Multipolar Intuitionistic Fuzzy Set with Finite Degree and Its Application in BCK/BCI-Algebras

Abstract

When events occur in everyday life, it is sometimes advantageous to approach them in two directions to find a solution for them. As a mathematical tool to handle these things, we can consider the intuitionistic fuzzy set. However, when events are complex and the key to a solution cannot be easily found, we feel the need to approach them for hours and from various directions. As mathematicians, we wish we had the mathematical tools that apply to these processes. If these mathematical tools were developed, we would be able to apply them to algebra, topology, graph theory, etc., from a close point of view, and we would be able to apply these research results to decision-making and/or coding theory, etc., from a distant point of view. In light of this view, the purpose of this study is to introduce the notion of a multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set), and to apply it to algebraic structure, in particular, a BCK/BCI-algebra. The notions of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra are introduced, and related properties are investigated. Relations between a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal are discussed. Characterizations of a k-polar intuitionistic fuzzy subalgebra/ideal are provided, and conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal are provided. In a BCI-algebra, relations between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal are discussed. A characterization of a closed k-polar intuitionistic fuzzy ideal is considered, and conditions for a k-polar intuitionistic fuzzy ideal to be closed are provided.

1. Introduction

In abstract algebra, an algebraic structure on a set F is a collection of finitary operations on F, and the set F with this structure is also called an algebra. BCK/BCI-algebra are algebraic structures, introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCK and BCI logics. In 1966, Iseki et al. [1,2,3] introduced BCK and BCI-algebras as algebraic structures which describe fragments of proposition calculus that include implication known as BCK and BCI logics. BCK/BCI-algebras have been applied in many areas, in particular BCK-algebra is applied to coding theory (see [4,5,6]). In the fuzzy set which is introduced by Zadeh [7], the membership degree is expressed by only one function, the so-called truth function. When there is an election to elect members of a city, the issue of voters choosing candidates is explained by a fuzzy set. However, by means of the fuzzy set we cannot consider the number that corresponds to the part of electorate who have not voted for the elected member. We also feel the need to consider voters who have not voted for elected members of the national assembly. This part also can be expressed using fuzzy sets. If we use fuzzy sets to express this part, we can show the part of the electorate who have not voted at all and the corresponding number. As such, we can think of an intuitionistic fuzzy set, which was introduced by Atanassove [8], as a good mathematical tool to represent both voters who have supported a candidate and those who have not. intuitionistic fuzzy set are a generalization of a fuzzy set, and use a membership function and nonmembership function. The membership (respectively, nonmembership) function represents the truth (respectively, false) part. An m-polar fuzzy set, an extension of a bipolar fuzzy set, was introduced by Chen et al. [9] in 2014. After that, this concept was applied to BCK/BCI-algebra (see [10,11,12]), graph theory (see [13,14,15,16]), and decision making problems (see [17,18,19,20]).

In this paper, we introduce the notion of multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set) and apply it to BCK/BCI-algebras. We introduce the concepts of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra, and investigate several properties. We investigate relations between a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal, and provide a characterization of a k-polar intuitionistic fuzzy subalgebra/ideal. We provide conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal. In a BCI-algebra, We consider the relationship between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal, and discuss the characterization of a closed k-polar intuitionistic fuzzy ideal. We consult conditions for a k-polar intuitionistic fuzzy ideal to be a closed k-polar intuitionistic fuzzy ideal in a BCI-algebra.

2. Preliminaries

In this section, we describe the known contents of BCK/BCI-algebra as necessary for the development of this paper.

If a set F has a special element 0 and a binary operation * satisfying the conditions:

(I)

$(\forall \tau ,\varsigma ,\varrho \in F)$$\left(\right((\tau \ast \varsigma )\ast (\tau \ast \varrho ))\ast (\varrho \ast \varsigma )=0),$

(II)

$(\forall \tau ,\varsigma \in F)$$\left(\right(\tau \ast (\tau \ast \varsigma ))\ast \varsigma =0),$

(III)

$(\forall \tau \in F)$$(\tau \ast \tau =0),$

(IV)

$(\forall \tau ,\varsigma \in F)$$(\tau \ast \varsigma =0,\varsigma \ast \tau =0\Rightarrow \tau =\varsigma )$,

then we say that F is a BCI-algebra. If a BCI-algebra F satisfies the following identity:

(V)

$(\forall \tau \in F)$$(0\ast \tau =0),$

then F is called a BCK-algebra. A BCI-algebra F is said to be p-semisimple if its p-radical is $\left\{0\right\}$.

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

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

(1)

$$\begin{array}{c}(\forall \tau ,\varsigma ,\varrho \in F)\left(\tau \le \varsigma \Rightarrow \tau \ast \varrho \le \varsigma \ast \varrho ,\varrho \ast \varsigma \le \varrho \ast \tau \right),\hfill \end{array}$$

(2)

$$\begin{array}{c}(\forall \tau ,\varsigma ,\varrho \in F)\left((\tau \ast \varsigma )\ast \varrho =(\tau \ast \varrho )\ast \varsigma \right)\hfill \end{array}$$

(3)

where $\tau \le \varsigma$ if and only if $\tau \ast \varsigma =0.$ In a BCI-algebra F, the following equalities are valid.

$$\begin{array}{cc}\hfill & (\forall \tau ,\varsigma \in F)\left(\tau \ast (\tau \ast (\tau \ast \varsigma \left)\right)=\tau \ast \varsigma \right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & (\forall \tau ,\varsigma \in F)\left(0\ast (\tau \ast \varsigma )=(0\ast \tau )\ast (0\ast \varsigma )\right).\hfill \end{array}$$

(5)

A subset S of a BCK/BCI-algebra F is called a subalgebra of F if $\tau \ast \varsigma \in S$ for all $\tau ,\varsigma \in S.$ A subset I of a BCK/BCI-algebra F is called an ideal of F if it satisfies:

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

(6)

$$\begin{array}{c}(\forall \tau \in F)\left(\forall \varsigma \in I\right)\left(\tau \ast \varsigma \in I\Rightarrow \tau \in I\right).\hfill \end{array}$$

(7)

See the books [21,22] for more information on BCK/BCI-algebras.

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

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

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

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

By a k-polar fuzzy set on a set F (see [9]), we mean a function $\widehat{\mathcal{E}}:F\to {[0,1]}^k$ where k is a natural number. The membership value of every element $w\in F$ is denoted by

$$\begin{array}{c}\hfill \widehat{\mathcal{E}}\left(w\right)=\left(({\pi }_1\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_2\circ \widehat{\mathcal{E}})\left(w\right),\cdots ,({\pi }_k\circ \widehat{\mathcal{E}})\left(w\right)\right),\end{array}$$

where ${\pi }_i:{[0,1]}^k\to [0,1]$ is the i-th projection for all $i=1,2,\cdots ,k$.

