m-Polar fuzzy ideals of BCK/BCI-algebras

Imai and Iséki (1966), Iséki (1966) An algebra $(X\text{;}\ast ,0)$ of type (2, 0) is called a BCI-algebra if it satisfies the following axioms:

1. $\left(\right(x\ast y)\ast (x\ast z\left)\right)\ast (z\ast y)=0$ ,

2. $(x\ast (x\ast y\left)\right)\ast y=0$ ,

3. $x\ast x=0$ ,

4. $x\ast y=0$ and $y\ast x=0$ imply $x=y$ .

for all $x,y,z\in X$ . If a BCI-algebra X satisfies $0\ast x=0$ for all $x\in X$ , then X is called a BCK-algebra. We can define a partial ordering ⩽ by $$(\forall x\in X)(x⩽y\iff x\ast y=0)\text{.}$$

Xi (1991) A non-empty subset I of a BCK/BCI-algebra X is called a subalgebra of X if $x\ast y\in I$ for all $x,y\in I$ .

Xi (1991) A non-empty subset S of a BCK/BCI-algebra X is called an ideal of X if it satisfies the following:

Meng (1991) A non-empty subset S of a BCK-algebra X is called a commutative ideal of X if it satisfies (4) and

Meng and Jun (1994) An ideal S of a BCK-algebra X is commutative if and only if the following assertion is valid:

Chen et al. (2014) An m-polar fuzzy set $\widehat{Q}$ on a non-empty set X is a mapping $\widehat{Q}:X\to {[0,1]}^m$ . The membership value of every element $x\in X$ is denoted by $$\widehat{Q}\left(x\right)=(p_1\circ \widehat{Q}(x),p_2\circ \widehat{Q}(x),\dots ,p_m\circ \widehat{Q}(x\left)\right)$$ where $p_i\circ \widehat{Q}:{[0,1]}^m\to [0,1]$ is defined the i-th projection mapping.

An m-polar fuzzy set $\widehat{Q}$ of X is called an m-polar fuzzy subalgebra if the following assertion is valid:

Let $X=\{0,a,b,c\}$ be a BCK-algebra with the Cayley table which is appeared in Table 1.

Define a 4-polar fuzzy set $\widehat{Q}:X\to {[0,1]}^4$ by: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}(0.3,0.4,0.5,0.8),\hfill & \mathrm{if}x=0\hfill \\ (0.2,0.3,0.4,0.3),\hfill & \mathrm{if}x=a\hfill \\ (0.1,0.2,0.3,0.2),\hfill & \mathrm{if}x=b\hfill \\ (0.2,0.3,0.5,0.5),\hfill & \mathrm{if}x=c\text{.}\hfill \end{array}\right.$$ It is routine to verify that $\widehat{Q}$ is a 4-polar fuzzy subalgebra of X.

Let $\widehat{Q}$ be an m-polar fuzzy set of X. Then $\widehat{Q}$ is an m-polar fuzzy subalgebra of X if and only if ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}\ne \varphi$ is a subalgebra of X for all $\widehat{\sigma }=({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_m)\in {[0,1]}^m$ .

Assume that $\widehat{Q}$ is an m-polar fuzzy subalgebra of X and let $\widehat{\sigma }\in {[0,1]}^m$ be such that ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}\ne \varnothing$ . Let $x,y\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ . Then $\widehat{Q}\left(x\right)⩾\widehat{\sigma }$ and $\widehat{Q}\left(y\right)⩾\widehat{\sigma }$ . It follows from (8) that $\widehat{Q}(x\ast y)⩾\inf \left\{\widehat{Q}\right(x),\widehat{Q}(y\left)\right\}⩾\widehat{\sigma }$ , so that $x\ast y\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ . Hence, ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ is a subalgebra of X.

