\ge \gamma \mathrm{or}{A}_F(v)\ge \gamma ,\mathrm{that}\mathrm{is},u\in L_F^{\in }(A;\gamma )\mathrm{or}v\in L_F^{\in }(A;\gamma ).\end{array}$$

Hence $\{x,y\}\cap L_T^{\in }(A;\alpha )\ne \varnothing$, $\{a,b\}\cap L_I^{\in }(A;\beta )\ne \varnothing$, and $\{u,v\}\cap L_F^{\in }(A;\gamma )\ne \varnothing$. Therefore $L_T^{\in }(A;\alpha )$, $L_I^{\in }(A;\beta )$, and $L_F^{\in }(A;\gamma )$ are S-energetic subsets of X. ☐

Definition 4.

A NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK/BCI$-algebra X is called an $(\in ,$ $\in )$- neutrosophic ideal of X if the following assertions are valid:

$$\begin{array}{c}(\forall x\in X)\left(\begin{array}{c}x\in U_T^{\in }(A;\alpha )\Rightarrow 0\in U_T^{\in }(A;\alpha )\hfill \\ x\in U_I^{\in }(A;\beta )\Rightarrow 0\in U_I^{\in }(A;\beta )\hfill \\ x\in U_F^{\in }(A;\gamma )\Rightarrow 0\in U_F^{\in }(A;\gamma )\hfill \end{array}\right),\hfill \end{array}$$

(18)

$$\begin{array}{c}(\forall x,y\in X)\left(\begin{array}{c}x\ast y\in U_T^{\in }(A;{\alpha }_x),y\in U_T^{\in }(A;{\alpha }_y)\Rightarrow x\in U_T^{\in }(A;{\alpha }_x\wedge {\alpha }_y)\hfill \\ x\ast y\in U_I^{\in }(A;{\beta }_x),y\in U_I^{\in }(A;{\beta }_y)\Rightarrow x\in U_I^{\in }(A;{\beta }_x\wedge {\beta }_y)\hfill \\ x\ast y\in U_F^{\in }(A;{\gamma }_x),y\in U_F^{\in }(A;{\gamma }_y)\Rightarrow x\in U_F^{\in }(A;{\gamma }_x\vee {\gamma }_y)\hfill \end{array}\right),\hfill \end{array}$$

(19)

for all $\alpha ,\beta ,{\alpha }_x,{\alpha }_y,{\beta }_x,{\beta }_y\in (0,1]$ and $\gamma ,{\gamma }_x,{\gamma }_y\in [0,1)$.

Theorem 7.

A NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK/BCI$-algebra X is an $(\in ,$ $\in )$-neutrosophic ideal of X if and only if $A=(A_T,$ $A_I,$ $A_F)$ satisfies

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{A}_T(0)\ge A_T(x)\ge A_T(x\ast y)\wedge A_T(y)\hfill \\ A_I(0)\ge A_I(x)\ge A_I(x\ast y)\wedge A_I(y)\hfill \\ A_F(0)\le A_F(x)\le A_F(x\ast y)\vee A_F(y)\hfill \end{array}\right).\end{array}$$

(20)

Proof. 

Assume that Equation (20) is valid, and let $x\in U_T^{\in }(A;\alpha )$, $a\in U_I^{\in }(A;\beta )$, and $u\in U_F^{\in }(A;\gamma )$ for any $x,a,u\in X$, $\alpha ,\beta \in (0,1]$ and $\gamma \in [0,1)$. Then $A_T(0)\ge A_T(x)\ge \alpha$, $A_I(0)\ge A_I(a)\ge \beta$, and $A_F(0)\le A_F(u)\le \gamma$. Hence $0\in U_T^{\in }(A;\alpha )$, $0\in U_I^{\in }(A;\beta )$, and $0\in U_F^{\in }(A;\gamma )$, and thus Equation (18) is valid. Let $x,y,a,b,u,v\in X$ be such that $x\ast y\in U_T^{\in }(A;{\alpha }_x)$, $y\in U_T^{\in }(A;{\alpha }_y)$, $a\ast b\in U_I^{\in }(A;{\beta }_a)$, $b\in U_I^{\in }(A;{\beta }_b)$, $u\ast v\in U_F^{\in }(A;{\gamma }_u)$, and $v\in U_F^{\in }(A;{\gamma }_v)$ for all ${\alpha }_x,{\alpha }_y,{\beta }_a,{\beta }_b\in (0,1]$ and ${\gamma }_u,{\gamma }_v\in [0,1)$. Then $A_T(x\ast y)\ge {\alpha }_x$, $A_T(y)\ge {\alpha }_y$, $A_I(a\ast b)\ge {\beta }_a$, $A_I(b)\ge {\beta }_b$, $A_F(u\ast v)\le {\gamma }_u$, and $A_F(v)\le {\gamma }_v$. It follows from Equation (20) that

