Neutrosophic Quadruple BCI-Positive Implicative Ideals

Abstract

By considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part $\left(a\right)$ and an unknown part $(bT,cI,dF)$ where $T,$ $I,$ F have their usual neutrosophic logic meanings and $a,b,c,d$ are real or complex numbers, Smarandache introduced the concept of neutrosophic quadruple numbers. Using the concept of neutrosophic quadruple numbers based on a set, Jun et al. constructed neutrosophic quadruple BCK/BCI-algebras and implicative neutrosophic quadruple BCK-algebras. The notion of a neutrosophic quadruple BCI-positive implicative ideal is introduced, and several properties are dealt with in this article. We establish the relationship between neutrosophic quadruple ideal and neutrosophic quadruple BCI-positive implicative ideal. Given nonempty subsets I and J of a BCI-algebra, conditions for the neutrosophic quadruple $(I,J)$-set to be a neutrosophic quadruple BCI-positive implicative ideal are provided.

1. Introduction

A BCK/BCI-algebra is a class of logical algebras introduced by K. Iséki (see [1,2]) and was extensively investigated by several researchers. Neutrosophic algebraic structures in BCK/BCI-algebras are discussed in the papers [3,4,5,6,7,8,9,10]. Smarandache introduced the notion of neutrosophic sets with wide applications in sciences (see [11,12,13]), which is a more general stage to extend the notions of classic set, (intuitionistic) fuzzy set and interval valued (intuitionistic) fuzzy set. Smarandache [14] introduced the concept of neutrosophic quadruple numbers by considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part $\left(a\right)$ and an unknown part $(bT,cI,dF)$, where $T,$ $I,$ F have their usual neutrosophic logic meanings and $a,b,c,d$ are real or complex numbers. Using the notion of neutrosophic quadruple numbers based on a set, Jun et al. [15] constructed neutrosophic quadruple BCK/BCI-algebras and implicative neutrosophic quadruple BCK-algebras (see also [16]).

In this paper, we introduce the notion of a neutrosophic quadruple BCI-positive implicative ideal, and investigate several properties. We consider relations between neutrosophic quadruple ideal and neutrosophic quadruple BCI-positive implicative ideal. Given nonempty subsets I and J of a BCI-algebra U, we provide conditions for the neutrosophic quadruple $(I,J)$-set to be a neutrosophic quadruple BCI-positive implicative ideal.

2. Preliminaries

A BCI-algebra is a set U with a special element 0 and a binary operation * that satisfies the following conditions:

(I)

$(\forall p,q,r\in U)$$\left(\right((p\ast q)\ast (p\ast r))\ast (r\ast q)=0),$

(II)

$(\forall p,q\in U)$$\left(\right(p\ast (p\ast q))\ast q=0),$

(III)

$(\forall p\in U)$$(p\ast p=0),$

(IV)

$(\forall p,q\in U)$$(p\ast q=0,q\ast p=0\Rightarrow p=q).$

If a BCI-algebra U satisfies the following identity:

(V)

$(\forall p\in U)$$(0\ast p=0),$

then U is called a BCK-algebra. In a BCK/BCI-algebra U, the following conditions are valid.

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

(1)

$$\begin{array}{c}(\forall p,q,r\in U)\left(p\le q\Rightarrow p\ast r\le q\ast r,r\ast q\le r\ast p\right),\hfill \end{array}$$

(2)

$$\begin{array}{c}(\forall p,q,r\in U)\left((p\ast q)\ast r=(p\ast r)\ast q\right),\hfill \end{array}$$

(3)

$$\begin{array}{c}(\forall p,q,r\in U)\left((p\ast r)\ast (q\ast r)\le p\ast q\right)\hfill \end{array}$$

(4)

where $p\le q$ if and only if $p\ast q=0.$

Any BCI-algebra U satisfies the following conditions (see [17]):

$$\begin{array}{cc}\hfill & (\forall p,q\in U)(p\ast (p\ast (p\ast q))=p\ast q),\hfill \end{array}$$

(5)

$$\begin{array}{c}(\forall p,q\in U)(0\ast (p\ast q)=(0\ast p)\ast (0\ast q\left)\right),\hfill \end{array}$$