Given a k-polar fuzzy set on a set F, we consider the set

$$\begin{array}{c}\hfill U(\widehat{\mathcal{E}};\widehat{t}):=\{w\in F\mid \widehat{\mathcal{E}}\left(w\right)\ge \widehat{t}\},\end{array}$$

(8)

that is,

$$\begin{array}{c}\hfill U(\widehat{\mathcal{E}};\widehat{t}):=\{w\in F\mid ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge t_i,i=1,2,\cdots ,k\},\end{array}$$

(9)

which is called a k-polar level set of $\widehat{\mathcal{E}}$.

Definition 1.

([10] [Definition 3.1]) A k-polar fuzzy set $\widehat{\mathcal{E}}$ on a BCK/BCI-algebra F is called a k-polar fuzzy subalgebra of F if the following condition is valid.

$$\begin{array}{c}\hfill (\forall w,y\in F)\left(\widehat{\mathcal{E}}(w\ast y)\ge \widehat{\mathcal{E}}\left(w\right)\wedge \widehat{\mathcal{E}}\left(y\right)\right),\end{array}$$

(10)

that is,

$$\begin{array}{c}\hfill (\forall w,y\in F)\left(({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\right)\end{array}$$

(11)

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

Definition 2.

([10] [Definition 3.7]) A k-polar fuzzy set $\widehat{\mathcal{E}}$ on a BCK/BCI-algebra F is called a k-polar fuzzy ideal of F if the following conditions are valid.

$$\begin{array}{cc}\hfill & (\forall w\in F)\left(\widehat{\mathcal{E}}\left(0\right)\ge \widehat{\mathcal{E}}\left(w\right)\right),\hfill \end{array}$$

(12)

$$\begin{array}{c}\hfill (\forall w,y\in F)\left(\widehat{\mathcal{E}}\left(w\right)\ge \widehat{\mathcal{E}}(w\ast y)\wedge \widehat{\mathcal{E}}\left(y\right)\right),\end{array}$$

(13)

that is,

$$\begin{array}{cc}\hfill & (\forall w\in F)\left(({\pi }_i\circ \widehat{\mathcal{E}})\left(0\right)\ge ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\right),\hfill \end{array}$$

(14)

$$\begin{array}{c}\hfill (\forall w,y\in F)\left(({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\right)\end{array}$$

(15)

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

3. k-Polar Intuitionistic Fuzzy Subalgebras/Ideals

In this section, we consider the finite polarity of intuitionistic fuzzy set, and apply it to BCK/BCI-algebras. We introduce the notions of a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal of a BCK/BCI-algebra, and discuss their characterizations.

Definition 3.

A multipolar intuitionistic fuzzy set with finite degree k (briefly, k-polar intuitionistic fuzzy set) over a universe F is a mapping

$$\begin{array}{c}\hfill (\widehat{\mathcal{E}},\widehat{\mathcal{K}}):F\to {[0,1]}^k\times {[0,1]}^k,w\mapsto (\widehat{\mathcal{E}}\left(w\right),\widehat{\mathcal{K}}\left(w\right))\end{array}$$

(16)

where $\widehat{\mathcal{E}}:F\to {[0,1]}^k$ and $\widehat{\mathcal{K}}:F\to {[0,1]}^k$ are k-polar fuzzy sets over a universe F such that $\widehat{\mathcal{E}}\left(w\right)+\widehat{\mathcal{K}}\left(w\right)\le \widehat{1}$ for all $w\in F$, that is,

$$\begin{array}{c}\hfill ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)+({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\le 1\end{array}$$

for all $w\in F$ and $i=1,2,\cdots ,k$.

Given a k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over a universe F, we consider the sets

$$\begin{array}{c}\hfill U(\widehat{\mathcal{E}},\widehat{t}):=\{w\in F\mid \widehat{\mathcal{E}}\left(w\right)\ge \widehat{t}\}\mathrm{and}L(\widehat{\mathcal{K}},\widehat{s}):=\{w\in F\mid \widehat{\mathcal{K}}\left(w\right)\le \widehat{s}\},\end{array}$$

(17)

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 $\widehat{t}+\widehat{s}\le \widehat{1}$, that is,

$$\begin{array}{c}\hfill U(\widehat{\mathcal{E}},\widehat{t}):=\{w\in F\mid ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge t_i\mathrm{for}\mathrm{all}i=1,2,\cdots ,k\}\end{array}$$

and

$$\begin{array}{c}\hfill L(\widehat{\mathcal{K}},\widehat{s}):=\{w\in F\mid ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\le s_i\mathrm{for}\mathrm{all}i=1,2,\cdots ,k\}\end{array}$$

which is called a k-polar upper (resp., lower) level set of $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$. It is clear that $U(\widehat{\mathcal{E}},\widehat{t})={\bigcap }_{i=1}^{k}U{(\widehat{\mathcal{E}},\widehat{t})}^i$ and $L(\widehat{\mathcal{K}},\widehat{s})={\bigcap }_{i=1}^{k}L{(\widehat{\mathcal{K}},\widehat{s})}^i$ where

$$\begin{array}{c}\hfill U{(\widehat{\mathcal{E}},\widehat{t})}^i=\{w\in F\mid ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge t_i\}\mathrm{and}L{(\widehat{\mathcal{K}},\widehat{s})}^i=\{w\in F\mid ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\le s_i\}.\end{array}$$

If there is no specific mention from here, F stands for a BCK/BCI-algebra.

Definition 4.

A k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over F is called a k-polar intuitionistic fuzzy subalgebra of F if the following assertion is valid.

$$\begin{array}{c}\hfill (\forall w,y\in F)\left(\begin{array}{c}\widehat{\mathcal{E}}(w\ast y)\ge \widehat{\mathcal{E}}\left(w\right)\wedge \widehat{\mathcal{E}}\left(y\right)\hfill \\ \widehat{\mathcal{K}}(w\ast y)\le \widehat{\mathcal{K}}\left(w\right)\vee \widehat{\mathcal{K}}\left(y\right)\hfill \end{array}\right),\end{array}$$

(18)

that is,

$$({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}$$

and

$$({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}$$

for all $w,y\in F$ and $i=1,2,\cdots ,k$.

Example 1.

Let $F=\{0,a,b,c\}$ be a set with the binary operation “*” which is given in

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

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

.