$$\begin{array}{c}{A}_T(x)\ge A_T(x\ast y)\wedge A_T(y)\ge {\alpha }_x\wedge {\alpha }_y,\hfill \\ A_I(a)\ge A_I(a\ast b)\wedge A_I(b)\ge {\beta }_a\wedge {\beta }_b,\hfill \\ A_F(u)\le A_F(u\ast v)\vee A_F(v)\le {\gamma }_u\vee {\gamma }_v.\hfill \end{array}$$

Hence $x\in U_T^{\in }(A;{\alpha }_x\wedge {\alpha }_y)$, $a\in U_I^{\in }(A;{\beta }_a\wedge {\beta }_b)$, and $u\in U_F^{\in }(A;{\gamma }_u\vee {\gamma }_v)$. Therefore $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic ideal of X.

Conversely, let $A=(A_T,$ $A_I,$ $A_F)$ be an $(\in ,$ $\in )$-neutrosophic ideal of X. If there exists $x_0\in X$ such that $A_T(0)<A_T(x_0)$, then $x_0\in U_T^{\in }(A;\alpha )$ and $0\notin U_T^{\in }(A;\alpha )$, where $\alpha =A_T(x_0)$. This is a contradiction, and thus $A_T(0)\ge A_T(x)$ for all $x\in X$. Assume that $A_T(x_0)<A_T(x_0\ast y_0)\wedge A_T(y_0)$ for some $x_0,y_0\in X$. Taking $\alpha :=A_T(x_0\ast y_0)\wedge A_T(y_0)$ implies that $x_0\ast y_0\in U_T^{\in }(A;\alpha )$ and $y_0\in U_T^{\in }(A;\alpha )$; but $x_0\notin U_T^{\in }(A;\alpha )$. This is a contradiction, and thus $A_T(x)\ge A_T(x\ast y)\wedge A_T(y)$ for all $x,y\in X$. Similarly, we can verify that $A_I(0)\ge A_I(x)\ge A_I(x\ast y)\wedge A_I(y)$ for all $x,y\in X$. Now, suppose that $A_F(0)>A_F(a)$ for some $a\in X$. Then $a\in U_F^{\in }(A;\gamma )$ and $0\notin U_F^{\in }(A;\gamma )$ by taking $\gamma =A_F(a)$. This is impossible, and thus $A_F(0)\le A_F(x)$ for all $x\in X$. Suppose there exist $a_0,b_0\in X$ such that $A_F(a_0)>A_F(a_0\ast b_0)\vee A_F(b_0)$, and take $\gamma :=A_F(a_0\ast b_0)\vee A_F(b_0)$. Then $a_0\ast b_0\in U_F^{\in }(A;\gamma )$, $b_0\in U_F^{\in }(A;\gamma )$, and $a_0\notin U_F^{\in }(A;\gamma )$, which is a contradiction. Thus $A_F(x)\le A_F(x\ast y)\vee A_F(y)$ for all $x,y\in X$. Therefore $A=(A_T,$ $A_I,$ $A_F)$ satisfies Equation (20). ☐

Lemma 2.

Every $(\in ,$ $\in )$-neutrosophic ideal $A=(A_T,$ $A_I,$ $A_F)$ of a $BCK/BCI$-algebra X satisfies

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(x\le y\Rightarrow A_T(x)\ge A_T(y),A_I(x)\ge A_I(y),A_F(x)\le A_F(y)\right).\end{array}$$

(21)

Proof. 