Conversely, assume that ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ is a subalgebra of X. Suppose that there exist $a,b\in X$ such that $\widehat{Q}(a\ast b)<\inf \left\{\widehat{Q}\right(a),\widehat{Q}(b\left)\right\}$ . Then there exists $\widehat{\alpha }=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m)\in {[0,1]}^m$ such that $\widehat{Q}(a\ast b)<\widehat{\alpha }⩽\inf \left\{\widehat{Q}\right(a),\widehat{Q}(b\left)\right\}$ . It follows that $a,b\in {\widehat{Q}}_{\left[\widehat{\alpha }\right]}$ , but $a\ast b\notin {\widehat{Q}}_{\left[\widehat{\alpha }\right]}$ . This is a contradiction, and so $\widehat{Q}(x\ast y)⩾\inf \left\{\widehat{Q}\right(x),\widehat{Q}(y\left)\right\}$ for all $x,y\in X$ . Therefore $\widehat{Q}$ is an m-polar fuzzy subalgebra of X. □

If $\widehat{Q}$ is an m-polar fuzzy subalgebra of X, then ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}^s\ne \varnothing$ is a subalgebra of X for all $\widehat{\sigma }\in {[0,1]}^m$ .

Straightforward.  □

Every m-polar fuzzy subalgebra $\widehat{Q}$ of X satisfies the following inequality:

Note that $x\ast x=0$ for all $x\in X$ . Using (8), we have $\widehat{Q}\left(0\right)=\widehat{Q}(x\ast x)⩾\inf \left\{\widehat{Q}\right(x),\widehat{Q}(x\left)\right\}=\widehat{Q}\left(x\right)$ for all $x\in X$ .  □

If every m-polar fuzzy subalgebra $\widehat{Q}$ of X satisfies the following inequality:

Let $x\in X$ . Using (1) and (10), we have $\widehat{Q}\left(x\right)=\widehat{Q}(x\ast 0)⩾\widehat{Q}\left(0\right)$ . It follows from Lemma 3.5 that $\widehat{Q}\left(x\right)=\widehat{Q}\left(0\right)$ .  □

An m-polar fuzzy set $\widehat{Q}$ of X is called an m-polar fuzzy ideal if the following assertion is valid:

Let $X=\{0,a,1,2,3\}$ be a BCI-algebra with the Cayley table which is appeared in Table 2.

Define a 4-polar fuzzy set $\widehat{Q}:X\to {[0,1]}^4$ by: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}(0.5,0.6,0.6,0.7),\hfill & \mathrm{if}x=0\hfill \\ (0.4,0.5,0.5,0.7),\hfill & \mathrm{if}x=a\hfill \\ (0.2,0.3,0.3,0.2),\hfill & \mathrm{if}x=1,3\hfill \\ (0.3,0.4,0.4,0.5),\hfill & \mathrm{if}x=2\text{.}\hfill \end{array}\right.$$ It is routine to verify that $\widehat{Q}$ is a 4-polar fuzzy ideal of X.

If $\widehat{Q}$ is an m-polar fuzzy ideal of X, then

Let $x,y\in X$ be such that $x⩽y$ . Then $x\ast y=0$ , and so $\widehat{Q}\left(x\right)⩾\inf \left\{\widehat{Q}\right(x\ast y),\widehat{Q}(y\left)\right\}=\inf \left\{\widehat{Q}\right(0),\widehat{Q}(y\left)\right\}=\widehat{Q}\left(y\right)$ . This completes the proof.  □

Let $\widehat{Q}$ be an m-polar fuzzy ideal of X. Then the following are equivalent:

1. $\left(\forall x,y\in X\right)\left(\widehat{Q}\left(x*y\right)⩾\widehat{Q}\left(\left(x*y\right)*y\right)\right)$ ,

2. $(\forall x,y,z\in X)\left(\widehat{Q}\right((x\ast z)\ast (y\ast z))⩾\widehat{Q}((x\ast y)\ast z\left)\right)$ .

Assume that (i) is valid and let $x,y,z\in X$ . Since $$\begin{array}{cc}\hfill \left(\right(x\ast (y\ast z))\ast z)\ast z=& \left(\right(x\ast z)\ast (y\ast z\left)\right)\ast z\hfill \\ \hfill ⩽& (x\ast y)\ast z\text{.}\hfill \end{array}$$ It follows from Proposition 3.9 that $\widehat{Q}\left(\left(\left(x*z\right)*\left(y*z\right)\right)*z\right)⩾\widehat{Q}\left(\left(x*y\right)*z\right)$ . Using (2) and (i), we have $$\begin{array}{cc}\hfill \widehat{Q}\left(\right(x\ast z)\ast (y\ast z\left)\right)=& \widehat{Q}\left(\right(x\ast (y\ast z))\ast z)\hfill \\ \hfill ⩾& \widehat{Q}\left(\right((x\ast (y\ast z\left)\right)\ast z)\ast z)\hfill \\ \hfill ⩾& \widehat{Q}\left(\right(x\ast y)\ast z)\text{.}\hfill \end{array}$$ Conversely, suppose that (ii) holds. If we use z instead of y in (ii), then $$\begin{array}{cc}\hfill \widehat{Q}(x\ast z)=& \widehat{Q}\left(\right(x\ast z)\ast 0)\hfill \\ \hfill =& \widehat{Q}\left(\right(x\ast z)\ast (z\ast z\left)\right)\hfill \\ \hfill ⩾& \widehat{Q}\left(\right(x\ast z)\ast z)\hfill \end{array}$$ for all $x,z\in X$ by using (III) and (1).  □