(6)

$$\begin{array}{c}(\forall p,q\in U)(0\ast (0\ast (p\ast q))=(0\ast q)\ast (0\ast p\left)\right).\hfill \end{array}$$

(7)

By a subalgebra of a BCK/BCI-algebra U, we mean a nonempty subset S of U such that $p\ast q\in S$ for all $p,q\in S.$ We say that a subset G of a BCK/BCI-algebra U is an ideal of U if it satisfies:

$$\begin{array}{cc}\hfill & 0\in G,\hfill \end{array}$$

(8)

$$\begin{array}{c}(\forall p\in U)\left(\forall q\in G\right)\left(p\ast q\in G\Rightarrow p\in G\right).\hfill \end{array}$$

(9)

A subset G of a BCI-algebra U is called a BCI-positive implicative ideal of U (see [18,19]) if it satisfies (8) and

$$\begin{array}{c}\hfill (\forall p,q,r\in U)\left(\left(\right(p\ast r)\ast r)\ast (q\ast r)\in G,q\in G\Rightarrow p\ast r\in G\right),\end{array}$$

(10)

For further information regarding BCK/BCI-algebras and neutrosophic set theory, we refer the reader to the books [17,20] and the site [21] respectively. We will use neutrosophic quadruple numbers based on a set instead of real or complex numbers.

Let U be a set. A neutrosophic quadruple U-number is an ordered quadruple $(a,pT,qI,rF)$, where $a,p,q,r\in U$ and $T,$ $I,$ F have their usual neutrosophic logic meanings (see [15]).

The set of all neutrosophic quadruple U-numbers which is denoted by $\mathcal{N}\left(U\right)$, that is,

$$\begin{array}{c}\hfill \mathcal{N}\left(U\right):=\left\{\right(a,pT,qI,rF)\mid a,p,q,r\in U\},\end{array}$$

is called the neutrosophic quadruple set based on U. In particular, if U is a BCK/BCI-algebra, then a neutrosophic quadruple U-number is called a neutrosophic quadruple BCK/BCI-number and $\mathcal{N}\left(U\right)$ is called the neutrosophic quadruple BCK/BCI-set.

We define a binary operation ⊡ on the neutrosophic quadruple BCK/BCI-set $\mathcal{N}\left(U\right)$ by

$$\begin{array}{c}\hfill (a,pT,qI,rF)⊡(b,uT,vI,wF)=(a\ast b,(p\ast u)T,(q\ast v)I,(z\ast w\left)F\right)\end{array}$$

for all $(a,pT,qI,rF),$ $(b,uT,vI,wF)\in \mathcal{N}\left(U\right)$. Given $a_1,a_2,a_3,a_4\in U$, the neutrosophic quadruple BCK/BCI-number $(a_1,a_{2}T,a_{3}I,a_{4}F)$ is denoted by $\tilde{a}$, that is,

$$\begin{array}{c}\hfill \tilde{a}=(a_1,a_{2}T,a_{3}I,a_{4}F),\end{array}$$

and the neutrosophic quadruple BCK/BCI-number $(0,0T,0I,0F)$ is denoted by $\tilde{0}$, that is,

$$\begin{array}{c}\hfill \tilde{0}=(0,0T,0I,0F),\end{array}$$

which is called the zero neutrosophic quadruple BCK/BCI-number. Then $(\mathcal{N}\left(U\right);⊡,\tilde{0})$ is a BCK/BCI-algebra (see [15]), which is called neutrosophic quadruple BCK/BCI-algebra, and it is simply denoted by $\mathcal{N}\left(U\right)$.

We define an order relation “≪” and the equality “=” on the neutrosophic quadruple BCK/BCI-algebra $\mathcal{N}\left(U\right)$ as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\tilde{p}\ll \tilde{q}\iff p_i\le q_i\mathrm{for}i=1,2,3,4,\hfill \\ \tilde{p}=\tilde{q}\iff p_i=q_i\mathrm{for}i=1,2,3,4\hfill \end{array}\end{array}$$

for all $\tilde{p}=(p_1,p_{2}T,p_{3}I,p_{4}F),\tilde{q}=(q_1,q_{2}T,q_{3}I,q_{4}F)\in \mathcal{N}\left(U\right)$. It is easy to verify that “≪” is an equivalence relation on $\mathcal{N}\left(U\right)$.