Let $x,y\in X$ be such that $x\le y$. Then $x\ast y=0$, and thus

$$\begin{array}{c}{A}_T(x)\ge A_T(x\ast y)\wedge A_T(y)=A_T(0)\wedge A_T(y)=A_T(y),\hfill \\ A_I(x)\ge A_I(x\ast y)\wedge A_I(y)=A_I(0)\wedge A_I(y)=A_I(y),\hfill \\ A_F(x)\le A_F(x\ast y)\vee A_F(y)=A_F(0)\vee A_F(y)=A_F(y),\hfill \end{array}$$

by Equation (20). This completes the proof. ☐

Theorem 8.

A NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK$-algebra X is an $(\in ,$ $\in )$-neutrosophic ideal of X if and only if $A=(A_T,$ $A_I,$ $A_F)$ satisfies

$$\begin{array}{c}\hfill (\forall x,y,z\in X)\left(\begin{array}{c}x\ast y\le z\Rightarrow \left\{\begin{array}{c}{A}_T(x)\ge A_T(y)\wedge A_T(z)\hfill \\ A_I(x)\ge A_I(y)\wedge A_I(z)\hfill \\ A_F(x)\le A_F(y)\vee A_F(z)\hfill \end{array}\right.\hfill \end{array}\right)\end{array}$$

(22)

Proof. 

Let $A=(A_T,$ $A_I,$ $A_F)$ be an $(\in ,$ $\in )$-neutrosophic ideal of X, and let $x,y,z\in X$ be such that $x\ast y\le z$. Using Theorem 7 and Lemma 2, we have

$$\begin{array}{c}{A}_T(x)\ge A_T(x\ast y)\wedge A_T(y)\ge A_T(y)\wedge A_T(z),\hfill \\ A_I(x)\ge A_I(x\ast y)\wedge A_I(y)\ge A_I(y)\wedge A_I(z),\hfill \\ A_F(x)\le A_F(x\ast y)\vee A_F(y)\le A_F(y)\vee A_F(z).\hfill \end{array}$$

Conversely, assume that $A=(A_T,$ $A_I,$ $A_F)$ satisfies Equation (22). Because $0\ast x\le x$ for all $x\in X$, it follows from Equation (22) that

$$\begin{array}{c}{A}_T(0)\ge A_T(x)\wedge A_T(x)=A_T(x),\hfill \\ A_I(0)\ge A_I(x)\wedge A_I(x)=A_I(x),\hfill \\ A_F(0)\le A_F(x)\vee A_F(x)=A_F(x),\hfill \end{array}$$

for all $x\in X$. Because $x\ast (x\ast y)\le y$ for all $x,y\in X$, we have

$$\begin{array}{c}{A}_T(x)\ge A_T(x\ast y)\wedge A_T(y),\hfill \\ A_I(x)\ge A_I(x\ast y)\wedge A_I(y),\hfill \\ A_F(x)\le A_F(x\ast y)\vee A_F(y),\hfill \end{array}$$

for all $x,y\in X$ by Equation (22). It follows from Theorem 7 that $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic ideal of X. ☐

Theorem 9.

If $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic ideal of a $BCK/BCI$-algebra X, then the lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic subsets of X where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Let $x,a,u\in X$, $\alpha ,\beta \in (0,1]$, and $\gamma \in [0,1)$ be such that $x\in L_T^{\in }(A;\alpha )$, $a\in L_I^{\in }(A;\beta )$, and $u\in L_F^{\in }(A;\gamma )$. Using Theorem 7, we have

$$\begin{array}{c}\alpha \ge A_T(x)\ge A_T(x\ast y)\wedge A_T(y),\hfill \\ \beta \ge A_I(a)\ge A_I(a\ast b)\wedge A_I(b),\hfill \\ \gamma \le A_F(u)\le A_F(u\ast v)\vee A_F(v),\hfill \end{array}$$

for all $y,b,v\in X$. It follows that

$$\begin{array}{c}\hfill A_T(x\ast y)\le \alpha \mathrm{or}{A}_T(y)\le \alpha ,\mathrm{that}\mathrm{is},x\ast y\in L_T^{\in }(A;\alpha )\mathrm{or}y\in L_T^{\in }(A;\alpha );\end{array}$$