Then $(F,\ast ,0)$ is a BCK-algebra (see [22]). Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy set over F given by

$$\begin{array}{c}(\widehat{\mathcal{E}},\widehat{\mathcal{K}}):F\to {[0,1]}^4\times {[0,1]}^4,\hfill \\ w\mapsto \left\{\begin{array}{cc}\left((0.6,0.2),(0.67,0.12),(0.77,0.21),(0.56,0.27)\right)\hfill & \mathrm{if}w=0,\hfill \\ \left((0.5,0.2),(0.23,0.41),(0.47,0.42),(0.32,0.35)\right)\hfill & \mathrm{if}w=a,\hfill \\ \left((0.3,0.4),(0.54,0.44),(0.56,0.39),(0.32,0.27)\right)\hfill & \mathrm{if}w=b,\hfill \\ \left((0.3,0.5),(0.43,0.32),(0.38,0.33),(0.56,0.35)\right)\hfill & \mathrm{if}w=c.\hfill \end{array}\right.\hfill \end{array}$$

It is routine to verify that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a four-polar intuitionistic fuzzy subalgebra of F.

We consider characterizations of a k-polar intuitionistic fuzzy subalgebra.

Theorem 1.

A k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over F is a k-polar intuitionistic fuzzy subalgebra of F if and only if $\widehat{\mathcal{E}}$ and ${\widehat{\mathcal{K}}}^c$ are k-polar fuzzy subalgebras of F where ${\widehat{\mathcal{K}}}^c=1-\widehat{\mathcal{K}}$, i.e., ${({\pi }_i\circ \widehat{\mathcal{K}})}^c=1-({\pi }_i\circ \widehat{\mathcal{K}})$.

Proof. 

Assume that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F. Then it is clear that $\widehat{\mathcal{E}}$ is a k-polar fuzzy subalgebra of F. For any $w,y\in F$, we have

$$\begin{array}{cc}{({\pi }_i\circ \widehat{\mathcal{K}})}^c(w\ast y)\hfill & =1-({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y)\ge 1-max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & =min\{1-({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right),1-({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & =min\{{({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(w\right),{({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(y\right)\}.\hfill \end{array}$$

Thus ${\widehat{\mathcal{K}}}^c$ is a k-polar fuzzy subalgebra of F.

Conversely, suppose that $\widehat{\mathcal{E}}$ and ${\widehat{\mathcal{K}}}^c$ are k-polar fuzzy subalgebras of F and let $w,y\in F$. Using (10), we have $\widehat{\mathcal{E}}(w\ast y)\ge \widehat{\mathcal{E}}\left(w\right)\wedge \widehat{\mathcal{E}}\left(y\right)$. Moreover, we get

$$\begin{array}{cc}1-({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y)\hfill & ={({\pi }_i\circ \widehat{\mathcal{K}})}^c(w\ast y)\ge min\{{({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(w\right),{({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(y\right)\}\hfill \\ & =min\{1-({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right),1-({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & =1-max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\},\hfill \end{array}$$

that is, $({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F. □

Theorem 2.

A k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over F is a k-polar intuitionistic fuzzy subalgebra of F if and only if its k-polar upper level set $U(\widehat{\mathcal{E}},\widehat{t})$ and k-polar lower level set $L(\widehat{\mathcal{K}},\widehat{s})$ are subalgebras of F for all $\widehat{s},\widehat{t}\in {[0,1]}^k$ with $\widehat{s}+\widehat{t}\le \widehat{1}$.

Proof. 

Assume that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F. Let $w,y\in U(\widehat{\mathcal{E}},\widehat{t})$ and $a,b\in L(\widehat{\mathcal{K}},\widehat{s})$. Then $({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge t_i$, $({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\ge t_i$, $({\pi }_i\circ \widehat{\mathcal{K}})\left(a\right)\le s_i$ and $({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)\le s_i$ for all $i=1,2,\cdots ,k$. It follows that

$$({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\ge t_i$$

and

$$({\pi }_i\circ \widehat{\mathcal{K}})(a\ast b)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(a\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)\}\le s_i$$

for all $i=1,2,\cdots ,k$. Hence $w\ast y\in {\bigcap }_{i=1}^{k}U{(\widehat{\mathcal{E}},\widehat{t})}^i=U(\widehat{\mathcal{E}},\widehat{t})$ and $a\ast b\in {\bigcap }_{i=1}^{k}l{(\widehat{\mathcal{K}},\widehat{s})}^i=L(\widehat{\mathcal{K}},\widehat{s})$. Therefore $U(\widehat{\mathcal{E}},\widehat{t})$ and $L(\widehat{\mathcal{K}},\widehat{s})$ are subalgebras of F.

Conversely let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy set over F such that its k-polar upper level set $U(\widehat{\mathcal{E}},\widehat{t})$ and k-polar lower level set $L(\widehat{\mathcal{K}},\widehat{s})$ are subalgebras of F for all $\widehat{s},\widehat{t}\in {[0,1]}^k$ with $\widehat{s}+\widehat{t}\le \widehat{1}$. Suppose that condition (18) is not valid. Then $({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)<min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}$ or $({\pi }_j\circ \widehat{\mathcal{K}})(a\ast b)>max\{({\pi }_j\circ \widehat{\mathcal{K}})\left(a\right),({\pi }_j\circ \widehat{\mathcal{K}})\left(b\right)\}$ for some $w,y,a,b\in F$ and $i,j\in [0,1]$. Then $w,y\in U{(\widehat{\mathcal{E}},\widehat{t})}^i$ and $w\ast y\notin U{(\widehat{\mathcal{E}},\widehat{t})}^i$ for $i=min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}$. Moreover, $a,b\in L{(\widehat{\mathcal{K}},\widehat{s})}^j$ and $a\ast b\notin L{(\widehat{\mathcal{K}},\widehat{s})}^j$ for $j=max\{({\pi }_j\circ \widehat{\mathcal{K}})\left(a\right),({\pi }_j\circ \widehat{\mathcal{K}})\left(b\right)\}$. This is a contradiction, and thus $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F. □

Proposition 1.

Every k-polar intuitionistic fuzzy subalgebra $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ of F satisfies the following assertion.

$$\begin{array}{c}\hfill (\forall w\in F)(\widehat{\mathcal{E}}\left(0\right)\ge \widehat{\mathcal{E}}\left(w\right),\widehat{\mathcal{K}}\left(0\right)\le \widehat{\mathcal{K}}\left(w\right)),\end{array}$$

(19)

that is, $({\pi }_i\circ \widehat{\mathcal{E}})\left(0\right)\ge ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)$ and $({\pi }_i\circ \widehat{\mathcal{K}})\left(0\right)\le ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)$ for $i=1,2,\cdots ,k$.

Proof. 

It is straightforward by putting $w=y$ in (18). □

Proposition 2.

In a BCI-algebra, every k-polar intuitionistic fuzzy subalgebra $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ of F satisfies the following assertion.

$$\begin{array}{c}\hfill (\forall w\in F)(\widehat{\mathcal{E}}(0\ast w)\ge \widehat{\mathcal{E}}\left(w\right),\widehat{\mathcal{K}}(0\ast w)\le \widehat{\mathcal{K}}\left(w\right)),\end{array}$$

(20)

that is, $({\pi }_i\circ \widehat{\mathcal{E}})(0\ast w)\ge ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)$ and $({\pi }_i\circ \widehat{\mathcal{K}})(0\ast w)\le ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)$ for $i=1,2,\cdots ,k$.