Let U be a BCK/BCI-algebra. Given nonempty subsets I and J of U, consider the set

$$\begin{array}{c}\hfill \mathcal{N}(I,J):=\left\{\right(a,pT,qI,rF)\in \mathcal{N}(U)\mid a,p\in I\&q,r\in J\},\end{array}$$

which is called the neutrosophic quadruple $(I,J)$-set.

The neutrosophic quadruple $(I,J)$-set $\mathcal{N}(I,J)$ with $I=J$ is denoted by $\mathcal{N}\left(I\right)$, and it is called the neutrosophic quadruple I-set.

3. Neutrosophic Quadruple BCI-Positive Implicative Ideals

In what follows, let U and $\mathcal{N}\left(U\right)$ be a BCI-algebra and a neutrosophic quadruple BCI-algebra, respectively, unless otherwise specified.

Definition 1.

Given nonempty subsets I and J of U, if $\mathcal{N}(I,J)$ is a BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$, we say $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Example 1.

Consider a BCI-algebra $U=\{0,1,a\}$ with the binary operation *, which is given in Table 1.

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

*	0	1	a
0	0	0	a
1	1	0	a
a	a	a	0

Then the neutrosophic quadruple BCI-algebra $\mathcal{N}\left(U\right)$ has 81 elements. If we take $I=\{0,a\}$ and $J=\{0,a\}$, then

$$\begin{array}{c}\hfill \mathcal{N}(I,J)=\{\tilde{0},{\tilde{\beta }}_1,{\tilde{\beta }}_2,{\tilde{\beta }}_3,{\tilde{\beta }}_4,{\tilde{\beta }}_5,{\tilde{\beta }}_6,{\tilde{\beta }}_7,{\tilde{\beta }}_8,{\tilde{\beta }}_9,{\tilde{\beta }}_{10},{\tilde{\beta }}_{11},{\tilde{\beta }}_{12},{\tilde{\beta }}_{13},{\tilde{\beta }}_{14},{\tilde{\beta }}_{15}\}\end{array}$$

and it is routine to check that $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ where

$$\tilde{0}=(0,0T,0I,0F),{\tilde{\beta }}_1=(0,0T,0I,aF),{\tilde{\beta }}_2=(0,0T,aI,0F),{\tilde{\beta }}_3=(0,0T,aI,aF),$$

$${\tilde{\beta }}_4=(0,aT,0I,0F),{\tilde{\beta }}_5=(0,aT,0I,aF),{\tilde{\beta }}_6=(0,aT,aI,0F),{\tilde{\beta }}_7=(0,aT,aI,aF),$$

$${\tilde{\beta }}_8=(a,0T,0I,0F),{\tilde{\beta }}_9=(a,0T,0I,aF),{\tilde{\beta }}_{10}=(a,0T,aI,0F),{\tilde{\beta }}_{11}=(a,0T,aI,aF),$$

$${\tilde{\beta }}_{12}=(a,aT,0I,0F),{\tilde{\beta }}_{13}=(a,aT,0I,aF),{\tilde{\beta }}_{14}=(a,aT,aI,0F),{\tilde{\beta }}_{15}=(a,aT,aI,aF).$$

Proposition 1.

Given nonempty subsets I and J of U, the neutrosophic quadruple BCI-positive implicative ideal $\mathcal{N}(I,J)$ of $\mathcal{N}\left(U\right)$ satisfies the following assertions.

$$\begin{array}{cc}\hfill & (\forall \tilde{p},\tilde{q},\tilde{r}\in \mathcal{N}\left(U\right))(((\tilde{p}⊡\tilde{r})⊡\tilde{r})⊡(\tilde{q}⊡\tilde{r})\in \mathcal{N}(I,J)\Rightarrow (\tilde{p}⊡\tilde{q})⊡\tilde{r}\in \mathcal{N}(I,J)),\hfill \end{array}$$

(11)

$$\begin{array}{c}(\forall \tilde{p},\tilde{q}\in \mathcal{N}\left(U\right))(((\tilde{p}⊡\tilde{q})⊡\tilde{q})⊡(\tilde{0}⊡\tilde{q})\in \mathcal{N}(I,J)\Rightarrow \tilde{p}⊡\tilde{q}\in \mathcal{N}(I,J)).\hfill \end{array}$$

(12)

Proof. 