$$\begin{array}{c}\hfill A_I(a\ast b)\le \beta \mathrm{or}{A}_I(b)\le \beta ,\mathrm{that}\mathrm{is},a\ast b\in L_T^{\in }(A;\beta )\mathrm{or}b\in L_T^{\in }(A;\beta );\end{array}$$

and

$$\begin{array}{c}\hfill A_F(u\ast v)\ge \gamma \mathrm{or}{A}_F(v)\ge \gamma ,\mathrm{that}\mathrm{is},u\ast v\in L_T^{\in }(A;\gamma )\mathrm{or}v\in L_T^{\in }(A;\gamma ).\end{array}$$

Hence $\{y,x\ast y\}\cap L_T^{\in }(A;\alpha )$, $\{b,a\ast b\}\cap L_I^{\in }(A;\beta )$, and $\{v,u\ast v\}\cap L_F^{\in }(A;\gamma )$ are nonempty, and therefore $L_T^{\in }(A;\alpha )$, $L_I^{\in }(A;\beta )$ and $L_F^{\in }(A;\gamma )$ are I-energetic subsets of X. ☐

Corollary 2.

If $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic ideal of a $BCK/BCI$-algebra X, then the strong lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic subsets of X where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Straightforward. ☐

Theorem 10.

Let $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic ideal of a $BCK$-algebra X, then

(1)

the (strong) upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are right stable where $\mathsf{\Phi }\in \{T,I,F\}$;

(2)

the (strong) lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are right vanished where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

(1) Let $x\in X$, $a\in U_T^{\in }(A;\alpha )$, $b\in U_I^{\in }(A;\beta )$, and $c\in U_F^{\in }(A;\gamma )$. Then $A_T(a)\ge \alpha$, $A_I(b)\ge \beta$, and $A_F(c)\le \gamma$. Because $a\ast x\le a$, $b\ast x\le b$, and $c\ast x\le c$, it follows from Lemma 2 that $A_T(a\ast x)\ge A_T(a)\ge \alpha$, $A_I(b\ast x)\ge A_I(b)\ge \beta$, and $A_F(c\ast x)\le A_F(c)\le \gamma$; that is, $a\ast x\in U_T^{\in }(A;\alpha )$, $b\ast x\in U_I^{\in }(A;\beta )$, and $c\ast x\in U_F^{\in }(A;\gamma )$. Hence the upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are right stable where $\mathsf{\Phi }\in \{T,I,F\}$. Similarly, the strong upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are right stable where $\mathsf{\Phi }\in \{T,I,F\}$.

(2) Assume that $x\ast y\in L_T^{\in }(A;\alpha )$, $a\ast b\in L_I^{\in }(A;\beta )$, and $c\ast d\in L_F^{\in }(A;\gamma )$ for any $x,y,a,b,c,d\in X$. Then $A_T(x\ast y)\le \alpha$, $A_I(a\ast b)\le \beta$, and $A_F(c\ast d)\ge \gamma$. Because $x\ast y\le x$, $a\ast b\le a$, and $c\ast d\le c$, it follows from Lemma 2 that $\alpha \ge A_T(x\ast y)\ge A_T(x)$, $\beta \ge A_I(a\ast b)\ge A_I(a)$, and $\gamma \le A_F(c\ast d)\le A_F(c)$; that is, $x\in L_T^{\in }(A;\alpha )$, $a\in L_I^{\in }(A;\beta )$, and $c\in L_F^{\in }(A;\gamma )$. Therefore the lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are right vanished where $\mathsf{\Phi }\in \{T,I,F\}$. In a similar way, we know that the strong lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are right vanished where $\mathsf{\Phi }\in \{T,I,F\}$. ☐

Definition 5.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. Then $(\alpha ,\beta ,\gamma )$ is called a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$ if the following assertion is valid:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}x\in U_T^{\in }(A;\alpha )\Rightarrow A_T(x\ast y)\vee A_T(y)\ge \alpha ,\hfill \\ x\in U_I^{\in }(A;\beta )\Rightarrow A_I(x\ast y)\vee A_I(y)\ge \beta ,\hfill \\ x\in U_F^{\in }(A;\gamma )\Rightarrow A_F(x\ast y)\wedge A_F(y)\le \gamma \hfill \end{array}\right).\end{array}$$

(23)

Example 4.

