A p-Ideal in BCI-Algebras Based on Multipolar Intuitionistic Fuzzy Sets

Abstract

In 2020, Kang, Song and Jun introduced the notion of multipolar intuitionistic fuzzy set with finite degree, which is a generalization of intuitionistic fuzzy set, and they applied it to BCK/BCI-algebras. In this paper, we used this notion to study p-ideals of BCI-algebras. The notion of k-polar intuitionistic fuzzy p-ideals in BCI-algebras is introduced, and several properties were investigated. An example to illustrate the k-polar intuitionistic fuzzy p-ideal is given. The relationship between k-polar intuitionistic fuzzy ideal and k-polar intuitionistic fuzzy p-ideal is displayed. A k-polar intuitionistic fuzzy p-ideal is found to be k-polar intuitionistic fuzzy ideal, and an example to show that the converse is not true is provided. The notions of p-ideals and k-polar $(\in ,\in )$-fuzzy p-ideal in BCI-algebras are used to study the characterization of k-polar intuitionistic p-ideal. The concept of normal k-polar intuitionistic fuzzy p-ideal is introduced, and its characterization is discussed. The process of eliciting normal k-polar intuitionistic fuzzy p-ideal using k-polar intuitionistic fuzzy p-ideal is provided.

1. Introduction

BCI-algebras were introduced by Iséki [1] as the algebraic counterpart of the BCI-logic. BCI-algebras are a generalization of BCK-algebras, and they originated from two sources: set theory and propositional calculi. See the books [2,3] for more information on BCK/BCI-algebras. Fuzzy sets were first introduced by Zadeh [4], in which the membership degree is represented by only one function—the truth function. Intuitionistic fuzzy sets, which were introduced by Atanassov (see [5,6]), are a generalization of fuzzy sets. As an extension of the bipolar fuzzy set, Chen et al. [7] introduced an m-polar fuzzy set in 2014, and then this concept was applied to certain algebraic structures as BCK/BCI algebras, graph theory and decision making problem. For BCK/BCI-algebras, see [8,9,10], for graph theory, see [11,12,13,14] and see [15,16,17,18] for decision making problems. Al-Masarwah and Ahmad discussed the notion of m-polar fuzzy sets with applications in BCK/BCI-algebras. They introduced the notions of m-polar fuzzy subalgebras and m-polar fuzzy (closed, commutative) ideals and gave characterizations of m-polar fuzzy subalgebras and m-polar fuzzy (commutative) ideals. They considered relations between m-polar fuzzy subalgebras, m-polar fuzzy ideals and m-polar fuzzy commutative ideals (see [8]). Using the notion of multipolar fuzzy point, Mohseni Takallo et al. [9] studied p-ideals of BCI-algebras. In [19], Kang et al. introduced the notion of multipolar intuitionistic fuzzy set with finite degree as a generalization of intuitionistic fuzzy set, and applied it to BCK/BCI-algebras. They introduced the concepts of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra, and investigated their relations and characterizations. In a BCI-algebra, they considered the relationship between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal, and discussed the characterization of a closed k-polar intuitionistic fuzzy ideal. They consulted conditions for a k-polar intuitionistic fuzzy ideal to be a closed k-polar intuitionistic fuzzy ideal in a BCI-algebra. The aim of this manuscript was to use Kang et al.’s notion so called multipolar intuitionistic fuzzy set for studying p-ideal in BCI-algebras. This is a generalization of multipolar fuzzy p-ideals of BCI-algebras which is studied in [9]. We introduce the concept of k-polar intuitionistic fuzzy p-ideals in BCI-algebras, and then we study several properties. We first give an example to illustrate the k-polar intuitionistic fuzzy p-ideal. We consider the relationship between k-polar intuitionistic fuzzy ideal and k-polar intuitionistic fuzzy p-ideal. We first prove that every k-polar intuitionistic fuzzy p-ideal is a k-polar intuitionistic fuzzy ideal, and then give an example to show that the converse is not true in general. We use the notion of p-ideals in BCI-algebras to study the characterization of k-polar intuitionistic fuzzy p-ideal. We also use the notion of k-polar $(\in ,\in )$-fuzzy p-ideal in BCI-algebras to study the characterization of k-polar intuitionistic fuzzy p-ideal. We define the concept of normal k-polar intuitionistic fuzzy p-ideal, and discuss its characterization. We look at the process of eliciting normal k-polar intuitionistic fuzzy p-ideal from a given k-polar intuitionistic fuzzy p-ideal.

2. Preliminaries

If a set U has a special element 0 and a binary operation ∗ satisfying the conditions:

(I)

$(\forall \omega ,\upsilon ,\tau \in U)$$\left(\right((\omega \ast \upsilon )\ast (\omega \ast \tau ))\ast (\tau \ast \upsilon )=0),$

(II)

$(\forall \omega ,\upsilon \in U)$$\left(\right(\omega \ast (\omega \ast \upsilon ))\ast \upsilon =0),$

(III)

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

(IV)

$(\forall \omega ,\upsilon \in U)$$(\omega \ast \upsilon =0,\upsilon \ast \omega =0\Rightarrow \omega =\upsilon )$,

then it is said that U is a BCI-algebra. If a BCI-algebra U satisfies the following identity:

(V)

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

then U is called a $BCK$-algebra.

Any BCK/BCI-algebra U satisfies the following conditions:

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

(1)

$$\begin{array}{c}(\forall \omega ,\upsilon ,\tau \in U)\left((\omega \ast \upsilon )\ast \tau =(\omega \ast \tau )\ast \upsilon \right).\hfill \end{array}$$

(2)

A subset I of a BCI-algebra U is called

• a subalgebra of U if $\omega \ast \upsilon \in I$ for all $\omega ,\upsilon \in I.$
• an ideal of U if it satisfies:

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

  (3)

  $$\begin{array}{c}\hfill (\forall \omega \in U)\left(\forall \upsilon \in I\right)\left(\omega \ast \upsilon \in I\Rightarrow \omega \in I\right).\end{array}$$

  (4)
• a p-ideal of U (see [20]) if it satisfies Equation (3) and

  $$\begin{array}{c}\hfill (\forall \omega ,\upsilon ,\tau \in U)\left((\omega \ast \tau )\ast (\upsilon \ast \tau )\in I,\upsilon \in I\Rightarrow \omega \in I\right).\end{array}$$

  (5)

Let $\{b_i\mid i\in \mathsf{\Gamma }\}$ be a family of real numbers where $\mathsf{\Gamma }$ is any index set and we define

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

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

If $\mathsf{\Gamma }=\{1,2\}$, we will also use $b_1\vee b_2$ and $b_1\wedge b_2$ instead of $\bigvee \{b_i\mid i\in \mathsf{\Gamma }\}$ and $\bigwedge \{b_i\mid i\in \mathsf{\Gamma }\}$, respectively.

Let k be a natural number and ${[0,1]}^k$ denote the k-Cartesian product of $[0,1]$, that is,

$$\begin{array}{c}\hfill {[0,1]}^k=[0,1]\times [0,1]\times \cdots \times [0,1]\end{array}$$

in which $[0,1]$ is repeated k times. The order “≤” in ${[0,1]}^k$ is given by the pointwise order.

By a k-polar fuzzy set on a set U (see [7]), we mean a function $\widehat{\xi }:U\to {[0,1]}^k$ where k is a natural number. The membership value of every element $z\in U$ is denoted by

$$\begin{array}{c}\hfill \widehat{\xi }\left(z\right)=\left(({\mathrm{proj}}_1\circ \widehat{\xi })\left(z\right),({\mathrm{proj}}_2\circ \widehat{\xi })\left(z\right),\cdots ,({\mathrm{proj}}_k\circ \widehat{\xi })\left(z\right)\right),\end{array}$$

where ${\mathrm{proj}}_i:{[0,1]}^k\to [0,1]$ is the i-th projection for all $i=1,2,\cdots ,k$ and ∘ is the composition of functions.