Let $\mathcal{N}(I,J)$ be a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ for any nonempty subsets I and J of U. Assume that $((\tilde{p}⊡\tilde{r})⊡\tilde{r})⊡(\tilde{q}⊡\tilde{r})\in \mathcal{N}(I,J)$ for all $\tilde{p},\tilde{q},\tilde{r}\in \mathcal{N}\left(U\right)$. Since

$$\begin{array}{cc}\hfill (((\tilde{p}⊡\tilde{q})⊡\tilde{r})⊡\tilde{r})⊡(\tilde{0}⊡\tilde{r})& =(((\tilde{p}⊡\tilde{r})⊡\tilde{r})⊡\tilde{q})⊡((\tilde{q}⊡\tilde{q})⊡\tilde{r})\hfill \\ & =(((\tilde{p}⊡\tilde{r})⊡\tilde{r})⊡\tilde{q})⊡((\tilde{q}⊡\tilde{r})⊡\tilde{q})\hfill \\ & \le ((\tilde{p}⊡\tilde{r})⊡\tilde{r})⊡(\tilde{q}⊡\tilde{r}),\hfill \end{array}$$

we have $(((\tilde{p}⊡\tilde{q})⊡\tilde{r})⊡\tilde{r})⊡(\tilde{0}⊡\tilde{r})\in \mathcal{N}(I,J)$. Since $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal, it follows that $(\tilde{p}⊡\tilde{q})⊡\tilde{r}\in \mathcal{N}(I,J))$. Hence (11) is valid. If we take $\tilde{q}=\tilde{0}$ and $\tilde{r}=\tilde{q}$ in (11), then we get (12). □

We consider relations between neutrosophic quadruple ideal and neutrosophic quadruple BCI-positive implicative ideal.

Theorem 1.

For any nonempty subsets I and J of U, if $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$, then it is a neutrosophic quadruple ideal of $\mathcal{N}\left(U\right)$.

Proof. 

Assume that $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$. Let $\tilde{p}=(p_1,$ $p_{2}T,$ $p_{3}I,$ $p_{4}F)$ and $\tilde{q}=(q_1,$ $q_{2}T,$ $q_{3}I,$ $q_{4}F)$ be elements of $\mathcal{N}\left(U\right)$ such that $\tilde{q}\in \mathcal{N}(I,J)$ and $\tilde{p}⊡\tilde{q}\in \mathcal{N}(I,J)$. Then

$$\begin{array}{c}\hfill ((\tilde{p}⊡\tilde{0})⊡\tilde{0})⊡(\tilde{q}⊡\tilde{0})=\tilde{p}⊡\tilde{q}\in \mathcal{N}(I,J),\end{array}$$

which implies that $\tilde{p}=\tilde{p}⊡\tilde{0}\in \mathcal{N}(I,J)$. Therefore $\mathcal{N}(I,J)$ is a neutrosophic quadruple ideal of $\mathcal{N}\left(U\right)$. □

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

Example 2.

Consider a BCI-algebra $U=\{0,1,a\}$ with the binary operation *, which is given in Table 2.

Table 2. Cayley table for the binary operation “*”.

*	0	1	a
0	0	0	0
1	1	0	0
a	a	1	0

Then the neutrosophic quadruple BCI-algebra $\mathcal{N}\left(U\right)$ has 81 elements. If we take $I=\left\{0\right\}$ and $J=\left\{0\right\}$, then $\mathcal{N}(I,J)=\left\{\tilde{0}\right\}$ is a neutrosophic quadruple ideal of $\mathcal{N}\left(U\right)$. But it is not a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ since

$$\begin{array}{c}\hfill (((a,aT,aI,aF)⊡(1,1T,1I,1F))⊡(1,1T,1I,1F))⊡(\tilde{0}⊡(1,1T,1I,1F))=\tilde{0}\in \mathcal{N}(I,J)\end{array}$$

and $(a,aT,aI,aF)⊡(1,1T,1I,1F)=(1,1T,1I,1F)\notin \mathcal{N}(I,J)$.