(1) In Example 2, $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$.

(2) Consider a $BCI$-algebra $X=\{0,1,a,b,c\}$ with the binary operation * that is given in

Table 7. Cayley table for the binary operation “*”.

*	0	1	a	b	c
0	0	0	a	b	c
1	1	0	a	b	c
a	a	a	0	c	b
b	b	b	c	0	a
c	c	c	b	a	0

(see [14]).

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in X that is given in

Table 8. Tabulation representation of $A=(A_T,$ $A_I,$ $A_F)$.

X	${\mathit{A}}_{\mathit{T}}(\mathit{x})$	${\mathit{A}}_{\mathit{I}}(\mathit{x})$	${\mathit{A}}_{\mathit{F}}(\mathit{x})$
0	$0.33$	$0.38$	$0.77$
1	$0.44$	$0.48$	$0.66$
a	$0.55$	$0.68$	$0.44$
b	$0.66$	$0.58$	$0.44$
c	$0.66$	$0.68$	$0.55$

.

It is routine to check that $(\alpha ,\beta ,\gamma )\in (0.33,1]\times (0.38,1]\times [0,0.77)$ is a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$.

Lemma 3.

If a NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK/BCI$-algebra X satisfies the condition of Equation (14), then

$$\begin{array}{c}\hfill (\forall x\in X)\left(A_T(0)\le A_T(x),A_I(0)\le A_I(x),A_F(0)\ge A_F(x)\right).\end{array}$$

(24)

Proof. 

Straightforward. ☐

Theorem 11.

If a NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK$-algebra X satisfies the condition of Equation (14), then every neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$.

Proof. 

Let $(\alpha ,\beta ,\gamma )$ be a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$. Let $x,y,a,b,u,v\in X$ be such that $x\ast y\in U_T^{\in }(A;\alpha )$, $a\ast b\in U_I^{\in }(A;\beta )$, and $u\ast v\in U_F^{\in }(A;\gamma )$. It follows from Equations (23), (3), (III), and (V) and Lemma 3 that

$$\begin{array}{cc}\hfill \alpha & \le A_T((x\ast y)\ast x)\vee A_T(x)=A_T((x\ast x)\ast y)\vee A_T(x)\hfill \\ & =A_T(0\ast y)\vee A_T(x)=A_T(0)\vee A_T(x)=A_T(x),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \beta & \le A_I((a\ast b)\ast a)\vee A_I(a)=A_I((a\ast a)\ast b)\vee A_I(a)\hfill \\ & =A_I(0\ast b)\vee A_I(a)=A_I(0)\vee A_I(a)=A_I(a),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \gamma & \ge A_F((u\ast v)\ast u)\wedge A_F(u)=A_F((u\ast u)\ast v)\wedge A_F(u)\hfill \\ & =A_F(0\ast v)\wedge A_F(u)=A_F(0)\wedge A_F(u)=A_F(u).\hfill \end{array}$$

Hence $A_T(x)\vee A_T(y)\ge A_T(x)\ge \alpha$, $A_I(a)\vee A_I(b)\ge A_I(a)\ge \beta$, and $A_F(u)\wedge A_F(v)\le A_F(u)\le \gamma$. Therefore $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$. ☐

Given a NS $A=(A_T,$ $A_I,$ $A_F)$ in a $BCK/BCI$-algebra X, any upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X may not be I-energetic where $\mathsf{\Phi }\in \{T,I,F\}$, as seen in the following example.

Example 5.

Consider a $BCK$-algebra $X=\{0,1,2,3,4\}$ with the binary operation * that is given in

Table 9. Cayley table for the binary operation “*”.

*	0	1	2	3	4
0	0	0	0	0	0
1	1	0	0	0	0
2	2	1	0	1	0
3	3	1	1	0	0
4	4	2	1	2	0

(see [14]).