Proof. 

It is straightforward by using (19). □

Given a k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over F and $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$, we consider the sets:

$$\begin{array}{c}\hfill Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right):=\{w\in F\mid \widehat{\mathcal{E}}\left(w\right)+\widehat{t}>\widehat{1}\}\end{array}$$

and

$$\begin{array}{c}\hfill Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right):=\{w\in F\mid \widehat{\mathcal{K}}\left(w\right)+\widehat{s}<\widehat{1}\}.\end{array}$$

Then $Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)={\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{E}},\widehat{t})}{\left(F\right)}^i$ and $Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)={\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{K}},\widehat{s})}{\left(F\right)}^i$ where

$$\begin{array}{c}\hfill Q_{(\widehat{\mathcal{E}},\widehat{t})}{\left(F\right)}^i:=\{w\in F\mid ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)+t_i>1\}\end{array}$$

and

$$\begin{array}{c}\hfill Q_{(\widehat{\mathcal{K}},\widehat{s})}{\left(F\right)}^i:=\{w\in F\mid ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)+s_i<1\}\end{array}$$

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

Theorem 3.

If $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F, then the sets $Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ are subalgebras of F for all $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$.

Proof. 

Let $w,y\in Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)={\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{E}},\widehat{t})}{\left(F\right)}^i$ and $a,b\in Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)={\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{K}},\widehat{s})}{\left(F\right)}^i$. Then

$({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)+t_i>1$, $({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)+t_i>1$, $({\pi }_i\circ \widehat{\mathcal{K}})\left(a\right)+s_i<1$ and $({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)+s_i<1$.

It follows from (18) that

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)+t_i\hfill & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}+t_i\hfill \\ & =min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)+t_i,({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)+t_i\}>1\hfill \end{array}$$

and

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{K}})(a\ast b)+s_i\hfill & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(a\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)\}+s_i\hfill \\ & =max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(a\right)+s_i,({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)+s_i\}<1\hfill \end{array}$$

for all $i=1,2,\cdots ,k$. Hence $w\ast y\in {\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{E}},\widehat{t})}{\left(F\right)}^i$$=Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $a\ast b\in {\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{K}},\widehat{s})}{\left(F\right)}^i$$=Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$. Therefore $Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ are subalgebras of F for all $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$. □

Definition 5.

A k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over F is called a k-polar intuitionistic fuzzy ideal of F if it satisfies the condition (19) and

$$\begin{array}{c}\hfill (\forall w,y\in F)\left(\begin{array}{c}\widehat{\mathcal{E}}\left(w\right)\ge \widehat{\mathcal{E}}(w\ast y)\wedge \widehat{\mathcal{E}}\left(y\right)\hfill \\ \widehat{\mathcal{K}}\left(w\right)\le \widehat{\mathcal{K}}(w\ast y)\vee \widehat{\mathcal{K}}\left(y\right)\hfill \end{array}\right),\end{array}$$

(21)

that is,

$$({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}$$

and

$$({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}$$

for all $w,y\in F$ and $i=1,2,\cdots ,k$.

Example 2.

Let $F=\{0,a,b,c,d\}$ be a set with the binary operation “*” which is given in

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

*	0	a	b	c	d
0	0	0	0	0	0
a	a	0	a	0	a
b	b	b	0	b	0
c	c	c	c	0	c
d	d	d	d	d	0

.

Then $(F,\ast ,0)$ is a BCK-algebra (see [22]). Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a four-polar intuitionistic fuzzy set over F given by

$$\begin{array}{c}(\widehat{\mathcal{E}},\widehat{\mathcal{K}}):F\to {[0,1]}^4\times {[0,1]}^4,\hfill \\ w\mapsto \left\{\begin{array}{cc}\left((0.65,0.21),(0.59,0.32),(0.72,0.18),(0.61,0.27)\right)\hfill & \mathrm{if}w=0,\hfill \\ \left((0.56,0.42),(0.39,0.41),(0.72,0.23),(0.42,0.27)\right)\hfill & \mathrm{if}w=a,\hfill \\ \left((0.34,0.37),(0.49,0.50),(0.48,0.39),(0.61,0.33)\right)\hfill & \mathrm{if}w=b,\hfill \\ \left((0.47,0.38),(0.19,0.48),(0.53,0.46),(0.32,0.27)\right)\hfill & \mathrm{if}w=c,\hfill \\ \left((0.31,0.41),(0.29,0.37),(0.37,0.43),(0.52,0.41)\right)\hfill & \mathrm{if}w=d.\hfill \end{array}\right.\hfill \end{array}$$

It is routine to verify that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a four-polar intuitionistic fuzzy ideal of F.

Proposition 3.

If $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F, then $\widehat{\mathcal{E}}$ is order-reversing and $\widehat{\mathcal{K}}$ is order-preserving.

Proof. 

Straightforward. □

Proposition 4.

Every k-polar intuitionistic fuzzy ideal $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ of F satisfies the following assertion.

$$\begin{array}{c}\hfill (\forall w,y,u\in X)\left(\begin{array}{c}w\ast y\le u\Rightarrow \left\{\begin{array}{c}\widehat{\mathcal{E}}\left(w\right)\ge \widehat{\mathcal{E}}\left(y\right)\wedge \widehat{\mathcal{E}}\left(u\right)\hfill \\ \widehat{\mathcal{K}}\left(w\right)\le \widehat{\mathcal{K}}\left(y\right)\vee \widehat{\mathcal{K}}\left(u\right)\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$

(22)

that is,

$$({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(u\right)\}$$

and

$$({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(u\right)\}$$

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

Proof. 

For any $i=1,2,\cdots ,k$, let $w,y,u\in F$ be such that $w\ast y\le u$. Then $(w\ast y)\ast u=0$, and so

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\hfill & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\hfill \\ & \ge min\{min\{({\pi }_i\circ \widehat{\mathcal{E}})((w\ast y)\ast u),({\pi }_i\circ \widehat{\mathcal{E}})\left(u\right)\},({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\hfill \\ & =min\{min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(0\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(u\right)\},({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\hfill \\ & =min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(u\right)\}\hfill \end{array}$$

and

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\hfill & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & \le max\{max\{({\pi }_i\circ \widehat{\mathcal{E}})((w\ast y)\ast u),({\pi }_i\circ \widehat{\mathcal{E}})\left(u\right)\},({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\hfill \\ & =max\{max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(0\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(u\right)\},({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & =max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(u\right)\}.\hfill \end{array}$$

This completes the proof. □

We provide characterizations of k-polar intuitionistic fuzzy ideal.

Theorem 4.

For a k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over F, the following are equivalent.

(i)

$(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F.

(ii)

The sets $Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ are ideals of F for all $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$.

Proof. 