Given nonempty subsets I and J of U, we provide conditions for the set $\mathcal{N}(I,J)$ to be a neutrosophic quadruple BCI-positive implicative ideal.

Theorem 2.

If I and J are BCI-positive implicative ideal of U, then $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Proof. 

Assume that I and J are BCI-positive implicative ideal of U. Obviously $\tilde{0}\in \mathcal{N}(I,J)$. Let $\tilde{p}=(p_1,$ $p_{2}T,$ $p_{3}I,$ $p_{4}F)$, $\tilde{q}=(q_1,$ $q_{2}T,$ $q_{3}I,$ $q_{4}F)$ and $\tilde{r}=(r_1,$ $r_{2}T,$ $r_{3}I,$ $r_{4}F)$ be elements of $\mathcal{N}\left(U\right)$ such that $\tilde{q}\in \mathcal{N}(I,J)$ and $((\tilde{p}⊡\tilde{r})⊡\tilde{r})⊡(\tilde{q}⊡\tilde{r})\in \mathcal{N}(I,J)$. Then $q_i\in I$ and $q_j\in J$ for $i=1,2$ and $j=3,4$. Also

$$\begin{array}{cc}\hfill ((\tilde{p}⊡\tilde{r})⊡\tilde{r})⊡(\tilde{q}⊡\tilde{r})& =(((p_1,p_{2}T,p_{3}I,p_{4}F)⊡(r_1,r_{2}T,r_{3}I,r_{4}F))⊡(r_1,r_{2}T,r_{3}I,r_{4}F))\hfill \\ & ⊡((q_1,q_{2}T,q_{3}I,q_{4}F)⊡(r_1,r_{2}T,r_{3}I,r_{4}F))\hfill \\ & =((p_1\ast r_1,(p_2\ast r_2)T,(p_3\ast r_3)I,(p_4\ast r_4)F)⊡(r_1,r_{2}T,r_{3}I,r_{4}F))\hfill \\ & ⊡(q_1\ast r_1,(q_2\ast r_2)T,(q_3\ast r_3)I,(q_4\ast r_4)F)\hfill \\ & =\left(((p_1\ast r_1)\ast r_1,((p_2\ast r_2)\ast r_2)T,((p_3\ast r_3)\ast r_3)I,((p_4\ast r_4)\ast r_4)F)\right)\hfill \\ & ⊡(q_1\ast r_1,(q_2\ast r_2)T,(q_3\ast r_3)I,(q_4\ast r_4)F)\hfill \\ & =(((p_1\ast r_1)\ast r_1)\ast (q_1\ast r_1),(((p_2\ast r_2)\ast r_2)\ast (q_2\ast r_2))T,\hfill \\ & (((p_3\ast r_3)\ast r_3)\ast (q_3\ast r_3))I,(((p_4\ast r_4)\ast r_4)\ast (q_4\ast r_4))F)\hfill \\ & \in \mathcal{N}(I,J),\hfill \end{array}$$

and so $((p_i\ast r_i)\ast r_i)\ast (q_i\ast r_i)\in I$ and $((p_j\ast r_j)\ast r_j)\ast (q_j\ast r_j)\in J$ for $i=1,2$ and $j=3,4$. it follows from (10) that $p_i\ast r_i\in I$ and $p_j\ast r_j\in J$ for $i=1,2$ and $j=3,4$. Hence

$$\begin{array}{cc}\hfill \tilde{p}⊡\tilde{r}& =(p_1,p_{2}T,p_{3}I,p_{4}F)⊡(r_1,r_{2}T,r_{3}I,r_{4}F)\hfill \\ & =(p_1\ast r_1,(p_2\ast r_2)T,(p_3\ast r_3)I,(p_4\ast r_4)F)\in \mathcal{N}(I,J).\hfill \end{array}$$

Therefore $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$. □

Corollary 1.

If I is a BCI-positive implicative ideal of U, then $\mathcal{N}\left(I\right)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Theorem 3.