A k-polar fuzzy set $\widehat{\xi }$ on a BCK/BCI-algebra U is called a k-polar fuzzy ideal of U (see [8]) if the following conditions are valid.

$$\begin{array}{cc}\hfill & (\forall z\in U)\left(\widehat{\xi }\left(0\right)\ge \widehat{\xi }\left(z\right)\right),\hfill \end{array}$$

(6)

$$\begin{array}{c}\hfill (\forall z,x\in U)\left(\widehat{\xi }\left(z\right)\ge \widehat{\xi }(z\ast x)\wedge \widehat{\xi }\left(x\right)\right).\end{array}$$

(7)

By a k-polar fuzzy point on a set U, we mean a k-polar fuzzy set $\widehat{\xi }$ on U of the form

$$\begin{array}{c}\hfill \widehat{\xi }\left(x\right)=\left\{\begin{array}{cc}\widehat{r}=(r_1,r_2,\cdots ,r_k)\in {(0,1]}^k\hfill & \mathrm{if}x=z,\hfill \\ \widehat{0}=(0,0,\cdots ,0)\hfill & \mathrm{if}x\ne z,\hfill \end{array}\right.\end{array}$$

(8)

and it is denoted by $z_{\widehat{r}}$ where z is a given element of U. We say that z is the support of $z_{\widehat{r}}$ and $\widehat{r}$ is the value of $z_{\widehat{r}}$.

We say that a k-polar fuzzy point $z_{\widehat{r}}$ is contained in a k-polar fuzzy set $\widehat{\xi }$, denoted by $z_{\widehat{r}}\in \widehat{\xi }$, if $\widehat{\xi }\left(z\right)\ge \widehat{r}$, that is, $({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)\ge r_i$ for all $i=1,2,\cdots ,k$.

A k-polar fuzzy set $\widehat{\xi }$ on a BCI-algebra U is called a k-polar $(\in ,\in )$-fuzzy p-ideal of U (see [9]) if it satisfies

$$\begin{array}{cc}\hfill & (\forall z\in U)(\forall \widehat{r}\in {[0,1]}^k)\left(z_{\widehat{r}}\in \widehat{\xi }\Rightarrow 0_{\widehat{r}}\in \widehat{\xi }\right),\hfill \end{array}$$

(9)

$$\begin{array}{c}\hfill (\forall z,x,y\in U)(\forall \widehat{r},\widehat{t}\in {[0,1]}^k)\left({((z\ast y)\ast (x\ast y))}_{\widehat{r}}\in \widehat{\xi },x_{\widehat{t}}\in \widehat{\xi }\Rightarrow z_{inf\{\widehat{r},\widehat{t}\}}\in \widehat{\xi }\right).\end{array}$$

(10)

It is easy to show that Condition (10) is equivalent to the following condition.

$$\begin{array}{c}\hfill (\forall z,x,y\in U)\left(\widehat{\xi }\left(z\right)\ge \widehat{\xi }((z\ast y)\ast (x\ast y))\wedge \widehat{\xi }\left(x\right)\right).\end{array}$$

(11)

A multipolar intuitionistic fuzzy set with finite degree k (briefly, k-pIF set) over a set U (see [19]) is a mapping

$$\begin{array}{c}\hfill (\widehat{\xi },\widehat{\varrho }):U\to {[0,1]}^k\times {[0,1]}^k,z\mapsto (\widehat{\xi }\left(z\right),\widehat{\varrho }\left(z\right))\end{array}$$

(12)

where $\widehat{\xi }:U\to {[0,1]}^k$ and $\widehat{\varrho }:U\to {[0,1]}^k$ are k-polar fuzzy sets over a set U such that $\widehat{\xi }\left(z\right)+\widehat{\varrho }\left(z\right)\le \widehat{1}$ for all $z\in U$, that is, $({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)+({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)\le 1$ for all $z\in U$ and $i=1,2,\cdots ,k$. We know that if the multipolar intuitionistic fuzzy set has degree 1, then it is an intuitionistic fuzzy set. So, the intuitionistic fuzzy set is a special case of the multipolar intuitionistic fuzzy set. From this point of view, multipolar intuitionistic fuzzy set is a generalization of intuitionistic fuzzy set.

Given a k-pIF set $(\widehat{\xi },\widehat{\varrho })$ over a set U, we consider the sets

$$\begin{array}{c}\hfill U(\widehat{\xi },\widehat{t}):=\{z\in U\mid \widehat{\xi }\left(z\right)\ge \widehat{t}\}\mathrm{and}L(\widehat{\varrho },\widehat{s}):=\{z\in U\mid \widehat{\varrho }\left(z\right)\le \widehat{s}\},\end{array}$$

(13)

where $\widehat{t}=(t_1,t_2,\cdots ,t_k)\in {[0,1]}^k$ and $\widehat{s}=(s_1,s_2,\cdots ,s_k)\in {[0,1]}^k$ with $\widehat{t}+\widehat{s}\le \widehat{1}$, which is called a k-polar upper (resp., lower) level set of $(\widehat{\xi },\widehat{\varrho })$ where "+" is the componentwise operation in ${[0,1]}^k$, that is, $t_i+s_i\le 1$ for all $i=1,2,\cdots ,k$. It is clear that $U(\widehat{\xi },\widehat{t})={\bigcap }_{i=1}^{k}U{(\widehat{\xi },\widehat{t})}^i$ and $L(\widehat{\varrho },\widehat{s})={\bigcap }_{i=1}^{k}L{(\widehat{\varrho },\widehat{s})}^i$ where

$$\begin{array}{c}\hfill U{(\widehat{\xi },\widehat{t})}^i=\{z\in U\mid ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)\ge t_i\}\mathrm{and}L{(\widehat{\varrho },\widehat{s})}^i=\{z\in U\mid ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)\le s_i\}.\end{array}$$

A k-pIF set $(\widehat{\xi },\widehat{\varrho })$ over U is called a k-polar intuitionistic fuzzy ideal (briefly, k-pIF ideal) of U (see [19]) if it satisfies the conditions

$$\begin{array}{c}\hfill (\forall z\in U)(\widehat{\xi }\left(0\right)\ge \widehat{\xi }\left(z\right),\widehat{\varrho }\left(0\right)\le \widehat{\varrho }\left(z\right)),\end{array}$$

(14)

that is, $({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)\ge ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)$ and $({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\le ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)$ for $i=1,2,\cdots ,k$. and

$$\begin{array}{c}\hfill (\forall z,x\in U)\left(\begin{array}{c}\widehat{\xi }\left(z\right)\ge \widehat{\xi }(z\ast x)\wedge \widehat{\xi }\left(x\right)\hfill \\ \widehat{\varrho }\left(z\right)\le \widehat{\varrho }(z\ast x)\vee \widehat{\varrho }\left(x\right)\hfill \end{array}\right).\end{array}$$

(15)

3. k-Polar Intuitionistic Fuzzy p-Ideals

In this section, let U be a BCI-algebra unless otherwise stated.

Definition 1.

A k-pIF set $(\widehat{\xi },\widehat{\varrho })$ over U is called a k-polar intuitionistic fuzzy p-ideal (briefly, k-pIF p-ideal) of U if it satisfies Condition (14) and

$$\begin{array}{c}\hfill (\forall z,x,y\in U)\left(\begin{array}{c}\widehat{\xi }\left(z\right)\ge \widehat{\xi }((z\ast x)\ast (y\ast x))\wedge \widehat{\xi }\left(y\right)\hfill \\ \widehat{\varrho }\left(z\right)\le \widehat{\varrho }((z\ast x)\ast (y\ast x))\vee \widehat{\varrho }\left(y\right)\hfill \end{array}\right).\end{array}$$

(16)

Example 1.