Assume that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F. It is clear that $0\in Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $0\in Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$. Let $w,y,a,b\in F$ be such that $w\ast y\in Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$, $y\in Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$, $a\ast b\in Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ and $b\in Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$. Then $\widehat{\mathcal{E}}(w\ast y)+\widehat{t}>\widehat{1}$, $\widehat{\mathcal{E}}\left(y\right)+\widehat{t}>\widehat{1}$, $\widehat{\mathcal{K}}(a\ast b)+\widehat{s}<\widehat{1}$ and $\widehat{\mathcal{K}}\left(b\right)+\widehat{s}<\widehat{1}$, that is, $({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)+t_i>1$, $({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)+t_i>1$. $({\pi }_i\circ \widehat{\mathcal{K}})(a\ast b)+s_i<1$ and $({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)+s_i<1$ for all $i=1,2,\cdots ,k$. It follows that

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)+t_i\hfill & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}+t_i\hfill \\ & =min\{({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)+t_i,({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)+t_i\}>1\hfill \end{array}$$

and

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{K}})\left(a\right)+s_i\hfill & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})(a\ast b),({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)\}+s_i\hfill \\ & =max\{({\pi }_i\circ \widehat{\mathcal{K}})(a\ast b)+s_i,({\pi }_i\circ \widehat{\mathcal{K}})\left(b\right)+s_i\}<1\hfill \end{array}$$

for all $i=1,2,\cdots ,k$. Hence $w\in {\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{E}},\widehat{t})}{\left(F\right)}^i=Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $a\in {\bigcap }_{i=1}^k{Q}_{(\widehat{\mathcal{K}},\widehat{s})}{\left(F\right)}^i=Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$. Therefore $Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ are ideals of F for all $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$.

Conversely suppose that (ii) is valid. If $\widehat{\mathcal{E}}\left(0\right)<\widehat{\mathcal{E}}\left(w\right)$ or $\widehat{\mathcal{K}}\left(0\right)>\widehat{\mathcal{K}}\left(a\right)$ for some $w,a\in F$, then $\widehat{\mathcal{E}}\left(0\right)+\widehat{t}\le \widehat{1}<\widehat{\mathcal{E}}\left(w\right)+\widehat{t}$ or $\widehat{\mathcal{K}}\left(0\right)+\widehat{s}\ge \widehat{1}>\widehat{\mathcal{K}}\left(a\right)+\widehat{s}$ for some $(\widehat{t},\widehat{s})\in {(0,1]}^k\times {[0,1)}^k$. Thus $0\notin Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ or $0\notin Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ which is a contradiction. Hence $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ satisfies the condition (19). Suppose that $\widehat{\mathcal{E}}\left(a\right)<\widehat{\mathcal{E}}(a\ast b)\wedge \widehat{\mathcal{E}}\left(b\right)$ for some $a,b\in F$. Then $\widehat{\mathcal{E}}\left(a\right)+\widehat{t}\le \widehat{1}<\widehat{\mathcal{E}}(a\ast b)\wedge \widehat{\mathcal{E}}\left(b\right)+\widehat{t}$ for some $\widehat{t}\in {(0,1]}^k$. It follows that $a\ast b\in Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ and $b\in Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ which implies that $a\in Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ since $Q_{(\widehat{\mathcal{E}},\widehat{t})}\left(F\right)$ is an ideal of F. Hence $\widehat{\mathcal{E}}\left(a\right)+\widehat{t}>\widehat{1}$ which is a contradiction. If $\widehat{\mathcal{K}}\left(w\right)>\widehat{\mathcal{K}}(w\ast y)\vee \widehat{\mathcal{K}}\left(y\right)$ for some $w,y\in F$, then $\widehat{\mathcal{K}}\left(w\right)+\widehat{s}\ge \widehat{1}>\widehat{\mathcal{E}}(w\ast y)\vee \widehat{\mathcal{E}}\left(y\right)+\widehat{s}$ for some $\widehat{s}\in {[0,1)}^k$. Thus $w\ast y\in Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ and $y\in Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$. Since $Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$ is an ideal of F, it follows that $w\in Q_{(\widehat{\mathcal{K}},\widehat{s})}\left(F\right)$, that is, $\widehat{\mathcal{K}}\left(w\right)+\widehat{s}<\widehat{1}$. This is a contradiction. This shows that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ satisfies the condition (21). Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F. □

Theorem 5.

A k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over F is a k-polar intuitionistic fuzzy ideal of F if and only if $\widehat{\mathcal{E}}$ and ${\widehat{\mathcal{K}}}^c$ are k-polar fuzzy ideals of F where ${\widehat{\mathcal{K}}}^c=1-\widehat{\mathcal{K}}$, i.e., ${({\pi }_i\circ \widehat{\mathcal{K}})}^c=1-({\pi }_i\circ \widehat{\mathcal{K}})$.

Proof. 

Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy ideal of F. It is clear that $\widehat{\mathcal{E}}$ is a k-polar fuzzy ideal of F. Let $w,y\in F$. Then

$${({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(0\right)=1-({\pi }_i\circ \widehat{\mathcal{K}})\left(0\right)\ge 1-({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)={({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(w\right)$$

and

$$\begin{array}{cc}{({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(w\right)\hfill & =1-({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\ge 1-max\{({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & =min\{1-({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),1-({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & =min\{{({\pi }_i\circ \widehat{\mathcal{K}})}^c(w\ast y),{({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(y\right)\}.\hfill \end{array}$$

Thus ${\widehat{\mathcal{K}}}^c$ is a k-polar fuzzy ideal of F.

Conversely, suppose that $\widehat{\mathcal{E}}$ and ${\widehat{\mathcal{K}}}^c$ are k-polar fuzzy ideals of F. For any $w,y\in F$, we have $({\pi }_i\circ \widehat{\mathcal{E}})\left(0\right)\ge ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)$, $({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}$, $1-({\pi }_i\circ \widehat{\mathcal{K}})\left(0\right)={({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(0\right)\ge {({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(w\right)=1-({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)$, i.e., $({\pi }_i\circ \widehat{\mathcal{K}})\left(0\right)\le ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)$ and

$$\begin{array}{cc}1-({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\hfill & ={({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(w\right)\ge min\{{({\pi }_i\circ \widehat{\mathcal{K}})}^c(w\ast y),{({\pi }_i\circ \widehat{\mathcal{K}})}^c\left(y\right)\}\hfill \\ & =min\{1-({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),1-({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & =1-max\{({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\},\hfill \end{array}$$

that is, $({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F. □

The following corollary is an immediate consequence of Theorem 5.

Corollary 1.

Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy set over F. Then $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F if and only if the necessary operator $\square (\widehat{\mathcal{E}},\widehat{\mathcal{K}})=(\widehat{\mathcal{E}},{\widehat{\mathcal{E}}}^c)$ and the possibility operator $\diamond (\widehat{\mathcal{E}},\widehat{\mathcal{K}})=({\widehat{\mathcal{K}}}^c,\widehat{\mathcal{K}})$ of $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ are k-polar intuitionistic fuzzy ideals of F.