Let I and J be ideals of U which satisfies the following condition.

$$\begin{array}{c}\hfill (\forall p,q\in U)\left(\right((p\ast q)\ast q)\ast (0\ast q)\in I\cap J\Rightarrow p\ast q\in I\cap J).\end{array}$$

(13)

Then $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Proof. 

Obviously $\tilde{0}\in \mathcal{N}(I,J)$. Let $\tilde{p}=(p_1,$ $p_{2}T,$ $p_{3}I,$ $p_{4}F)$, $\tilde{q}=(q_1,$ $q_{2}T,$ $q_{3}I,$ $q_{4}F)$ and $\tilde{r}=(r_1,$ $r_{2}T,$ $r_{3}I,$ $r_{4}F)$ be elements of $\mathcal{N}\left(U\right)$ such that $\tilde{r}\in \mathcal{N}(I,J)$ and $((\tilde{p}⊡\tilde{q})⊡\tilde{q})⊡(\tilde{r}⊡\tilde{q})\in \mathcal{N}(I,J)$. Then $r_1,r_2\in I$, $r_3,r_4\in J$ and

$$\begin{array}{cc}\hfill ((\tilde{p}⊡\tilde{q})⊡\tilde{q})⊡(\tilde{r}⊡\tilde{q})& =(((p_1\ast q_1)\ast q_1)\ast (r_1\ast q_1),(((p_2\ast q_2)\ast q_2)\ast (r_2\ast q_2))T,\hfill \\ & (((p_3\ast q_3)\ast q_3)\ast (r_3\ast q_3))I,(((p_4\ast q_4)\ast q_4)\ast (r_4\ast q_4))F)\hfill \\ & \in \mathcal{N}(I,J),\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill ((p_i\ast q_i)\ast q_i)\ast (r_i\ast q_i)\in I\mathrm{and}((p_j\ast q_j)\ast q_j)\ast (r_j\ast q_j)\in J\end{array}$$

(14)

for $i=1,2$ and $j=3,4$. Note that

$$\begin{array}{c}(((p_k\ast q_k)\ast q_k)\ast (0\ast q_k))\ast (((p_k\ast q_k)\ast q_k)\ast (r_k\ast q_k))\hfill \\ \le (r_k\ast q_k)\ast (0\ast q_k)\le r_k\ast 0=r_k\hfill \end{array}$$

for $k=1,2,3,4$ by (I), (1) and (4). Since I and J are ideals of U, it follows that

$$\begin{array}{c}\hfill \begin{array}{c}(((p_i\ast q_i)\ast q_i)\ast (0\ast q_i))\ast (((p_i\ast q_i)\ast q_i)\ast (r_i\ast q_i))\in I,\hfill \\ (((p_j\ast q_j)\ast q_j)\ast (0\ast q_j))\ast (((p_j\ast q_j)\ast q_j)\ast (r_j\ast q_j))\in J\hfill \end{array}\end{array}$$

(15)

for i = 1, 2 and j = 3, 4. Combining (14) and (15), we get

$$((p_i\ast q_i)\ast q_i)\ast (0\ast q_i)\in I\mathrm{and}((p_j\ast q_j)\ast q_j)\ast (0\ast q_j)\in J$$

for i = 1, 2 and j = 3, 4. Using (13) implies that p_i ∗ q _i ∈ I and p_j ∗ q_j ∈ J for i = 1, 2 and j = 3, 4. Thus

$$\begin{array}{cc}\hfill \tilde{p}⊡\tilde{q}& =(p_1,p_{2}T,p_{3}I,p_{4}F)⊡(q_1,q_{2}T,q_{3}I,q_{4}F)\hfill \\ & =(p_1\ast q_1,(p_2\ast q_2)T,(p_3\ast q_3)I,(p_4\ast q_4)F)\in \mathcal{N}(I,J).\hfill \end{array}$$

Therefore $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$. □

Corollary 2.

Let I be an ideal of U which satisfies the following condition.

$$\begin{array}{c}\hfill (\forall p,q\in U)\left(\right((p\ast q)\ast q)\ast (0\ast q)\in I\Rightarrow p\ast q\in I).\end{array}$$

(16)

Then $\mathcal{N}\left(I\right)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Theorem 4.