Let $U=\{0,x,a,b\}$ be a set with a binary operation ∗ which is given in Table 1.

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

∗	0	x	a	b
0	0	x	a	b
x	x	0	b	a
a	a	b	0	x
b	b	a	x	0

Then, U is a BCI-algebra (see [2]). Let $(\widehat{\xi },\widehat{\varrho })$ be a 4-polar intuitionistic fuzzy set over U given by

$$\begin{array}{c}(\widehat{\xi },\widehat{\varrho }):U\to {[0,1]}^4\times {[0,1]}^4,\hfill \\ z\mapsto \left\{\begin{array}{cc}\left((0.8,0.67,0.9,0.56),(0.19,0.15,0.07,0.28)\right)\hfill & \mathrm{if}z=0,\hfill \\ \left((0.7,0.57,0.7,0.56),(0.19,0.24,0.07,0.35)\right)\hfill & \mathrm{if}z=x,\hfill \\ \left((0.5,0.37,0.4,0.32),(0.37,0.44,0.39,0.58)\right)\hfill & \mathrm{if}z=a,\hfill \\ \left((0.5,0.37,0.4,0.32),(0.37,0.44,0.39,\mathrm{0.\; 58})\right)\hfill & \mathrm{if}z=b.\hfill \end{array}\right.\hfill \end{array}$$

It is routine to check that $(\widehat{\xi },\widehat{\varrho })$ is a 4-polar intuitionistic fuzzy p-ideal of U.

Theorem 1.

Let I be a subset of U and let $({\widehat{\xi }}_I,{\widehat{\varrho }}_I)$ be a k-pIF set on U defined by

$$\begin{array}{c}\hfill {\widehat{\xi }}_I:U\to {[0,1]}^k,z\mapsto \left\{\begin{array}{cc}\widehat{1}\hfill & \mathrm{if}z\in I,\hfill \\ \widehat{0}\hfill & \mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\widehat{\varrho }}_I:U\to {[0,1]}^k,z\mapsto \left\{\begin{array}{cc}\widehat{0}\hfill & \mathrm{if}z\in I,\hfill \\ \widehat{1}\hfill & \mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

Then, $({\widehat{\xi }}_I,{\widehat{\varrho }}_I)$ is a k-pIF ideal p-ideal of U if and only if I is a p-ideal of U.

Proof. 

Straightforward. □

In the following theorem, we look at the relationship between k-pIF ideal and k-pIF p-ideal.

Theorem 2.

Every k-pIF p-ideal is a k-pIF ideal.

Proof. 

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF p-ideal of U. If we put $x=0$ in (16) and use (1), then

$$\begin{array}{ll}({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)\hfill & \ge min\{({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast 0)\ast (x\ast 0)),({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)\}\hfill \\ & =min\{({\mathrm{proj}}_i\circ \widehat{\xi })(z\ast x),({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)\}\hfill \end{array}$$

and

$$\begin{array}{ll}\hfill ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)& \le max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast 0)\ast (x\ast 0)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\}\hfill \\ & =max\{({\mathrm{proj}}_i\circ \widehat{\varrho })(z\ast x),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\}\hfill \end{array}$$

for all $z,x\in U$. Therefore $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF ideal of U. □

In the following example, we find that the converse of Theorem 2 is not true.

Example 2.

Let $U=\{0,x,b,c,d\}$ be a set with a binary operation ∗, which is given in Table 2.

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

∗	0	x	b	c	d
0	0	0	d	c	b
x	x	0	d	c	b
b	b	b	0	d	c
c	c	c	b	0	d
d	d	d	c	b	0

Then, U is a BCI-algebra (see [2]). Define a 3-polar intuitionistic fuzzy set $(\widehat{\xi },\widehat{\varrho })$ on U as follows:

$$\begin{array}{c}(\widehat{\xi },\widehat{\varrho }):U\to {[0,1]}^3\times {[0,1]}^3,\hfill \\ z\mapsto \left\{\begin{array}{cc}\left((0.6,0.7,0.9),(0.2,0.25,0.07)\right)\hfill & \mathrm{if}z=0,\hfill \\ \left((0.6,0.5,0.7),(0.3,0.25,0.17)\right)\hfill & \mathrm{if}z=x,\hfill \\ \left((0.2,0.3,0.4),(0.6,0.45,0.27)\right)\hfill & \mathrm{if}z=b,\hfill \\ \left((0.5,0.4,0.6),(0.4,0.35,0.37)\right)\hfill & \mathrm{if}z=c,\hfill \\ \left((0.2,0.3,0.4),(0.6,0.45,0.27)\right)\hfill & \mathrm{if}z=d.\hfill \end{array}\right.\hfill \end{array}$$

It is easy to confirm that $(\widehat{\xi },\widehat{\varrho })$ is a 3-polar intuitionistic fuzzy ideal of U. But it is not a 3-polar intuitionistic fuzzy p-ideal of U since

$$({\mathrm{proj}}_2\circ \widehat{\xi })\left(x\right)=0.5<0.7=min\{({\mathrm{proj}}_2\circ \widehat{\xi })((x\ast b)\ast (0\ast b)),({\mathrm{proj}}_2\circ \widehat{\xi })\left(0\right)\}$$

and/or

$$({\mathrm{proj}}_3\circ \widehat{\varrho })\left(x\right)=0.17>0.07=max\{({\mathrm{proj}}_3\circ \widehat{\varrho })((x\ast b)\ast (0\ast b)),({\mathrm{proj}}_3\circ \widehat{\varrho })\left(0\right)\}.$$

Proposition 1.

Every k-pIF p-ideal $(\widehat{\xi },\widehat{\varrho })$ of U satisfies the following inequalities.

$$\begin{array}{c}\hfill (\forall z\in U)(\widehat{\xi }\left(z\right)\ge \widehat{\xi }(0\ast (0\ast z)),\widehat{\varrho }\left(z\right)\le \widehat{\varrho }(0\ast (0\ast z))).\end{array}$$

(17)

Proof. 

If we change y to z and x to 0 in Equation (16), then

$$\begin{array}{cc}\hfill ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)& \ge min\{({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast z)\ast (0\ast z)),({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)\}\hfill \\ & =min\{({\mathrm{proj}}_i\circ \widehat{\xi })(0\ast (0\ast z)),({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)\}\hfill \\ & =({\mathrm{proj}}_i\circ \widehat{\xi })(0\ast (0\ast z))\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)& \le max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast z)\ast (0\ast z)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\}\hfill \\ & =max\{({\mathrm{proj}}_i\circ \widehat{\varrho })(0\ast (0\ast z)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\}\hfill \\ & =({\mathrm{proj}}_i\circ \widehat{\varrho })(0\ast (0\ast z))\hfill \end{array}$$

for all $z\in U$. □

Proposition 2.

Every k-pIF p-ideal $(\widehat{\xi },\widehat{\varrho })$ of U satisfies the following inequalities.

$$\begin{array}{c}\hfill (\forall z,x,y\in U)\left(\begin{array}{c}\widehat{\xi }(z\ast x)\le \widehat{\xi }((z\ast y)\ast (x\ast y))\hfill \\ \widehat{\varrho }(z\ast x)\ge \widehat{\varrho }((z\ast y)\ast (x\ast y))\hfill \end{array}\right).\end{array}$$

(18)

Proof. 