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in X that is given in

Table 10. Tabulation representation of $A=(A_T,$ $A_I,$ $A_F)$.

X	${\mathit{A}}_{\mathit{T}}(\mathit{x})$	${\mathit{A}}_{\mathit{I}}(\mathit{x})$	${\mathit{A}}_{\mathit{F}}(\mathit{x})$
0	$0.75$	$0.73$	$0.34$
1	$0.53$	$0.45$	$0.58$
2	$0.67$	$0.86$	$0.34$
3	$0.53$	$0.56$	$0.58$
4	$0.46$	$0.56$	$0.66$

.

Then $U_T^{\in }(A;0.6)=\{0,2\}$, $U_I^{\in }(A;0.7)=\{0,2\}$, and $U_F^{\in }(A;0.4)=\{0,2\}$. Because $2\in \{0,2\}$ and $\{1,2\ast 1\}\cap \{0,2\}=\varnothing$, we know that $\{0,2\}$ is not an I-energetic subset of X.

We now provide conditions for the upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets to be I-energetic where $\mathsf{\Phi }\in \{T,I,F\}$.

Theorem 12.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$, then the upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic subsets of X where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Let $x,a,u\in X$ and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$ such that $x\in U_T^{\in }(A;\alpha )$, $a\in U_I^{\in }(A;\beta )$, and $u\in U_F^{\in }(A;\gamma )$. Because $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$, it follows from Equation (23) that

$$\begin{array}{c}\hfill A_T(x\ast y)\vee A_T(y)\ge \alpha ,A_I(a\ast b)\vee A_I(b)\ge \beta ,\mathrm{and}{A}_F(u\ast v)\wedge A_F(v)\le \gamma \end{array}$$

for all $y,b,v\in X$. Hence

$$\begin{array}{c}\hfill A_T(x\ast y)\ge \alpha \mathrm{or}{A}_T(y)\ge \alpha ,\mathrm{that}\mathrm{is},x\ast y\in U_T^{\in }(A;\alpha )\mathrm{or}y\in U_T^{\in }(A;\alpha );\end{array}$$

$$\begin{array}{c}\hfill A_I(a\ast b)\ge \beta \mathrm{or}{A}_I(b)\ge \beta ,\mathrm{that}\mathrm{is},a\ast b\in U_I^{\in }(A;\beta )\mathrm{or}b\in U_I^{\in }(A;\beta );\end{array}$$

and

$$\begin{array}{c}\hfill A_F(u\ast v)\le \gamma \mathrm{or}{A}_F(v)\le \gamma ,\mathrm{that}\mathrm{is},u\ast v\in U_F^{\in }(A;\gamma )\mathrm{or}v\in U_F^{\in }(A;\gamma ).\end{array}$$

Hence $\{y,x\ast y\}\cap U_T^{\in }(A;\alpha )$, $\{b,a\ast b\}\cap U_I^{\in }(A;\beta )$, and $\{v,u\ast v\}\cap U_F^{\in }(A;\gamma )$ are nonempty, and therefore the upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic subsets of X where $\mathsf{\Phi }\in \{T,I,F\}$. ☐

Theorem 13.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ satisfies the following condition:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{A}_T(x)\le A_T(x\ast y)\vee A_T(y)\hfill \\ A_I(x)\le A_I(x\ast y)\vee A_I(y)\hfill \\ A_F(x)\ge A_F(x\ast y)\wedge A_F(y)\hfill \end{array}\right),\end{array}$$

(25)

then $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$.

Proof. 

Let $x,a,u\in X$ and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$ such that $x\in U_T^{\in }(A;\alpha )$, $a\in U_I^{\in }(A;\beta )$, and $u\in U_F^{\in }(A;\gamma )$. Using Equation (25), we obtain

$$\begin{array}{c}\alpha \le A_T(x)\le A_T(x\ast y)\vee A_T(y),\hfill \\ \beta \le A_I(a)\le A_I(a\ast b)\vee A_I(b),\hfill \\ \gamma \ge A_F(u)\ge A_F(u\ast v)\wedge A_F(v),\hfill \end{array}$$

for all $y,b,v\in X$. Therefore $(\alpha ,\beta ,\gamma )$ is a neutrosophic permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$. ☐

Combining Theorems 12 and 13, we have the following corollary.