Let $\widehat{Q}$ be an m-polar fuzzy set of X. Then $\widehat{Q}$ is an m-polar fuzzy ideal of X if and only if it satisfies

Assume that $\widehat{Q}$ is an m-polar fuzzy ideal of X. Let $\widehat{\sigma }=({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_m)\in {[0,1]}^m$ be such that ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}\ne \varnothing$ . Obviously, $0\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ . Let $x,y\in X$ be such that $x\ast y\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ and $y\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ . Then $\widehat{Q}\left(x*y\right)⩾\widehat{\sigma }$ and $\widehat{Q}\left(y\right)⩾\widehat{\sigma }$ . It follows from (11) that $\widehat{Q}\left(x\right)⩾\inf \left\{\widehat{Q}\left(x*y\right),\widehat{Q}\left(y\right)\right\}⩾\widehat{\sigma }$ , so that $x\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ . Hence, ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ is an ideal of X.

Conversely, suppose that (13) is valid. If there exists $h\in X$ such that $\widehat{Q}\left(0\right)<\widehat{Q}\left(h\right)$ , then $\widehat{Q}\left(0\right)<{\widehat{\sigma }}_h⩽\widehat{Q}\left(h\right)$ for some ${\widehat{\sigma }}_h=({\sigma }_{h_1},{\sigma }_{h_2},\dots ,{\sigma }_{h_m})\in {[0,1]}^m$ . Then $0\notin {\widehat{Q}}_{\left[{\widehat{\sigma }}_h\right]}$ which is a contradiction. Hence $\widehat{Q}\left(0\right)⩾\widehat{Q}\left(x\right)$ for all $x\in X$ . Now, assume that there exist $h,q\in X$ such that $\widehat{Q}\left(h\right)<\inf \left\{\widehat{Q}\right(h\ast q),\widehat{Q}(q\left)\right\}$ . Then there exists $\widehat{\beta }=({\beta }_1,{\beta }_2,\dots ,{\beta }_m)\in {[0,1]}^m$ such that $\widehat{Q}\left(h\right)<\widehat{\beta }⩽\inf \left\{\widehat{Q}\left(h*q\right),\widehat{Q}\left(q\right)\right\}$ . It follows that $h\ast q\in {\widehat{Q}}_{\left[\widehat{\beta }\right]}$ and $q\in {\widehat{Q}}_{\left[\widehat{\beta }\right]}$ , but $h\notin {\widehat{Q}}_{\left[\widehat{\beta }\right]}$ . This is impossible, and so $\widehat{Q}\left(x\right)⩾\inf \left\{\widehat{Q}\left(x*y\right),\widehat{Q}\left(y\right)\right\}$ for all $x,y\in X$ . Therefore, $\widehat{Q}$ is an m-polar fuzzy ideal of X. □

If $\widehat{Q}$ is an m-polar fuzzy ideal of X, then ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}^s\ne \varnothing$ is an ideal of X for all $\widehat{\sigma }\in {[0,1]}^m$ .

Straightforward.  □

Let $\omega$ be an element of X. If $\widehat{Q}$ is an m-polar fuzzy ideal of X, then the set $X_{\omega }$ is an ideal of X.