4. Relations between a k-Polar Intuitionistic Fuzzy Subalgebra and a k-Polar Intuitionistic Fuzzy Ideal

In this section, We provide a relation between a k-polar intuitionistic fuzzy subalgebra and a k-polar intuitionistic fuzzy ideal. We find an example to show that a k-polar intuitionistic fuzzy subalgebra may not be a k-polar intuitionistic fuzzy ideal in a BCK-algebra, and then we provide conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra. We define a concept of a closed k-polar intuitionistic fuzzy ideal in a BCI-algebra, and investigate relation between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal. We find conditions for a k-polar intuitionistic fuzzy ideal to be closed.

Theorem 6.

In a BCK-algebra, every k-polar intuitionistic fuzzy ideal is a k-polar intuitionistic fuzzy subalgebra.

Proof. 

Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy ideal of a BCK-algebra F. Since $w\ast y\le w$ for all $w,y\in F$, we have $({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)\ge ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)$ and $({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y)\le ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)$ for $i=1,2,\cdots ,k$ by Proposition 3. It follows from (21) that

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)\hfill & \ge ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\hfill \\ & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\}\hfill \end{array}$$

and

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y)\hfill & \le ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \\ & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\}\hfill \end{array}$$

for all $w,y\in F$ and $i=1,2,\cdots ,k$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F. □

In the following example, we can see that the converse of Theorem 6 does not have a role.

Example 3.

The four-polar intuitionistic fuzzy subalgebra $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ in Example 1 is not a four-polar intuitionistic fuzzy ideal of F since

$$({\pi }_2\circ \widehat{\mathcal{E}})\left(a\right)=0.23<0.54=min\{({\pi }_2\circ \widehat{\mathcal{E}})(a\ast b),({\pi }_2\circ \widehat{\mathcal{E}})\left(b\right)\}.$$

We provide conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal.

Theorem 7.

If a k-polar intuitionistic fuzzy subalgebra $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ of F satisfies the condition (22), then $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F.

Proof. 

By Proposition 1, $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ satisfies the condition (19). Since $w\ast (w\ast y)\le y$ for all $w,y\in F$, it follows from (22) that $\widehat{\mathcal{E}}\left(w\right)\ge \widehat{\mathcal{E}}(w\ast y)\wedge \widehat{\mathcal{E}}\left(y\right)$ and $\widehat{\mathcal{K}}\left(w\right)\le \widehat{\mathcal{K}}(w\ast y)\vee \widehat{\mathcal{K}}\left(y\right)$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F. □

In general, the inverse of Theorem 6 is not established even in a BCI-algebra. We want to give a condition for the inverse of Theorem 6 to be established in a BCI-algebra.

Theorem 8.

In a p-semisimple BCI-algebra, every k-polar intuitionistic fuzzy subalgebra is a k-polar intuitionistic fuzzy ideal.

Proof. 

Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy subalgebra of a p-semisimple BCI-algebra F. Then it satisfies the condition (19) by Proposition 1. Let $w,y\in F$. Note that $(y\ast w)\ast (0\ast w)\le y$. Since F is p-semisimple, every element is minimal. Hence $(y\ast w)\ast (0\ast w)=y$, which implies from (18) and (19) that

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right)\hfill & =({\pi }_i\circ \widehat{\mathcal{E}})((y\ast w)\ast (0\ast w))\hfill \\ & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(y\ast w),({\pi }_i\circ \widehat{\mathcal{E}})(0\ast w)\}\hfill \\ & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})(y\ast w),min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(0\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\}\}\hfill \\ & =min\{({\pi }_i\circ \widehat{\mathcal{E}})(y\ast w),({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\}\hfill \end{array}$$

and

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right)\hfill & =({\pi }_i\circ \widehat{\mathcal{K}})((y\ast w)\ast (0\ast w))\hfill \\ & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})(y\ast w),({\pi }_i\circ \widehat{\mathcal{K}})(0\ast w)\}\hfill \\ & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})(y\ast w),max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(0\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\}\}\hfill \\ & =max\{({\pi }_i\circ \widehat{\mathcal{K}})(y\ast w),({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\}\hfill \end{array}$$

for all $i=1,2,\cdots ,k$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F. □

Corollary 2.

Let F be a BCI-algebra in which any one of the following conditions is valid.

(1)

$(\forall w,y,u,a\in F)$$\left(\right(w\ast y)\ast (u\ast a)=(w\ast u)\ast (y\ast a\left)\right)$.

(2)

$(\forall w,y\in F)$$(0\ast (y\ast w)=w\ast y)$.

(3)

$(\forall w,y,u\in F)$$\left(\right(w\ast y)\ast (w\ast u)=u\ast y)$.

(4)

$(\forall w,y\in F)$$(w\ast y=0\Rightarrow w=y)$.

(5)

$F=\{0\ast w\mid w\in F\}$.

(6)

F satisfies the left cancellation law.

Then every k-polar intuitionistic fuzzy subalgebra is a k-polar intuitionistic fuzzy ideal.

The following example shows that Theorem 6 does not hold in a BCI-algebra.

Example 4.

Let $(Y,\ast ,0)$ be a BCI-algebra and $(\mathbb{Z},-,0)$ be the adjoint BCI-algebra of the additive group $(\mathbb{Z},+)$ of integers. Consider the BCI-algebra $F=Y\times \mathbb{Z}$ which is the direct product of Y and $\mathbb{Z}$. If we put $A=Y\times \mathbb{N}$ where $\mathbb{N}$ is the set of all nonnegative integers, then A is an ideal of F. Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy set over F given by

$$\begin{array}{c}\hfill (\widehat{\mathcal{E}},\widehat{\mathcal{K}}):F\to {[0,1]}^k\times {[0,1]}^k,w\mapsto \left\{\begin{array}{cc}(\widehat{t},\widehat{s})\hfill & \mathrm{if}w\in A,\hfill \\ (\widehat{0},\widehat{1})\hfill & \mathrm{if}w\notin A\hfill \end{array}\right.\end{array}$$

(23)

where $\widehat{t}=(t_1,t_2,\cdots ,t_k)$, $\widehat{s}=(s_1,s_2,\cdots ,s_k)\in {(0,1)}^k$ such that $\widehat{t}+\widehat{s}\le \widehat{1}$. Then $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F. If we take $w_0=(0,0)\in A$ and $w_1=(0,1)\in A$, then $w_0\ast w_1=(0,-1)\notin A$. Hence $\widehat{\mathcal{E}}(w_0\ast w_1)=\widehat{0}<\widehat{t}=\widehat{\mathcal{E}}\left(w_0\right)\wedge \widehat{\mathcal{E}}\left(w_1\right)$ and $\widehat{\mathcal{K}}(w_0\ast w_1)=\widehat{1}>\widehat{s}=\widehat{\mathcal{K}}\left(w_0\right)\vee \widehat{\mathcal{K}}\left(w_1\right)$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is not a k-polar intuitionistic fuzzy subalgebra of F.