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF p-ideal of U. Then, it is a k-pIF ideal of U by Theorem 2. For any $z,x,y\in U$, we have $\left(\right(z\ast y)\ast (x\ast y\left)\right)\ast (z\ast x)=0$. Hence

$$\begin{array}{c}({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y))\hfill \\ \ge min\{({\mathrm{proj}}_i\circ \widehat{\xi })(((z\ast y)\ast (x\ast y))\ast (z\ast x)),({\mathrm{proj}}_i\circ \widehat{\xi })(z\ast x)\}\hfill \\ =min\{({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right),({\mathrm{proj}}_i\circ \widehat{\xi })(z\ast x)\}=({\mathrm{proj}}_i\circ \widehat{\xi })(z\ast x)\hfill \end{array}$$

and

$$\begin{array}{c}({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y))\hfill \\ \le max\{({\mathrm{proj}}_i\circ \widehat{\varrho })(((z\ast y)\ast (x\ast y))\ast (z\ast x)),({\mathrm{proj}}_i\circ \widehat{\varrho })(z\ast x)\}\hfill \\ =max\{({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right),({\mathrm{proj}}_i\circ \widehat{\varrho })(z\ast x)\}=({\mathrm{proj}}_i\circ \widehat{\varrho })(z\ast x)\hfill \end{array}$$

for all $z,x,y\in U$. □

We provide conditions for a k-pIF ideal to be a k-pIF p-ideal.

Theorem 3.

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF ideal of U satisfying the condition

$$\begin{array}{c}\hfill (\forall z,x,y\in U)\left(\begin{array}{c}\widehat{\xi }(z\ast x)\ge \widehat{\xi }((z\ast y)\ast (x\ast y))\hfill \\ \widehat{\varrho }(z\ast x)\le \widehat{\varrho }((z\ast y)\ast (x\ast y))\hfill \end{array}\right).\end{array}$$

(19)

Then, it is a k-pIF p-ideal of U.

Proof. 

Using Equations (15) and (19), we have that

$$\begin{array}{c}\hfill \widehat{\xi }\left(z\right)\ge \widehat{\xi }(z\ast x)\wedge \widehat{\xi }\left(x\right)\ge \widehat{\xi }((z\ast y)\ast (x\ast y))\wedge \widehat{\xi }\left(x\right)\end{array}$$

and

$$\begin{array}{c}\hfill \widehat{\varrho }\left(z\right)\le \widehat{\varrho }(z\ast x)\vee \widehat{\varrho }\left(x\right)\le \widehat{\varrho }((z\ast y)\ast (x\ast y))\vee \widehat{\varrho }\left(x\right)\end{array}$$

for all $z,x,y\in U$. Therefore $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U. □

Lemma 1.

Every k-pIF ideal $(\widehat{\xi },\widehat{\varrho })$ of U satisfies the following inequalities.

$$\begin{array}{c}\hfill (\forall z\in U)(\widehat{\xi }\left(z\right)\le \widehat{\xi }(0\ast (0\ast z)),\widehat{\varrho }\left(z\right)\ge \widehat{\varrho }(0\ast (0\ast z))).\end{array}$$

(20)

Proof. 

For any $z,x\in U$, we obtain

$$\begin{array}{c}\hfill \widehat{\xi }(0\ast (0\ast z))\ge \widehat{\xi }((0\ast (0\ast z))\ast z)\wedge \widehat{\xi }\left(z\right)=\widehat{\xi }((0\ast z)\ast (0\ast z))\wedge \widehat{\xi }\left(z\right)=\widehat{\xi }\left(0\right)\wedge \widehat{\xi }\left(z\right)=\widehat{\xi }\left(z\right)\end{array}$$

and

$$\begin{array}{c}\hfill \widehat{\varrho }(0\ast (0\ast z))\le \widehat{\varrho }((0\ast (0\ast z))\ast z)\vee \widehat{\varrho }\left(z\right)=\widehat{\varrho }((0\ast z)\ast (0\ast z))\vee \widehat{\varrho }\left(z\right)=\widehat{\varrho }\left(0\right)\vee \widehat{\varrho }\left(z\right)=\widehat{\varrho }\left(z\right)\end{array}$$

by Equations (2), (3), (14) and (15). □

Theorem 4.

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF set over U. If $(\widehat{\xi },\widehat{\varrho })$ satisfies the following inequalities

$$\begin{array}{c}\hfill (\forall z\in U)(\widehat{\xi }\left(z\right)\ge \widehat{\xi }(0\ast (0\ast z)),\widehat{\varrho }\left(z\right)\le \widehat{\varrho }(0\ast (0\ast z))).\end{array}$$

(21)

Proof. 

For any $z,x,y\in U$ and $i=1,2,\cdots ,k$, we have

$$\begin{array}{cc}\hfill ({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y))& \le ({\mathrm{proj}}_i\circ \widehat{\xi })(0\ast (0\ast (z\ast y)\ast (x\ast y)))\hfill \\ & =({\mathrm{proj}}_i\circ \widehat{\xi })((0\ast x)\ast (0\ast y))\hfill \\ & =({\mathrm{proj}}_i\circ \widehat{\xi })(0\ast (0\ast (z\ast y)))\hfill \\ & \le ({\mathrm{proj}}_i\circ \widehat{\xi })(z\ast x),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y))& \ge ({\mathrm{proj}}_i\circ \widehat{\varrho })(0\ast (0\ast (z\ast y)\ast (x\ast y)))\hfill \\ & =({\mathrm{proj}}_i\circ \widehat{\varrho })((0\ast x)\ast (0\ast y))\hfill \\ & =({\mathrm{proj}}_i\circ \widehat{\varrho })(0\ast (0\ast (z\ast y)))\hfill \\ & \ge ({\mathrm{proj}}_i\circ \widehat{\varrho })(z\ast x),\hfill \end{array}$$

which imply that $\widehat{\xi }((z\ast y)\ast (x\ast y))\le \widehat{\xi }(z\ast x)$ and $\widehat{\varrho }((z\ast y)\ast (x\ast y))\ge \widehat{\varrho }(z\ast x)$ for all $z,x,y\in U$. Therefore $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U by Theorem 3. □

We consider characterizations of a k-pIF p-ideal.

Theorem 5.

Given a k-pIF set $(\widehat{\xi },\widehat{\varrho })$ over U, the following assertions are equivalent.

(i) 

$(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U.

(ii) 

The k-polar upper and lower level sets $U(\widehat{\xi },\widehat{r})$ and $L(\widehat{\varrho },\widehat{q})$ are p-ideals of U for all $(\widehat{r},\widehat{q})\in {[0,1]}^k\times {[0,1]}^k$ with $U(\widehat{\xi },\widehat{r})\ne \varnothing \ne L(\widehat{\varrho },\widehat{q})$.

Proof. 

Assume that $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U. It is clear that $0\in U(\widehat{\xi };\widehat{r})$ and $0\in L(\widehat{\varrho };\widehat{q})$ for any $\widehat{r}=(r_1,$$r_2,\cdots ,r_k{)\in (0,1]}^k$ and $\widehat{q}=(q_1,$$q_2,$$\cdots ,q_k{)\in (0,1]}^k$. Let $z,x,y,b,c,d\in U$ be such that $(z\ast y)\ast (x\ast y)\in U(\widehat{\xi };\widehat{r})$, $x\in U(\widehat{\xi };\widehat{r})$, $(b\ast d)\ast (c\ast d)\in L(\widehat{\varrho };\widehat{q})$ and $c\in L(\widehat{\varrho };\widehat{q})$. Then, $({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y))\ge r_i$, $({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)\ge r_i$, $({\mathrm{proj}}_i\circ \widehat{\varrho })((b\ast d)\ast (c\ast d))\le q_i$ and $({\mathrm{proj}}_i\circ \widehat{\varrho })\left(c\right)\le q_i$. It follows from Equations (16) that

$$\begin{array}{c}\hfill ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)\ge min\{({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)\}\ge r_i\end{array}$$

and

$$\begin{array}{c}\hfill ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(b\right)\le max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((b\ast d)\ast (c\ast d)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(c\right)\}\le q_i\end{array}$$

for $i=1,2,\cdots ,k$. Hence $z\in U(\widehat{\xi };\widehat{r})$ and $b\in L(\widehat{\varrho };\widehat{q})$ and therefore $U(\widehat{\xi };\widehat{r})$ and $L(\widehat{\varrho };\widehat{q})$ are p-ideals of U.