Obviously, $0\in X_{\omega }$ by (11). Let $x,y\in X$ be such that $x\ast y\in X_{\omega }$ and $y\in X_{\omega }$ . Then $\widehat{Q}\left(x*y\right)⩾\widehat{Q}\left(\omega \right)$ and $\widehat{Q}\left(y\right)⩾\widehat{Q}\left(\omega \right)$ . Since $\widehat{Q}$ is an m-polar fuzzy ideal of X, it follows from (11) that $\widehat{Q}\left(x\right)⩾\inf \left\{\widehat{Q}\left(x*y\right),\widehat{Q}\left(y\right)\right\}⩾\widehat{Q}\left(\omega \right)$ , so that $x\in X_{\omega }$ . Hence, $X_{\omega }$ is an ideal of X.  □

Let $\widehat{Q}$ be an m-polar fuzzy ideal of X. If X satisfies the following assertion:

Assume that (15) is valid in X. Then $\widehat{Q}(x\ast y)⩾\inf \left\{\widehat{Q}\right((x\ast y)\ast z),\widehat{Q}(z\left)\right\}=\inf \left\{\widehat{Q}\right(0),\widehat{Q}(z\left)\right\}=\widehat{Q}\left(z\right)$ for all $x,y,z\in X$ . It follows that $\widehat{Q}\left(x\right)⩾\inf \left\{\widehat{Q}\right(x\ast y),\widehat{Q}(y\left)\right\}⩾\inf \left\{\widehat{Q}\right(y),\widehat{Q}(z\left)\right\}$ for all $x,y,z\in X$ . This completes the proof.  □

For any BCK-algebra X, every m-polar fuzzy ideal is an m-polar fuzzy subalgebra.

Let $\widehat{Q}$ be an m-polar fuzzy ideal of a BCK-algebra X and let $x,y\in X$ . Then $$\begin{array}{cc}\hfill \widehat{Q}(x\ast y)⩾& \inf \left\{\widehat{Q}\right((x\ast y)\ast x),\widehat{Q}(x\left)\right\}\hfill \\ \hfill =& \inf \left\{\widehat{Q}\right((x\ast x)\ast y),\widehat{Q}(x\left)\right\}\hfill \\ \hfill =& \inf \left\{\widehat{Q}\right(0\ast y),\widehat{Q}(x\left)\right\}\hfill \\ \hfill =& \inf \left\{\widehat{Q}\right(0),\widehat{Q}(x\left)\right\}\hfill \\ \hfill ⩾& \inf \left\{\widehat{Q}\right(x),\widehat{Q}(y\left)\right\}\text{.}\hfill \end{array}$$ Therefore, $\widehat{Q}$ is an m-polar fuzzy subalgebra of X.  □

Consider a BCK-algebra $X=\{0,a,b,c\}$ which is given in Example 3.2. Define a 3-polar fuzzy set $\widehat{Q}:X\to {[0,1]}^3$ by: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}(0.3,0.7,0.8),\hfill & \mathrm{if}x=0,b\hfill \\ (0.1,0.4,0.5),\hfill & \mathrm{if}x=a,c\text{.}\hfill \end{array}\right.$$ Then $\widehat{Q}$ is a 3-polar fuzzy subalgebra of X. But it is not a 3-polar fuzzy ideal of X, since $\widehat{Q}\left(a\right)=(0.1,0.4,0.5)<(0.3,0.7,0.8)=\inf \left\{\widehat{Q}\right(a\ast b),\widehat{Q}(b\left)\right\}$ .

Consider a BCI-algebra $X=Y\times \mathbb{Z}$ , where $(Y,\ast ,0)$ is a BCI-algebra and $(\mathbb{Z},-,0)$ is the adjoint BCI-algebra of the additive group $(\mathbb{Z},+,0)$ of integers (see Huang, 2006). Let $A=Y\times \mathbb{N}$ , where $\mathbb{N}$ is the set of nonnegative integers. Define an m-polar fuzzy set $\widehat{Q}:X\to {[0,1]}^m$ as follows: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}(0.7,0.7,\dots ,0.7),\hfill & x\in A\hfill \\ (0.2,0.2,\dots ,0.2),\hfill & x\notin A\text{.}\hfill \end{array}\right.$$ Then $\widehat{Q}$ is an m-polar fuzzy ideal of X. If we take $x=(0,0)$ and $y=(0,1)$ , then $z=x\ast y=(0,0)\ast (0,1)=(0,-1)$ , and so $\widehat{Q}(x\ast y)=\widehat{Q}\left(z\right)=(0.2,0.2,\dots ,0.2)<(0.7,0.7,\dots ,0.7)=\inf \left\{\widehat{Q}\right(x),\widehat{Q}(y\left)\right\}$ . Therefore, $\widehat{Q}$ is not an m-polar fuzzy subalgebra of X.