Definition 6.

A k-polar intuitionistic fuzzy set $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ over a BCI-algebra F is called a closed k-polar intuitionistic fuzzy ideal of F if it satisfies (21) and

$$\begin{array}{c}(\forall w\in F)(\widehat{\mathcal{E}}(0\ast w)\ge \widehat{\mathcal{E}}\left(w\right),\widehat{\mathcal{K}}(0\ast w)\le \widehat{\mathcal{K}}\left(w\right)),\hfill \end{array}$$

(24)

that is, $({\pi }_i\circ \widehat{\mathcal{E}})(0\ast w)\ge ({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)$ and $({\pi }_i\circ \widehat{\mathcal{K}})(0\ast w)\le ({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)$ for all $w\in F$ and $i=1,2,\cdots ,k$.

Example 5.

Let $F=\{0,1,2,a,b\}$ be a set with the binary operation “*” which is given in

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

*	0	1	2	a	b
0	0	0	0	a	a
1	1	0	1	b	a
2	2	2	0	a	a
a	a	a	a	0	0
b	b	a	b	1	0

.

Then $(F,\ast ,0)$ is a BCI-algebra (see [21]). Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a three-polar intuitionistic fuzzy set over F given by

$$\begin{array}{c}\hfill (\widehat{\mathcal{E}},\widehat{\mathcal{K}}):F\to {[0,1]}^3\times {[0,1]}^3,w\mapsto \left\{\begin{array}{cc}\left((0.63,0.24),(0.67,0.12),(0.77,0.21)\right)\hfill & \mathrm{if}w=0,\hfill \\ \left((0.53,0.44),(0.33,0.28),(0.47,0.21)\right)\hfill & \mathrm{if}w=1,\hfill \\ \left((0.33,0.32),(0.54,0.18),(0.77,0.23)\right)\hfill & \mathrm{if}w=2,\hfill \\ \left((0.23,0.42),(0.44,0.43),(0.56,0.36)\right)\hfill & \mathrm{if}w=a,\hfill \\ \left((0.23,0.44),(0.33,0.43),(0.47,0.36)\right)\hfill & \mathrm{if}w=b.\hfill \end{array}\right.\end{array}$$

It is routine to verify that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a closed three-polar intuitionistic fuzzy ideal of F.

Theorem 9.

In a BCI-algebra, every closed k-polar intuitionistic fuzzy ideal is a k-polar intuitionistic fuzzy ideal.

Proof. 

Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a closed k-polar intuitionistic fuzzy ideal of a BCI-algebra F. If we put $w=0$ and $y=w$ in (21) and use (24), then we get (19). Hence $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy ideal of F. □

The converse of Theorem 9 is not true as seen in the following example.

Example 6.

Let $F=\{2^n\mid n\in \mathbb{Z}\}$ be a set with a binary operation “÷” (usual division). Then F is a BCI-algebra. Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a two-polar intuitionistic fuzzy set over F given by

$$\begin{array}{c}\hfill (\widehat{\mathcal{E}},\widehat{\mathcal{K}}):F\to {[0,1]}^2\times {[0,1]}^2,w\mapsto \left\{\begin{array}{cc}\left((t_1,s_1),(t_2,s_2)\right)\hfill & \mathrm{if}n\ge 0,\hfill \\ \left((m_1,r_1),(m_2,r_2)\right)\hfill & \mathrm{if}n<0\hfill \end{array}\right.\end{array}$$

where $t_i>m_i$ and $s_i<r_i$ for $i=1,2$. Then $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a two-polar intuitionistic fuzzy ideal of F, but it is not closed since $({\pi }_1\circ \widehat{\mathcal{E}})(1÷2^3)=({\pi }_1\circ \widehat{\mathcal{E}})\left(2^{-3}\right)=m_1<t_1=({\pi }_1\circ \widehat{\mathcal{E}})\left(2^3\right)$ and/or $({\pi }_2\circ \widehat{\mathcal{K}})(1÷2^5)=({\pi }_2\circ \widehat{\mathcal{K}})\left(2^{-5}\right)=r_2>s_2=({\pi }_2\circ \widehat{\mathcal{K}})\left(2^5\right)$.

We provide conditions for a k-polar intuitionistic fuzzy ideal to be closed.

Theorem 10.

Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy ideal of a BCI-algebra F. Then $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is closed if and only if it is a k-polar intuitionistic fuzzy subalgebra of F.

Proof. 

Assume that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a closed k-polar intuitionistic fuzzy ideal of a BCI-algebra F. Using (21), (3), (III) and (24), we have

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{E}})(w\ast y)\hfill & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})((w\ast y)\ast w),({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\}\hfill \\ & =min\{({\pi }_i\circ \widehat{\mathcal{E}})((w\ast w)\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\}\hfill \\ & =min\{({\pi }_i\circ \widehat{\mathcal{E}})(0\ast y),({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\}\hfill \\ & \ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(y\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\}\hfill \end{array}$$

and

$$\begin{array}{cc}({\pi }_i\circ \widehat{\mathcal{K}})(w\ast y)\hfill & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})((w\ast y)\ast w),({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\}\hfill \\ & =max\{({\pi }_i\circ \widehat{\mathcal{K}})((w\ast w)\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\}\hfill \\ & =max\{({\pi }_i\circ \widehat{\mathcal{K}})(0\ast y),({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\}\hfill \\ & \le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(y\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\}\hfill \end{array}$$

for all $w,y\in F$ and $i=1,2,\cdots ,k$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F.

Conversely suppose that $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a k-polar intuitionistic fuzzy subalgebra of F. Then

$$\begin{array}{c}\hfill ({\pi }_i\circ \widehat{\mathcal{E}})(0\ast w)\ge min\{({\pi }_i\circ \widehat{\mathcal{E}})\left(0\right),({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\}=({\pi }_i\circ \widehat{\mathcal{E}})\left(w\right)\end{array}$$

and

$$\begin{array}{c}\hfill ({\pi }_i\circ \widehat{\mathcal{K}})(0\ast w)\le max\{({\pi }_i\circ \widehat{\mathcal{K}})\left(0\right),({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\}=({\pi }_i\circ \widehat{\mathcal{K}})\left(w\right)\end{array}$$

for all $w\in F$ and $i=1,2,\cdots ,k$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a closed k-polar intuitionistic fuzzy ideal of F. □

Let F be a BCI-algebra and $B\left(F\right):=\{w\in F\mid 0\le w\}$. For any $n\in \mathbb{N}$ and $w\in F$, we define $x^n$ by

$$\begin{array}{c}\hfill w^1=w\mathrm{and}{w}^{n+1}=w\ast (0\ast w^n).\end{array}$$

Given an element w of a BCI-algebra F, if $w^n\in B\left(F\right)$ for some $n\in \mathbb{N}$, then we say that w is of finite periodic (see [23]), and we denote its period $\left|w\right|$ by

$$\begin{array}{c}\hfill \left|w\right|=min\{n\in \mathbb{N}\mid w^n\in B\left(F\right)\}.\end{array}$$

Otherwise, w is of infinite period and denoted by $\left|w\right|=\infty$.