Conversely, suppose that the k-polar upper and lower level sets $U(\widehat{\xi },\widehat{r})$ and $L(\widehat{\varrho },\widehat{q})$ are p-ideals of U for all $(\widehat{r},\widehat{q})\in {[0,1]}^k\times {[0,1]}^k$ with $U(\widehat{\xi },\widehat{r})\ne \varnothing \ne L(\widehat{\varrho },\widehat{q})$. If $\widehat{\xi }\left(0\right)<\widehat{\xi }\left(b\right)$ for some $b\in U$, then $b\in U(\widehat{\xi };\widehat{r})$ and $0\notin U(\widehat{\xi };\widehat{r})$ where $\widehat{r}:=\widehat{\xi }\left(b\right)$. This is a contradiction, and so $\widehat{\xi }\left(0\right)\ge \widehat{\xi }\left(z\right)$ for all $z\in U$. If $\widehat{\varrho }\left(0\right)>\widehat{\varrho }\left(c\right)$ for some $c\in U$, then $({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)>({\mathrm{proj}}_i\circ \widehat{\varrho })\left(c\right)$ for $i=1,2,\cdots ,k$. If we take $q_i:=({\mathrm{proj}}_i\circ \widehat{\varrho })\left(c\right)$ for $i=1,2,\cdots ,k$, then $c\in L{(\widehat{\varrho },\widehat{q})}^i$ and $0\notin L{(\widehat{\varrho },\widehat{q})}^i$ for $i=1,2,\cdots ,k$. Thus $c\in {\bigcap }_{i=1}^{k}L{(\widehat{\varrho },\widehat{q})}^i=L(\widehat{\varrho },\widehat{q})$ and $0\notin L(\widehat{\varrho },\widehat{q})$, which is a contradiction; hence $\widehat{\varrho }\left(0\right)\le \widehat{\varrho }\left(z\right)$ for all $z\in U$. Now, suppose that there exist $b,c,d\in U$ such that $\widehat{\xi }\left(b\right)<\widehat{\xi }((b\ast d)\ast (c\ast d))\wedge \widehat{\xi }\left(c\right)$ or $\widehat{\varrho }\left(b\right)>\widehat{\varrho }((b\ast d)\ast (c\ast d))\vee \widehat{\varrho }\left(c\right)$. If we take

$$\widehat{r}:=\widehat{\xi }((b\ast d)\ast (c\ast d))\wedge \widehat{\xi }\left(c\right)$$

and

$$\widehat{q}:=\widehat{\varrho }((b\ast d)\ast (c\ast d))\vee \widehat{\varrho }\left(c\right),$$

then

$$(b\ast d)\ast (c\ast d)\in U(\widehat{\xi };\widehat{r})\mathrm{and}c\in U(\widehat{\xi };\widehat{r})$$

or

$$(b\ast d)\ast (c\ast d)\in L(\widehat{\varrho },\widehat{q})\mathrm{and}c\in L(\widehat{\varrho },\widehat{q}).$$

Since $U(\widehat{\xi };\widehat{r})$ and $L(\widehat{\varrho },\widehat{q})$ are p-ideals of U by assumption, it follows that $b\in U(\widehat{\xi };\widehat{r})$ or $b\in L(\widehat{\varrho };\widehat{q})$. Hence $\widehat{\xi }\left(b\right)\ge \widehat{r}=\widehat{\xi }((b\ast d)\ast (c\ast d))\wedge \widehat{\xi }\left(c\right)$ or $\widehat{\varrho }\left(b\right)\le \widehat{q}=\widehat{\varrho }((b\ast d)\ast (c\ast d))\vee \widehat{\varrho }\left(c\right)$, which is a contradiction. Thus $\widehat{\xi }\left(z\right)\ge \widehat{\xi }((z\ast y)\ast (x\ast y))\wedge \widehat{\xi }\left(x\right)$ and $\widehat{\varrho }\left(z\right)\le \widehat{\varrho }((z\ast y)\ast (x\ast y))\vee \widehat{\varrho }\left(x\right)$ for all $z,x,y\in U$; therefore $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U. □

Given a k-pIF set $(\widehat{\xi },\widehat{\varrho })$ over U and $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$, we consider the sets:

$$\begin{array}{c}\hfill R_{(\widehat{\xi },\widehat{t})}\left(U\right):=\{z\in U\mid \widehat{\xi }\left(z\right)+\widehat{t}>\widehat{1}\}\end{array}$$

and

$$\begin{array}{c}\hfill R_{(\widehat{\varrho },\widehat{s})}\left(U\right):=\{z\in U\mid \widehat{\varrho }\left(z\right)+\widehat{s}<\widehat{1}\}.\end{array}$$

Then, $R_{(\widehat{\xi },\widehat{t})}\left(U\right)={\bigcap }_{i=1}^k{R}_{(\widehat{\xi },\widehat{t})}{\left(U\right)}^i$ and $R_{(\widehat{\varrho },\widehat{s})}\left(U\right)={\bigcap }_{i=1}^k{R}_{(\widehat{\varrho },\widehat{s})}{\left(U\right)}^i$ where

$$\begin{array}{c}\hfill R_{(\widehat{\xi },\widehat{t})}{\left(U\right)}^i:=\{z\in U\mid ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)+t_i>1\}\end{array}$$

and

$$\begin{array}{c}\hfill R_{(\widehat{\varrho },\widehat{s})}{\left(U\right)}^i:=\{z\in U\mid ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)+s_i<1\}\end{array}$$

for $i=1,2,\cdots ,k$.

Theorem 6.

Given a k-pIF set $(\widehat{\xi },\widehat{\varrho })$ over U, the following assertions are equivalent.

(i) 

$(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U.

(ii) 

The sets $R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ and $R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$ are p-ideals of U for all $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$ with $R_{(\widehat{\xi },\widehat{t})}\left(U\right)\ne \varnothing \ne R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$.

Proof. 

Assume that $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U. It is clear that $0\in R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ and $0\in R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$. Let $z,x,y,b,c,d\in U$ be such that $(z\ast y)\ast (x\ast y)\in R_{(\widehat{\xi },\widehat{t})}\left(U\right)$, $x\in R_{(\widehat{\xi },\widehat{t})}\left(U\right)$, $(b\ast d)\ast (c\ast d)\in R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$ and $c\in R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$. Then, $\widehat{\xi }((z\ast y)\ast (x\ast y))+\widehat{t}>\widehat{1}$, $\widehat{\xi }\left(x\right)+\widehat{t}>\widehat{1}$, $\widehat{\varrho }((b\ast d)\ast (c\ast d))+\widehat{s}<\widehat{1}$ and $\widehat{\varrho }\left(c\right)+\widehat{s}<\widehat{1}$. It follows that

$$\begin{array}{cc}\hfill ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)+t_i& \ge min\{({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)\}+t_i\hfill \\ & =min\{({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y))+t_i,({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)+t_i\}>1\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(b\right)+s_i& \le max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((b\ast d)\ast (c\ast d)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(c\right)\}+s_i\hfill \\ & =max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((b\ast d)\ast (c\ast d))+s_i,({\mathrm{proj}}_i\circ \widehat{\varrho })\left(c\right)+s_i\}<1\hfill \end{array}$$

for all $i=1,2,\cdots ,k$. Hence $z\in {\bigcap }_{i=1}^k{R}_{(\widehat{\xi },\widehat{t})}{\left(U\right)}^i=R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ and $b\in {\bigcap }_{i=1}^k{R}_{(\widehat{\varrho },\widehat{s})}{\left(U\right)}^i=R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$; therefore $R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ and $R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$ are p-ideals of U for all $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$.