Corollary 3.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ satisfies the condition of Equation (25), then the upper neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic subsets of X where $\mathsf{\Phi }\in \{T,I,F\}$.

Definition 6.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. Then $(\alpha ,\beta ,\gamma )$ is called a neutrosophic anti-permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$ if the following assertion is valid:

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}x\in L_T^{\in }(A;\alpha )\Rightarrow A_T(x\ast y)\wedge A_T(y)\le \alpha ,\hfill \\ x\in L_I^{\in }(A;\beta )\Rightarrow A_I(x\ast y)\wedge A_I(y)\le \beta ,\hfill \\ x\in L_F^{\in }(A;\gamma )\Rightarrow A_F(x\ast y)\vee A_F(y)\ge \gamma \hfill \end{array}\right).\end{array}$$

(26)

Theorem 14.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ satisfies the condition of Equation (19), then $(\alpha ,\beta ,\gamma )$ is a neutrosophic anti-permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$.

Proof. 

Let $x,a,u\in X$ be such that $x\in L_T^{\in }(A;\alpha )$, $a\in L_I^{\in }(A;\beta )$, and $u\in L_F^{\in }(A;\gamma )$. Then

$$\begin{array}{c}{A}_T(x\ast y)\wedge A_T(y)\le A_T(x)\le \alpha ,\hfill \\ A_I(a\ast b)\wedge A_I(b)\le A_I(a)\le \beta ,\hfill \\ A_F(u\ast v)\vee A_F(v)\ge A_F(u)\ge \gamma ,\hfill \end{array}$$

for all $y,b,v\in X$ by Equation (20). Hence $(\alpha ,\beta ,\gamma )$ is a neutrosophic anti-permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$. ☐

Theorem 15.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $(\alpha ,\beta ,\gamma )$ is a neutrosophic anti-permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$, then the lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic where $\mathsf{\Phi }\in \{T,I,F\}$.

Proof. 

Let $x\in L_T^{\in }(A;\alpha )$, $a\in L_I^{\in }(A;\beta )$, and $u\in L_F^{\in }(A;\gamma )$. Then $A_T(x\ast y)\wedge A_T(y)\le \alpha$, $A_I(a\ast b)\wedge A_I(b)\le \beta$, and $A_F(u\ast v)\vee A_F(v)\ge \gamma$ for all $y,b,v\in X$ by Equation (26). It follows that

$$\begin{array}{c}\hfill A_T(x\ast y)\le \alpha \mathrm{or}{A}_T(y)\le \alpha ,\mathrm{that}\mathrm{is},x\ast y\in L_T^{\in }(A;\alpha )\mathrm{or}y\in L_T^{\in }(A;\alpha );\end{array}$$

$$\begin{array}{c}\hfill A_I(a\ast b)\le \beta \mathrm{or}{A}_I(b)\le \beta ,\mathrm{that}\mathrm{is},a\ast b\in L_I^{\in }(A;\beta )\mathrm{or}b\in L_I^{\in }(A;\beta );\end{array}$$

and

$$\begin{array}{c}\hfill A_F(u\ast v)\ge \gamma \mathrm{or}{A}_F(v)\ge \gamma ,\mathrm{that}\mathrm{is},u\ast v\in L_F^{\in }(A;\gamma )\mathrm{or}v\in L_F^{\in }(A;\gamma ).\end{array}$$

Hence $\{y,x\ast y\}\cap L_T^{\in }(A;\alpha )$, $\{b,a\ast b\}\cap L_I^{\in }(A;\beta )$ and $\{v,u\ast v\}\cap L_F^{\in }(A;\gamma )$ are nonempty, and therefore the lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic where $\mathsf{\Phi }\in \{T,I,F\}$. ☐

Combining Theorems 14 and 15, we obtain the following corollary.

Corollary 4.