Let I and J be ideals of U which satisfies the following condition.

$$\begin{array}{c}\hfill (\forall p,q,r\in U)\left(\right((p\ast r)\ast (q\ast r))\ast r\in I\cap J\Rightarrow (p\ast q)\ast r\in I\cap J).\end{array}$$

(17)

Then $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Proof. 

Suppose that $\left(\right(p\ast q)\ast q)\ast (0\ast q)\in I\cap J$ for all $p,q\in U$. Then $\left(\right(p\ast q)\ast (0\ast q\left)\right)\ast q=\left(\right(p\ast q)\ast q)\ast (0\ast q)\in I\cap J$, which implies from (17) and (1) that $p\ast q=(p\ast 0)\ast q\in I\cap J$. Therefore $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ by Theorem 3. □

Corollary 3.

Let I be an ideal of U which satisfies the following condition.

$$\begin{array}{c}\hfill (\forall p,q,r\in U)\left(\right((p\ast r)\ast (q\ast r))\ast r\in I\Rightarrow (p\ast q)\ast r\in I).\end{array}$$

(18)

Then $\mathcal{N}\left(I\right)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Theorem 5.

Let I and J be subsets of U such that

$$\begin{array}{cc}\hfill & 0\in I\cap J,\hfill \end{array}$$

(19)

$$\begin{array}{c}(\forall p,q,r\in U)\left(\right(\left(\right(p\ast q)\ast q)\ast (0\ast q))\ast r\in I\cap J,r\in I\cap J\Rightarrow p\ast q\in I\cap J).\hfill \end{array}$$

(20)

Then $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Proof. 

If we take $q=0$ in (20) and use (1) and (III), then

$$\begin{array}{c}\hfill (\forall p,r\in U)(p\ast r\in I\cap J,r\in I\cap J\Rightarrow p\in I\cap J).\end{array}$$

Hence I and J are ideals of U. Assume that $\left(\right(p\ast q)\ast q)\ast (0\ast q)\in I\cap J$ for all $p,q\in U$. Then

$$\begin{array}{c}\hfill \left(\right((p\ast q)\ast q)\ast (0\ast q\left)\right)\ast 0=\left(\right(p\ast q)\ast q)\ast (0\ast q)\in I\cap J,\end{array}$$

It follows from (19) and (20) that $p\ast q\in I\cap J$. Consequently, $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ by Theorem 3. □

Corollary 4.

Let I be a subset of U such that

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

(21)

$$\begin{array}{c}(\forall p,q,r\in U)\left(\right(\left(\right(p\ast q)\ast q)\ast (0\ast q))\ast r\in I,r\in I\Rightarrow p\ast q\in I).\hfill \end{array}$$

(22)

Then $\mathcal{N}\left(I\right)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Theorem 6.

Let I, J, G and H be ideals of U such that $G\subseteq I$ and $H\subseteq J$. If G and H are BCI-positive implicative ideals of U, then $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Proof. 

Let $p,q,r\in U$ be such that $\left(\right(p\ast q)\ast q)\ast (0\ast q)\in I\cap J$. Then

$$\begin{array}{c}\left(\right((p\ast (\left(\right(p\ast q)\ast q)\ast (0\ast q)\left)\right)\ast q)\ast q)\ast (0\ast q)\hfill \\ =\left(\right((p\ast q)\ast q)\ast (0\ast q\left)\right)\ast \left(\right((p\ast q)\ast q)\ast (0\ast q\left)\right)\hfill \\ =0\in G\cap H,\hfill \end{array}$$

and so $(p\ast q)\ast \left(\right((p\ast q)\ast q)\ast (0\ast q\left)\right)=(p\ast (\left(\right(p\ast q)\ast q)\ast (0\ast q)\left)\right)\ast q\in G\cap H\subseteq I\cap J$ since G and H are BCI-positive implicative ideals of U. Thus $p\ast q\in I\cap J$, and therefore $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ by Theorem 3. □

Corollary 5.

Let I and G be ideals of U such that $G\subseteq I$. If G is a BCI-positive implicative ideal of U, then $\mathcal{N}\left(I\right)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Theorem 7.