Theorem 11.

Let F be a BCI-algebra in which every emenent is of finite period. Then every k-polar intuitionistic fuzzy ideal is closed.

Proof. 

Let $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ be a k-polar intuitionistic fuzzy ideal of F. For any $w\in F$, let $\left|w\right|=n$. Then $w^n\in B\left(F\right)$. Using (19), we have

$$\begin{array}{cc}\widehat{\mathcal{E}}((0\ast w^{n-1})\ast w)\hfill & =\widehat{\mathcal{E}}((0\ast (0\ast (0\ast w^{n-1})))\ast w)\hfill \\ & =\widehat{\mathcal{E}}((0\ast w)\ast (0\ast (0\ast w^{n-1})))\hfill \\ & =\widehat{\mathcal{E}}(0\ast (w\ast (0\ast w^{n-1})))\hfill \\ & =\widehat{\mathcal{E}}(0\ast w^n)=\widehat{\mathcal{E}}\left(0\right)\ge \widehat{\mathcal{E}}\left(w\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\widehat{\mathcal{K}}((0\ast w^{n-1})\ast w)\hfill & =\widehat{\mathcal{K}}((0\ast (0\ast (0\ast w^{n-1})))\ast w)\hfill \\ & =\widehat{\mathcal{K}}((0\ast w)\ast (0\ast (0\ast w^{n-1})))\hfill \\ & =\widehat{\mathcal{K}}(0\ast (w\ast (0\ast w^{n-1})))\hfill \\ & =\widehat{\mathcal{K}}(0\ast w^n)=\widehat{\mathcal{K}}\left(0\right)\le \widehat{\mathcal{K}}\left(w\right).\hfill \end{array}$$

It follows from (21) that

$$\begin{array}{c}\widehat{\mathcal{E}}(0\ast w^{n-1})\ge \widehat{\mathcal{E}}((0\ast w^{n-1})\ast w)\wedge \widehat{\mathcal{E}}\left(w\right)=\widehat{\mathcal{E}}\left(w\right),\hfill \\ \widehat{\mathcal{K}}(0\ast w^{n-1})\le \widehat{\mathcal{K}}((0\ast w^{n-1})\ast w)\vee \widehat{\mathcal{K}}\left(w\right)=\widehat{\mathcal{K}}\left(w\right).\hfill \end{array}$$

(25)

Note that

$$\begin{array}{cc}(0\ast w^{n-2})\ast w\hfill & =(0\ast (0\ast (0\ast w^{n-2})))\ast w\hfill \\ & =(0\ast w)\ast (0\ast (0\ast w^{n-2}))\hfill \\ & =0\ast (w\ast (0\ast w^{n-2}))\hfill \\ & =0\ast w^{n-1},\hfill \end{array}$$

which implies from (25) that

$$\begin{array}{c}\widehat{\mathcal{E}}((0\ast w^{n-2})\ast w)=\widehat{\mathcal{E}}(0\ast w^{n-1})\ge \widehat{\mathcal{E}}\left(w\right),\hfill \\ \widehat{\mathcal{K}}((0\ast w^{n-2})\ast w)=\widehat{\mathcal{K}}(0\ast w^{n-1})\le \widehat{\mathcal{K}}\left(w\right).\hfill \end{array}$$

Using (21), we have

$$\begin{array}{c}\widehat{\mathcal{E}}(0\ast w^{n-2})\ge \widehat{\mathcal{E}}((0\ast w^{n-2})\ast w)\wedge \widehat{\mathcal{E}}\left(w\right)=\widehat{\mathcal{E}}\left(w\right),\hfill \\ \widehat{\mathcal{K}}(0\ast w^{n-2})\le \widehat{\mathcal{K}}((0\ast w^{n-2})\ast w)\vee \widehat{\mathcal{K}}\left(w\right)=\widehat{\mathcal{K}}\left(w\right).\hfill \end{array}$$

Continuing this process, we obtain $\widehat{\mathcal{E}}(0\ast w)\ge \widehat{\mathcal{E}}\left(w\right)$ and $\widehat{\mathcal{K}}(0\ast w)\le \widehat{\mathcal{K}}\left(w\right)$ for all $w\in F$. Therefore $(\widehat{\mathcal{E}},\widehat{\mathcal{K}})$ is a closed k-polar intuitionistic fuzzy ideal of F. □

5. Conclusions

The intuitionistic fuzzy set, which was introduced by Atanassove, is a generalization of the fuzzy set, that is, Atanassov’s intuitionistic fuzzy sets are an extension of Zadeh’s fuzzy sets. The intuitionistic fuzzy set is a very useful mathematical tool for accessing an event in two directions simultaneously. The bipolar fuzzy set is a mathematical tool that can be useful for accessing a single event simultaneously on both the positive and negative aspects. By converting a negative approach to a positive approach in the bipolar fuzzy set and generalizing it, Chen et al. introduced an m-polar fuzzy set in 2014. Since then, this concept has been applied to graph theory, decision making problem, BCK/BCI-algebra, and other domains. In our lives, events that are difficult to solve even though we approach them in two directions at the same time are occurring in many ways. To solve this, we need a more diverse approach. In light of the fact that mathematics is a basic science that can be useful in several sciences, it is necessary to develop mathematical tools to address everyday difficulties that may arise in any case. Based on this need, we have introduced the multipolar intuitionistic fuzzy set with finite degree (briefly, k-polar intuitionistic fuzzy set), which generalized the intuitionistic fuzzy set in this paper, and have applied it to BCK/BCI-algebra. We have introduced the notions of a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal in a BCK/BCI-algebra, and have investigated several properties. We have discussed relations between a k-polar intuitionistic fuzzy subalgebra and a (closed) k-polar intuitionistic fuzzy ideal, and have provided a characterization of a k-polar intuitionistic fuzzy subalgebra/ideal. We have given conditions for a k-polar intuitionistic fuzzy subalgebra to be a k-polar intuitionistic fuzzy ideal. In a BCI-algebra, We have considered the relationship between a k-polar intuitionistic fuzzy ideal and a closed k-polar intuitionistic fuzzy ideal, and have discussed the characterization of a closed k-polar intuitionistic fuzzy ideal. We have consulted conditions for a k-polar intuitionistic fuzzy ideal to be a closed k-polar intuitionistic fuzzy ideal in a BCI-algebra.