Conversely suppose that (ii) is valid. If $\widehat{\xi }\left(0\right)<\widehat{\xi }\left(z\right)$ or $\widehat{\varrho }\left(0\right)>\widehat{\varrho }\left(b\right)$ for some $z,b\in U$, then $\widehat{\xi }\left(0\right)+\widehat{t}\le \widehat{1}<\widehat{\xi }\left(z\right)+\widehat{t}$ or $\widehat{\varrho }\left(0\right)+\widehat{s}\ge \widehat{1}>\widehat{\varrho }\left(b\right)+\widehat{s}$ for some $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$. Thus $0\notin R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ or $0\notin R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$ which is a contradiction. Hence $(\widehat{\xi },\widehat{\varrho })$ satisfies Condition (14). Suppose that $\widehat{\xi }\left(b\right)<\widehat{\xi }((b\ast d)\ast (c\ast d))\wedge \widehat{\xi }\left(c\right)$ for some $b,c\in U$. Then, $\widehat{\xi }\left(b\right)+\widehat{t}\le \widehat{1}<(\widehat{\xi }((b\ast d)\ast (c\ast d))\wedge \widehat{\xi }\left(c\right))+\widehat{t}=(\widehat{\xi }((b\ast d)\ast (c\ast d))+\widehat{t})\wedge (\widehat{\xi }\left(c\right)+\widehat{t})$ for some $\widehat{t}\in {(0,1]}^k$. It follows that $(b\ast d)\ast (c\ast d)\in R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ and $c\in R_{(\widehat{\xi },\widehat{t})}\left(U\right)$, which implies that $b\in R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ since $R_{(\widehat{\xi },\widehat{t})}\left(U\right)$ is a p-ideal of U; hence $\widehat{\xi }\left(b\right)+\widehat{t}>\widehat{1}$, which is a contradiction. If $\widehat{\varrho }\left(z\right)>\widehat{\varrho }((z\ast y)\ast (x\ast y))\vee \widehat{\varrho }\left(x\right)$ for some $z,x\in U$, then

$$\widehat{\varrho }\left(z\right)+\widehat{s}\ge \widehat{1}>(\widehat{\xi }((z\ast y)\ast (x\ast y))\vee \widehat{\xi }\left(x\right))+\widehat{s}=(\widehat{\xi }((z\ast y)\ast (x\ast y))+\widehat{s})\vee (\widehat{\xi }\left(x\right)+\widehat{s})$$

for some $\widehat{s}\in {[0,1)}^k$. Thus $(z\ast y)\ast (x\ast y)\in R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$ and $x\in R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$. Since $R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$ is a p-ideal of U, it follows that $z\in R_{(\widehat{\varrho },\widehat{s})}\left(U\right)$, that is, $\widehat{\varrho }\left(z\right)+\widehat{s}<\widehat{1}$. This is a contradiction. This shows that $(\widehat{\xi },\widehat{\varrho })$ satisfies Condition (16); therefore $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U. □

The following theorem shows the characterization of k-pIF p-ideal using k-polar $(\in ,\in )$-fuzzy p-ideal.

Theorem 7.

A k-pIF set $(\widehat{\xi },\widehat{\varrho })$ over U is a k-pIF p-ideal of U if and only if $\widehat{\xi }$ and ${\widehat{\varrho }}^c$ are k-polar $(\in ,\in )$-fuzzy p-ideals of U where ${\widehat{\varrho }}^c=1-\widehat{\varrho }$, i.e., ${({\mathrm{proj}}_i\circ \widehat{\varrho })}^c=1-({\mathrm{proj}}_i\circ \widehat{\varrho })$ for $i=1,2,\cdots ,k$.

Proof. 

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF p-ideal of U. It is clear that $\widehat{\xi }$ is a k-polar $(\in ,\in )$-fuzzy p-ideal of U. Let $z,x,y\in U$. Then,

$${({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(0\right)=1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\ge 1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)={({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(z\right)$$

and

$$\begin{array}{cc}\hfill {({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(z\right)& =1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)\ge 1-max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\}\hfill \\ & =min\{1-({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y)),1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\}\hfill \\ & =min\{{({\mathrm{proj}}_i\circ \widehat{\varrho })}^c((z\ast y)\ast (x\ast y)),{({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(x\right)\}.\hfill \end{array}$$

Thus ${\widehat{\varrho }}^c$ is a k-polar $(\in ,\in )$-fuzzy p-ideal of U.

Conversely, suppose that $\widehat{\xi }$ and ${\widehat{\varrho }}^c$ are k-polar $(\in ,\in )$-fuzzy p-ideals of U. For any $z,x\in U$, we have $({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)\ge ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)$, $({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)\ge min\{({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)\}$, $1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)={({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(0\right)\ge {({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(z\right)=1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)$, i.e., $({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\le ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)$ and

$$\begin{array}{cc}\hfill 1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)& ={({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(z\right)\ge min\{{({\mathrm{proj}}_i\circ \widehat{\varrho })}^c((z\ast y)\ast (x\ast y)),{({\mathrm{proj}}_i\circ \widehat{\varrho })}^c\left(x\right)\}\hfill \\ & =min\{1-({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y)),1-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\}\hfill \\ & =1-max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\},\hfill \end{array}$$

that is, $({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)\le max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\}$; therefore $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U. □

The following corollary is an immediate consequence of Theorem 7.

Corollary 1.

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF set over U. Then, $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U if and only if the necessary operator $\square (\widehat{\xi },\widehat{\varrho })=(\widehat{\xi },{\widehat{\xi }}^c)$ and the possibility operator $\diamond (\widehat{\xi },\widehat{\varrho })=({\widehat{\varrho }}^c,\widehat{\varrho })$ of $(\widehat{\xi },\widehat{\varrho })$ are k-pIF p-ideals of U.

Definition 2.

A k-pIF p-ideal $(\widehat{\xi },\widehat{\varrho })$ of U is said to be normal if there exists $z,x\in U$ such that $\widehat{\xi }\left(z\right)=\widehat{1}$ and $\widehat{\varrho }\left(x\right)=\widehat{0}$.

Example 3.

Consider the BCI-algebra $U=\{0,x,a,b\}$, which is given in Example 1. Let $(\widehat{\xi },\widehat{\varrho })$ be a 3-polar intuitionistic fuzzy set over U given by

$$\begin{array}{c}(\widehat{\xi },\widehat{\varrho }):U\to {[0,1]}^3\times {[0,1]}^3,\hfill \\ z\mapsto \left\{\begin{array}{cc}\left((1.00,1.00,1.00),(0.00,0.00,0.00)\right)\hfill & \mathrm{if}z=0,\hfill \\ \left((0.72,0.57,1.00),(0.00,0.24,0.35)\right)\hfill & \mathrm{if}z=x,\hfill \\ \left((0.52,0.37,0.32),(0.37,0.44,0.58)\right)\hfill & \mathrm{if}z=a,\hfill \\ \left((0.52,0.37,0.32),(0.37,0.44,0.58)\right)\hfill & \mathrm{if}z=b.\hfill \end{array}\right.\hfill \end{array}$$

It is routine to check that $(\widehat{\xi },\widehat{\varrho })$ is a normal 3-polar intuitionistic fuzzy p-ideal of U.

It is clear that if a k-pIF p-ideal $(\widehat{\xi },\widehat{\varrho })$ of U is normal, then $\widehat{\xi }\left(0\right)=\widehat{1}$ and $\widehat{\varrho }\left(0\right)=\widehat{0}$, that is, $({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)=1$ and $({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)=0$ for all $i=1,2,\cdots ,k$.

Lemma 2.

A k-pIF p-ideal $(\widehat{\xi },\widehat{\varrho })$ of U is normal if and only if $\widehat{\xi }\left(0\right)=\widehat{1}$ and $\widehat{\varrho }\left(0\right)=\widehat{0}$.