Let X be a BCI-algebra. An m-polar fuzzy ideal $\widehat{Q}$ of X is said to be closed if it is also an m-polar fuzzy subalgebra of X.

Consider a BCI-algebra $X=\{0,a,1,2,3\}$ which is given in Example 3.8. Define a 4-polar fuzzy set $\widehat{Q}:X\to {[0,1]}^4$ by: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}(0.5,0.6,0.8,0.9),\hfill & \mathrm{if}x=0\hfill \\ (0.3,0.4,0.6,0.7),\hfill & \mathrm{if}x=a\hfill \\ (0.2,0.3,0.5,0.6),\hfill & \mathrm{if}x=1,2,3\text{.}\hfill \end{array}\right.$$ Then $\widehat{Q}$ is a closed 4-polar fuzzy ideal of X.

Let X be a BCI-algebra and let $\widehat{Q}$ be an m-polar fuzzy set of X given as follows: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}\widehat{t}=(t_1,t_2,\dots ,t_m),\hfill & \mathrm{if}x\in X_{+}\hfill \\ \widehat{s}=(s_1,s_2,\dots ,s_m),\hfill & \mathit{otherwise}\hfill \end{array}\right.$$ where $\widehat{t},\widehat{s}\in {[0,1]}^m$ with $\widehat{t}>\widehat{s}$ and $X_{+}=\{x\in X|0⩽x\}$ . Then $\widehat{Q}$ is a closed m-polar fuzzy ideal of X.

Since $0\in X_{+}$ , we have $\widehat{Q}\left(0\right)=\widehat{t}=\left(t_1,t_2,\dots ,t_m\right)⩾\widehat{Q}\left(x\right)$ for all $x\in X$ . Let $x,y\in X$ . If $x\in X_{+}$ , then $$\widehat{Q}\left(x\right)=\widehat{t}=(t_1,t_2,\dots ,t_m)⩾\inf \left\{\widehat{Q}\right(x\ast y),\widehat{Q}(y\left)\right\}\text{.}$$ Assume that $x\notin X_{+}$ . If $x\ast y\in X_{+}$ , then $y\notin X_{+}$ ; and if $y\in X_{+}$ , then $x\ast y\notin X_{+}$ . In either case, we get $$\widehat{Q}\left(x\right)=\widehat{s}=(s_1,s_2,\dots ,s_m)=\inf \left\{\widehat{Q}\right(x\ast y),\widehat{Q}(y\left)\right\}\text{.}$$ For any $x,y\in X$ , if any one of x and y does not belong to $X_{+}$ , then $$\widehat{Q}\left(x*y\right)⩾\widehat{s}=\left(s_1,s_2,\dots ,s_m\right)=\inf \left\{\widehat{Q}\left(x\right),\widehat{Q}\left(y\right)\right\}\text{.}$$ If $x,y\in X_{+}$ , then $x\ast y\in X_{+}$ , and so $$\widehat{Q}(x\ast y)=\widehat{t}=(t_1,t_2,\dots ,t_m)=\inf \left\{\widehat{Q}\right(x),\widehat{Q}(y\left)\right\}\text{.}$$ Therefore, $\widehat{Q}$ is a closed m-polar fuzzy ideal of X.  □

Every closed m-polar fuzzy ideal $\widehat{Q}$ of a BCI-algebra X satisfies the following assertion:

For any $x\in X$ , we have $\widehat{Q}(0\ast x)⩾\inf \left\{\widehat{Q}\right(0),\widehat{Q}(x\left)\right\}⩾\inf \left\{\widehat{Q}\right(x),\widehat{Q}(x\left)\right\}=\widehat{Q}\left(x\right)$ . This completes the proof.  □

Let X be a BCI-algebra. If $\widehat{Q}$ is an m-polar fuzzy ideal of X that satisfies the condition (16), then $\widehat{Q}$ is an m-polar fuzzy subalgebra and hence is a closed m-polar fuzzy ideal of X.