Let $A=(A_T,$ $A_I,$ $A_F)$ be a NS in a $BCK/BCI$-algebra X and $(\alpha ,\beta ,\gamma )\in {\mathsf{\Lambda }}_T\times {\mathsf{\Lambda }}_I\times {\mathsf{\Lambda }}_F$, where ${\mathsf{\Lambda }}_T,$ ${\mathsf{\Lambda }}_I$, and ${\mathsf{\Lambda }}_F$ are subsets of $[0,1]$. If $A=(A_T,$ $A_I,$ $A_F)$ satisfies the condition of Equation (19), then the lower neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets of X are I-energetic where $\mathsf{\Phi }\in \{T,I,F\}$.

Theorem 16.

If $A=(A_T,$ $A_I,$ $A_F)$ is an $(\in ,$ $\in )$-neutrosophic subalgebra of a $BCK$-algebra X, then every neutrosophic anti-permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$ is a neutrosophic anti-permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$.

Proof. 

Let $(\alpha ,\beta ,\gamma )$ be a neutrosophic anti-permeable I-value for $A=(A_T,$ $A_I,$ $A_F)$. Let $x,y,a,b,u,v\in X$ be such that $x\ast y\in L_T^{\in }(A;\alpha )$, $a\ast b\in L_I^{\in }(A;\beta )$, and $u\ast v\in L_F^{\in }(A;\gamma )$. It follows from Equations (26), (3), (III), and (V) and Proposition 1 that

$$\begin{array}{cc}\hfill \alpha & \ge A_T((x\ast y)\ast x)\wedge A_T(x)=A_T((x\ast x)\ast y)\wedge A_T(x)\hfill \\ & =A_T(0\ast y)\wedge A_T(x)=A_T(0)\wedge A_T(x)=A_T(x),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \beta & \ge A_I((a\ast b)\ast a)\wedge A_I(a)=A_I((a\ast a)\ast b)\wedge A_I(a)\hfill \\ & =A_I(0\ast b)\wedge A_I(a)=A_I(0)\wedge A_I(a)=A_I(a),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \gamma & \le A_F((u\ast v)\ast u)\vee A_F(u)=A_F((u\ast u)\ast v)\vee A_F(u)\hfill \\ & =A_F(0\ast v)\vee A_F(u)=A_F(0)\vee A_F(u)=A_F(u).\hfill \end{array}$$

Hence $A_T(x)\wedge A_T(y)\le A_T(x)\le \alpha$, $A_I(a)\wedge A_I(b)\le A_I(a)\le \beta$, and $A_F(u)\vee A_F(v)\ge A_F(u)\ge \gamma$. Therefore $(\alpha ,\beta ,\gamma )$ is a neutrosophic anti-permeable S-value for $A=(A_T,$ $A_I,$ $A_F)$. ☐

4. Conclusions

Using the notions of subalgebras and ideals in $BCK/BCI$-algebras, Jun et al. [8] introduced the notions of energetic subsets, right vanished subsets, right stable subsets, and (anti-)permeable values in $BCK/BCI$-algebras, as well as investigated relations between these sets. As a more general platform that extends the concepts of classic and fuzzy sets, intuitionistic fuzzy sets, and interval-valued (intuitionistic) fuzzy sets, the notion of NS theory has been developed by Smarandache (see [1,2]) and has been applied to various parts: pattern recognition, medical diagnosis, decision-making problems, and so on (see [3,4,5,6]). In this article, we have introduced the notions of neutrosophic permeable S-values, neutrosophic permeable I-values, $(\in ,$ $\in )$-neutrosophic ideals, neutrosophic anti-permeable S-values, and neutrosophic anti-permeable I-values, which are motivated by the idea of subalgebras (s-values) and ideals (I-values), and have investigated their properties. We have considered characterizations of $(\in ,$ $\in )$-neutrosophic ideals and have discussed conditions for the lower (upper) neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets to be S- and I-energetic. We have provided conditions for a triple $(\alpha ,\beta ,\gamma )$ of numbers to be a neutrosophic (anti-)permeable S- or I-value, and have considered conditions for the upper (lower) neutrosophic ${\in }_{\mathsf{\Phi }}$-subsets to be right stable (right vanished) subsets. We have established relations between neutrosophic (anti-)permeable S- and I-values.