Proof. 

Straightforward. □

In the following theorem we look at the process of eliciting normal k-pIF p-ideal from a given k-pIF p-ideal.

Theorem 8.

If $(\widehat{\xi },\widehat{\varrho })$ is k-pIF p-ideal of U, then the k-pIF set ${(\widehat{\xi },\widehat{\varrho })}^{+}=({\widehat{\xi }}^{+},{\widehat{\varrho }}^{+})$ on U defined by

$$\begin{array}{c}{\widehat{\xi }}^{+}:U\to {[0,1]}^k,z\mapsto \widehat{1}+\widehat{\xi }\left(z\right)-\widehat{\xi }\left(0\right),\hfill \\ {\widehat{\varrho }}^{+}:U\to {[0,1]}^k,z\mapsto \widehat{\varrho }\left(z\right)-\widehat{\varrho }\left(0\right)\hfill \end{array}$$

(22)

is a normal k-pIF p-ideal of U containing $(\widehat{\xi },\widehat{\varrho })$.

Proof. 

Assume that $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF p-ideal of U. Then, $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF ideal of U by Theorem 2. For any $z,x\in U$, we have

$$({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)=1+({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)-({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)=1\ge ({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right),$$

$$({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)=({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)=0\le ({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right),$$

$$\begin{array}{cc}{({\mathrm{proj}}_i\circ \widehat{\xi })}^{+}\left(z\right)\hfill & =1+({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)-({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)\hfill \\ & \ge 1+min\{({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)\}-({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)\hfill \\ & =min\{1+({\mathrm{proj}}_i\circ \widehat{\xi })((z\ast y)\ast (x\ast y))-({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right),1+({\mathrm{proj}}_i\circ \widehat{\xi })\left(x\right)-({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)\}\hfill \\ & =min\{{({\mathrm{proj}}_i\circ \widehat{\xi })}^{+}((z\ast y)\ast (x\ast y)),{({\mathrm{proj}}_i\circ \widehat{\xi })}^{+}\left(x\right)\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {({\mathrm{proj}}_i\circ \widehat{\varrho })}^{+}\left(z\right)& =({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\hfill \\ & \le max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y)),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)\}-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\hfill \\ & =max\{({\mathrm{proj}}_i\circ \widehat{\varrho })((z\ast y)\ast (x\ast y))-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right),({\mathrm{proj}}_i\circ \widehat{\varrho })\left(x\right)-({\mathrm{proj}}_i\circ \widehat{\varrho })\left(0\right)\}\hfill \\ & =max\{{({\mathrm{proj}}_i\circ \widehat{\varrho })}^{+}((z\ast y)\ast (x\ast y)),{({\mathrm{proj}}_i\circ \widehat{\varrho })}^{+}\left(x\right)\}\hfill \end{array}$$

for all for $i=1,2,\cdots ,k$. Hence ${(\widehat{\xi },\widehat{\varrho })}^{+}$ is a k-pIF p-ideal of U and it is normal by Lemma 2. It is clear that $(\widehat{\xi },\widehat{\varrho })$ is contained in ${(\widehat{\xi },\widehat{\varrho })}^{+}$. □

Theorem 9.

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF p-ideal of U. Then, $(\widehat{\xi },\widehat{\varrho })$ is normal if and only if ${(\widehat{\xi },\widehat{\varrho })}^{+}=(\widehat{\xi },\widehat{\varrho })$, that is, ${\widehat{\xi }}^{+}=\widehat{\xi }$ and ${\widehat{\varrho }}^{+}=\widehat{\varrho }$.

Proof. 

The sufficiency is clear. Assume that $(\widehat{\xi },\widehat{\varrho })$ is normal. Then,

$$\begin{array}{c}{({\mathrm{proj}}_i\circ \widehat{\xi })}^{+}\left(z\right)=1+({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)-({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)=({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)\hfill \\ {({\mathrm{proj}}_i\circ \widehat{\varrho })}^{+}\left(z\right)=({\mathrm{proj}}_i\circ \widehat{\varrho })\left(z\right)-({\mathrm{proj}}_i\circ \widehat{\xi })\left(0\right)=({\mathrm{proj}}_i\circ \widehat{\xi })\left(z\right)\hfill \end{array}$$

for all $z\in U$ by Lemma 2. This completes the proof. □

Corollary 2.

Let $(\widehat{\xi },\widehat{\varrho })$ be a k-pIF p-ideal of U. If $(\widehat{\xi },\widehat{\varrho })$ is normal, then ${\left({(\widehat{\xi },\widehat{\varrho })}^{+}\right)}^{+}=(\widehat{\xi },\widehat{\varrho })$.

Theorem 10.

Let $(\widehat{\xi },\widehat{\varrho })$ be a non-constant normal k-pIF p-ideal of U, which is maximal in the poset of normal k-pIF p-ideals under set inclusion. Then, $\widehat{\xi }$ and $\widehat{\varrho }$ have the values $\widehat{0}$ and $\widehat{1}$ only.

Proof. 

Since $(\widehat{\xi },\widehat{\varrho })$ is normal, we have $\widehat{\xi }\left(0\right)=\widehat{1}$ and $\widehat{\varrho }\left(0\right)=\widehat{0}$ by Lemma 2. Let $z,x\in U$ be such that $\widehat{\xi }\left(z\right)\ne \widehat{1}$ and $\widehat{\varrho }\left(x\right)\ne \widehat{0}$. It is sufficient to show that $\widehat{\xi }\left(z\right)=\widehat{0}$ and $\widehat{\varrho }\left(x\right)=\widehat{1}$. If $\widehat{\xi }\left(z\right)\ne \widehat{0}$ and $\widehat{\varrho }\left(x\right)\ne \widehat{1}$, then there exists $b,c\in U$ such that $\widehat{0}<\widehat{\xi }\left(b\right)<\widehat{1}$ and $\widehat{0}<\widehat{\varrho }\left(c\right)<\widehat{1}$. Let ${(\widehat{\xi },\widehat{\varrho })}_{\ast }=({\widehat{\xi }}_{\ast },{\widehat{\varrho }}_{\ast })$ be a k-pIF set on U given by

$$\begin{array}{c}\hfill {\widehat{\xi }}_{\ast }:U\to {[0,1]}^k,z\mapsto \frac{1}{2}\left(\widehat{\xi }\left(z\right)+\widehat{\xi }\left(b\right)\right).\end{array}$$

and

$$\begin{array}{c}\hfill {\widehat{\varrho }}_{\ast }:U\to {[0,1]}^k,z\mapsto \frac{1}{2}\left(\widehat{\varrho }\left(z\right)+\widehat{\varrho }\left(c\right)\right).\end{array}$$