Note that $(x\ast y)\ast x⩽0\ast y$ for all $x,y\in X$ . Using Proposition 3.14 and the condition of Eq. (16), we have $$\widehat{Q}(x\ast y)⩾\inf \left\{\widehat{Q}\right(x),\widehat{Q}(0\ast y\left)\right\}⩾\inf \left\{\widehat{Q}\right(x),\widehat{Q}(y\left)\right\}\text{.}$$ Hence $\widehat{Q}$ is an m-polar fuzzy subalgebra of X and therefore $\widehat{Q}$ is a closed m-polar fuzzy ideal of X.  □

Let X be a BCK-algebra. An m-polar fuzzy set $\widehat{Q}$ of X is called an m-polar fuzzy commutative ideal of X if the following assertions are valid:

Consider a BCK-algebra $X=\{0,a,b,c\}$ which is given in Example 3.2. Define an m-polar fuzzy set $\widehat{Q}:X\to {[0,1]}^m$ by: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}\widehat{\alpha }=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m),\hfill & \mathrm{if}x=0\hfill \\ \widehat{\beta }=({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\hfill & \mathrm{if}x=a\hfill \\ \widehat{\gamma }=({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_m),\hfill & \mathrm{if}x=b,c,\hfill \end{array}\right.$$ where $\widehat{\alpha },\widehat{\beta },\widehat{\gamma }\in {[0,1]}^m$ and $\widehat{\alpha }>\widehat{\beta }>\widehat{\gamma }$ . It is routine to verify that $\widehat{Q}$ is an m-polar fuzzy commutative ideal of X.

Every m-polar fuzzy commutative ideal of a BCK-algebra X is an m-polar fuzzy ideal of X.

Let $\widehat{Q}$ be an m-polar fuzzy commutative ideal of a BCK-algebra X. For any $x,z\in X$ , we have $$\begin{array}{cc}\hfill \widehat{Q}\left(x\right)=& \widehat{Q}(x\ast (0\overrightarrow{\wedge }x\left)\right)\hfill \\ \hfill ⩾& \inf \left\{\widehat{Q}\right((x\ast 0)\ast z),\widehat{Q}(z\left)\right\}\hfill \\ \hfill =& \inf \left\{\widehat{Q}\right(x\ast z),\widehat{Q}(z\left)\right\}\text{.}\hfill \end{array}$$ Hence, $\widehat{Q}$ is an m-polar fuzzy ideal of X.  □

Let $X=\{0,1,2,3,4\}$ be a set with the Cayley table which is appeared in Table 3.

Then X is a BCK-algebra Meng (1991). Define an m-polar fuzzy set $\widehat{Q}:X\to {[0,1]}^m$ by: $$\widehat{Q}\left(x\right)=\left\{\begin{array}{cc}\widehat{\alpha }=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m),\hfill & \mathrm{if}x=0\hfill \\ \widehat{\beta }=({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\hfill & \mathrm{if}x=1\hfill \\ \widehat{\gamma }=({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_m),\hfill & \mathrm{if}x=2,3,4,\hfill \end{array}\right.$$ where $\widehat{\alpha },\widehat{\beta },\widehat{\gamma }\in {[0,1]}^m$ and $\widehat{\alpha }>\widehat{\beta }>\widehat{\gamma }$ . Then $\widehat{Q}$ is an m-polar fuzzy ideal of X. But it is not an m-polar fuzzy commutative ideal of X, since $\widehat{Q}\left(2*\left(3\overrightarrow{\wedge }2\right)\right)<\inf \left\{\widehat{Q}\left(\left(2*3\right)*0\right),\widehat{Q}\left(0\right)\right\}$ .

Let $\widehat{Q}$ be an m-polar fuzzy ideal of a BCK-algebra X. Then $\widehat{Q}$ is an m-polar fuzzy commutative ideal of X if and only if the following assertion is valid:

Assume that $\widehat{Q}$ is an m-polar fuzzy commutative ideal of a BCK-algebra X. Then assertion (19) is by taking $z=0$ in (18) and using (1) and (17); then we get (19).

Conversely, suppose that an m-polar fuzzy ideal $\widehat{Q}$ of a BCK-algebra X satisfies the condition (19). Then

In a commutative BCK-algebra X, every m-polar fuzzy ideal is an m-polar fuzzy commutative ideal.