Let I, J, G and H be ideals of U such that $G\subseteq I$, $H\subseteq J$ and

$$\begin{array}{c}\hfill (\forall p,q\in U)\left(\right((p\ast q)\ast q)\ast (0\ast q)\in G\cap H\Rightarrow p\ast q\in G\cap H).\end{array}$$

(23)

Then $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Proof. 

Let $p,q,r\in U$ be such that $r\in G\cap H$ and $\left(\right(p\ast q)\ast q)\ast (r\ast q)\in G\cap H$. Since

$$\begin{array}{c}\hfill \left(\right((p\ast q)\ast q)\ast (0\ast q\left)\right)\ast \left(\right((p\ast q)\ast q)\ast (r\ast q\left)\right)\le (r\ast q)\ast (0\ast q)\le r\ast 0=r\in G\cap H,\end{array}$$

we have $\left(\right(p\ast q)\ast q)\ast (0\ast q)\in G\cap H$. It follows from (23) that $p\ast q\in G\cap H$. Hence G and H are BCI-positive implicative ideals of U, and therefore $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ by Theorem 6. □

Corollary 6.

Let I and G be ideals of U such that $G\subseteq I$ and

$$\begin{array}{c}\hfill (\forall p,q\in U)\left(\right((p\ast q)\ast q)\ast (0\ast q)\in G\Rightarrow p\ast q\in G).\end{array}$$

(24)

Then $\mathcal{N}\left(I\right)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Theorem 8.

Let I, J, G and H be ideals of U such that $G\subseteq I$, $H\subseteq J$ and

$$\begin{array}{c}\hfill (\forall p,q\in U)\left(\right((p\ast r)\ast (q\ast r))\ast r\in G\cap H\Rightarrow (p\ast q)\ast r\in G\cap H).\end{array}$$

(25)

Then $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

Proof. 

Let $p,q\in U$ be such that $\left(\right(p\ast q)\ast q)\ast (0\ast q)\in G\cap H$. Then

$$\begin{array}{c}\hfill \left(\right(p\ast q)\ast (0\ast q\left)\right)\ast q=\left(\right(p\ast q)\ast q)\ast (0\ast q)\in G\cap H.\end{array}$$

It follows from (25) and (1) that $p\ast q=(p\ast 0)\ast q\in G\cap H$. Hence $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ by Theorem 7. □

Proof. 

If we put $q=0$ and $r=q$ in (25), then we have the condition (23). Hence $\mathcal{N}(I,J)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$ by Theorem 7. □

Corollary 7.

Let I and G be ideals of U such that $G\subseteq I$ and

$$\begin{array}{c}\hfill (\forall p,q\in U)\left(\right((p\ast r)\ast (q\ast r))\ast r\in G\Rightarrow (p\ast q)\ast r\in G).\end{array}$$

(26)

Then $\mathcal{N}\left(I\right)$ is a neutrosophic quadruple BCI-positive implicative ideal of $\mathcal{N}\left(U\right)$.

4. Conclusions

By considering an entry (i.e., a number, an idea, an object, etc.) which is represented by a known part $\left(a\right)$ and an unknown part $(bT,cI,dF)$ where $T,$ $I,$ F have their usual neutrosophic logic meanings and $a,b,c,d$ are real or complex numbers, Smarandache have introduced the concept of neutrosophic quadruple numbers. Using the notion of neutrosophic quadruple numbers based on a set (instead of real or complex numbers), Jun et al. have constructed neutrosophic quadruple BCK/BCI-algebras and implicative neutrosophic quadruple BCK-algebras. In this manuscript, we have introduced the concept of a neutrosophic quadruple BCI-positive implicative ideal, and investigated several properties. We have discussed relations between neutrosophic quadruple ideal and neutrosophic quadruple BCI-positive implicative ideal. Given nonempty subsets I and J of a BCI-algebra U, we have provided conditions for the neutrosophic quadruple $(I,J)$-set to be a neutrosophic quadruple BCI-positive implicative ideal. In the forthcoming research and papers, we will continue these ideas and will define new notions in several algebraic structures.