It is clear that ${(\widehat{\xi },\widehat{\varrho })}_{\ast }$ is well-defined. For any $z,x\in U$, we have

$$\begin{array}{c}\hfill {\widehat{\xi }}_{\ast }\left(0\right)=\frac{1}{2}\left(\widehat{\xi }\left(0\right)+\widehat{\xi }\left(b\right)\right)=\frac{1}{2}\left(\widehat{1}+\widehat{\xi }\left(b\right)\right)\ge \frac{1}{2}\left(\widehat{\xi }\left(z\right)+\widehat{\xi }\left(b\right)\right)={\widehat{\xi }}_{\ast }\left(z\right),\end{array}$$

$$\begin{array}{c}\hfill {\widehat{\varrho }}_{\ast }\left(0\right)=\frac{1}{2}\left(\widehat{\varrho }\left(0\right)+\widehat{\varrho }\left(c\right)\right)=\frac{1}{2}\left(\widehat{0}+\widehat{\varrho }\left(c\right)\right)\le \frac{1}{2}\left(\widehat{\varrho }\left(z\right)+\widehat{\varrho }\left(c\right)\right)={\widehat{\varrho }}_{\ast }\left(z\right),\end{array}$$

$$\begin{array}{cc}\hfill {\widehat{\xi }}_{\ast }\left(z\right)& =\frac{1}{2}\left(\widehat{\xi }\left(z\right)+\widehat{\xi }\left(b\right)\right)\ge \frac{1}{2}\left((\widehat{\xi }(z\ast x)\wedge \widehat{\xi }\left(x\right))+\widehat{\xi }\left(b\right)\right)\hfill \\ & =\frac{1}{2}((\widehat{\xi }(z\ast x)+\widehat{\xi }\left(b\right))\wedge (\widehat{\xi }\left(x\right)+\widehat{\xi }\left(b\right)))\hfill \\ & =\frac{1}{2}(\widehat{\xi }(z\ast x)+\widehat{\xi }\left(b\right))\wedge \frac{1}{2}(\widehat{\xi }\left(x\right)+\widehat{\xi }\left(b\right))\hfill \\ & ={\widehat{\xi }}_{\ast }(z\ast x)\wedge {\widehat{\xi }}_{\ast }\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\widehat{\varrho }}_{\ast }\left(z\right)& =\frac{1}{2}\left(\widehat{\varrho }\left(z\right)+\widehat{\varrho }\left(c\right)\right)\le \frac{1}{2}\left((\widehat{\varrho }(z\ast x)\vee \widehat{\varrho }\left(x\right))+\widehat{\varrho }\left(c\right)\right)\hfill \\ & =\frac{1}{2}((\widehat{\varrho }(z\ast x)+\widehat{\varrho }\left(c\right))\vee (\widehat{\varrho }\left(x\right)+\widehat{\varrho }\left(c\right)))\hfill \\ & =\frac{1}{2}(\widehat{\varrho }(z\ast x)+\widehat{\varrho }\left(c\right))\vee \frac{1}{2}(\widehat{\varrho }\left(x\right)+\widehat{\varrho }\left(c\right))\hfill \\ & ={\widehat{\varrho }}_{\ast }(z\ast x)\vee {\widehat{\varrho }}_{\ast }\left(x\right).\hfill \end{array}$$

Hence $(\widehat{\xi },\widehat{\varrho })$ is a k-pIF ideal of U. We have

$$\begin{array}{c}\hfill {\widehat{\xi }}_{\ast }\left(z\right)=\frac{1}{2}\left(\widehat{\xi }\left(z\right)+\widehat{\xi }\left(b\right)\right)\ge \frac{1}{2}\left(\widehat{\xi }(0\ast (0\ast z))+\widehat{\xi }\left(b\right)\right)={\widehat{\xi }}_{\ast }(0\ast (0\ast z))\end{array}$$

and

$$\begin{array}{c}\hfill {\widehat{\varrho }}_{\ast }\left(z\right)=\frac{1}{2}\left(\widehat{\varrho }\left(z\right)+\widehat{\varrho }\left(c\right)\right)\le \frac{1}{2}\left(\widehat{\varrho }(0\ast (0\ast z))+\widehat{\varrho }\left(c\right)\right)={\widehat{\varrho }}_{\ast }(0\ast (0\ast z))\end{array}$$

for all $z\in U$. Hence ${(\widehat{\xi },\widehat{\varrho })}_{\ast }$ is a k-pIF p-ideal of U by Theorem 4. Now, we get

$$\begin{array}{c}\hfill {\widehat{\xi }}_{\ast }^{+}\left(z\right)=\widehat{1}+{\widehat{\xi }}_{\ast }\left(z\right)-{\widehat{\xi }}_{\ast }\left(0\right)=\widehat{1}+\frac{1}{2}\left(\widehat{\xi }\left(z\right)+\widehat{\xi }\left(b\right)\right)-\frac{1}{2}\left(\widehat{\xi }\left(0\right)+\widehat{\xi }\left(b\right)\right)=\frac{1}{2}\left(\widehat{1}+\widehat{\xi }\left(z\right)\right),\end{array}$$

and

$$\begin{array}{c}\hfill {\widehat{\varrho }}_{\ast }^{+}\left(z\right)={\widehat{\varrho }}_{\ast }\left(z\right)-{\widehat{\varrho }}_{\ast }\left(0\right)=\frac{1}{2}\left(\widehat{\varrho }\left(z\right)+\widehat{\varrho }\left(c\right)\right)-\frac{1}{2}\left(\widehat{\varrho }\left(0\right)+\widehat{\varrho }\left(c\right)\right)=\frac{1}{2}\widehat{\varrho }\left(z\right),\end{array}$$

and so ${\widehat{\xi }}_{\ast }^{+}\left(0\right)=\frac{1}{2}\left(\widehat{1}+\widehat{\xi }\left(0\right)\right)=\widehat{1}$ and ${\widehat{\varrho }}_{\ast }^{+}\left(z\right)=\frac{1}{2}\widehat{\varrho }\left(0\right)=\widehat{0}$. Hence ${(\widehat{\xi },\widehat{\varrho })}_{\ast }$ is normal. Note that

$${\widehat{\xi }}_{\ast }^{+}\left(0\right)=\widehat{1}>{\widehat{\xi }}_{\ast }^{+}\left(b\right)=\frac{1}{2}\left(\widehat{1}+\widehat{\xi }\left(b\right)\right)>\widehat{\xi }\left(b\right)$$

and

$${\widehat{\varrho }}_{\ast }^{+}\left(0\right)=\widehat{0}<{\widehat{\varrho }}_{\ast }^{+}\left(c\right)=\frac{1}{2}\left(\widehat{0}+\widehat{\varrho }\left(c\right)\right)<\widehat{\varrho }\left(c\right).$$

Hence ${(\widehat{\xi },\widehat{\varrho })}_{\ast }^{+}$ is non-constant and $(\widehat{\xi },\widehat{\varrho })$ is not maximal, which is a contradiction; therefore $\widehat{\xi }$ and $\widehat{\varrho }$ have the values $\widehat{0}$ and $\widehat{1}$ only. □

4. Conclusions and Future Works

As a generalization of intuitionistic fuzzy set, Kang et al. [19] introduced the notion of multipolar intuitionistic fuzzy set with finite degree, and then they applied the notion to BCK/BCI-algebras. In this manuscript, we used Kang et al.’s multipolar intuitionistic fuzzy set to study p-ideal in BCI-algebras. We introduced the notion of k-polar intuitionistic fuzzy p-ideals (see Definition 1) in BCI-algebras, and then we studied several properties (See Proposition 1, Proposition 2). We gave an example to illustrate the k-polar intuitionistic fuzzy p-ideal (see Example 1), and considered the relationship between k-polar intuitionistic fuzzy ideal and k-polar intuitionistic fuzzy p-ideal. We have shown that every k-polar intuitionistic fuzzy p-ideal is a k-polar intuitionistic fuzzy ideal (see Theorem 2), and then provided an example to show that the converse is not true in general (see Example 2). We used the notion of p-ideals in BCI-algebras to study the characterization of k-polar intuitionistic fuzzy p-ideal (see Theorem 1, Theorem 5 and Theorem 6), and also used the notion of k-polar $(\in ,\in )$-fuzzy p-ideal in BCI-algebras to study the characterization of k-polar intuitionistic fuzzy p-ideal (see Theorem 7). We defined the concept of normal k-polar intuitionistic fuzzy p-ideal (see Definition 2), and discussed its characterization (see Lemma 2 and Theorem 9). We looked at the process of eliciting normal k-polar intuitionistic fuzzy p-ideal from a given k-polar intuitionistic fuzzy p-ideal (see Theorem 8). Our goal in the future is to apply the ideas and results of this paper to other forms of ideals, filters, etc. in BCK/BCI-algebras. We will also apply the ideas and results of this paper to other algebraic structures, for example, MV-algebras, EQ-algebras, equality algebras, hoops, etc.