Let $\widehat{Q}$ be an m-polar fuzzy ideal of a commutative BCK-algebra X. Using (I) and (2), we have $$\begin{array}{cc}\hfill \left(\right(x\ast \left(y\overrightarrow{\wedge }x\right))\ast ((x\ast y)\ast z\left)\right)\ast z=& \left(\right(x\ast \left(y\overrightarrow{\wedge }x\right))\ast z)\ast \left(\right(x\ast y)\ast z)\hfill \\ \hfill ⩽& (x\ast (y\overrightarrow{\wedge }x\left)\right)\ast (x\ast y)\hfill \\ \hfill =& \left(x\overrightarrow{\wedge }y\right)\ast \left(y\overrightarrow{\wedge }x\right)\hfill \\ \hfill =& 0,\hfill \end{array}$$ and so $\left(\right(x\ast \left(y\overrightarrow{\wedge }x\right))\ast ((x\ast y)\ast z\left)\right)\ast z=0$ , i.e., $(x\ast (y\overrightarrow{\wedge }x\left)\right)\ast \left(\right(x\ast y)\ast z)⩽z$ for all $x,y,z\in X$ . Since $\widehat{Q}$ is an m-polar fuzzy ideal, it follows from Proposition 3.14, $\widehat{Q}(x\ast (y\overrightarrow{\wedge }x\left)\right)⩾\inf \left\{\widehat{Q}\right((x\ast y)\ast z),\widehat{Q}(z\left)\right\}$ . Hence, $\widehat{Q}$ is an m-polar fuzzy commutative ideal of X.  □

Let $\widehat{Q}$ be an m-polar fuzzy set of a BCK-algebra X. Then $\widehat{Q}$ is an m-polar fuzzy commutative ideal of X if and only if it satisfies

Let $\widehat{Q}$ be an m-polar fuzzy commutative ideal of X. Then $\widehat{Q}$ is an m-polar fuzzy ideal of X, and so every non-empty $\sigma$ -level cut set ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ of $\widehat{Q}$ is an ideal of X. Let $x,y,z\in X$ be such that $(x\ast y)\ast z\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ and $z\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ . Then $\widehat{Q}\left(\left(x*y\right)*z\right)⩾\widehat{\sigma }$ and $\widehat{Q}\left(z\right)⩾\widehat{\sigma }$ . It follows from (18) that $$\widehat{Q}\left(x*\left(y\overrightarrow{\wedge }x\right)\right)⩾\inf \left\{\widehat{Q}\left(\left(x*y\right)*z\right),\widehat{Q}\left(z\right)\right\}⩾\widehat{\sigma },$$ so that $x\ast \left(y\overrightarrow{\wedge }x\right)\in {\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ . Hence ${\widehat{Q}}_{\left[\widehat{\sigma }\right]}$ is a commutative ideal of X.

Conversely, suppose that (21) is valid. Obviously, $\widehat{Q}\left(0\right)⩾\widehat{Q}\left(x\right)$ for all $x\in X$ . Let $\widehat{Q}\left(\right(x\ast y)\ast z)=\widehat{\alpha }=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m)$ and $\widehat{Q}\left(z\right)=\widehat{\beta }=({\beta }_1,{\beta }_2,\dots ,{\beta }_m)$ for all $x,y,z\in X$ . Then $(x\ast y)\ast z\in {\widehat{Q}}_{\left[\widehat{\alpha }\right]}$ and $z\in {\widehat{Q}}_{\left[\widehat{\beta }\right]}$ . Without loss of generality, we may assume that $\widehat{\alpha }⩽\widehat{\beta }$ . Then ${\widehat{Q}}_{\left[\widehat{\beta }\right]}\subseteq {\widehat{Q}}_{\left[\widehat{\alpha }\right]}$ , and so $z\in {\widehat{Q}}_{\left[\widehat{\alpha }\right]}$ . Since ${\widehat{Q}}_{\left[\widehat{\alpha }\right]}$ is a commutative ideal of X by hypothesis, we obtain that $x\ast \left(y\overrightarrow{\wedge }x\right)\in {\widehat{Q}}_{\left[\widehat{\alpha }\right]}$ , and so $$\widehat{Q}\left(x*\left(y\overrightarrow{\wedge }x\right)\right)⩾\widehat{\alpha }=\inf \left\{\widehat{\alpha },\widehat{\beta }\right\}=\inf \left\{\widehat{Q}\left(\left(x*y\right)*z\right),\widehat{Q}\left(z\right)\right\}\text{.}$$ Therefore, $\widehat{Q}$ is an m-polar fuzzy commutative ideal of X.  □