m-polar fuzzy q-ideals in BCI-algebras

(Huang, 2006) A BCI-algebra $X$ is quasi-associative if and only if $0*x=0*(0*x)$ for all $x\in X$ .

(Al-Masarwah and Ahmad, 2019, Definition 3.1) An m-polar fuzzy set $\widehat{\alpha }$ on a BCK/BCI-algebra $X$ is called an m-polar fuzzy subalgebra of $X$ if the following condition is valid.

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

Define a 4-polar fuzzy set $\widehat{\alpha }$ on $X$ as follows; $$\widehat{\alpha }:X\to {[0,1]}^4,x\mapsto \left\{\begin{array}{cc}(0.33,0.41,0.57,0.83)\hfill & \mathrm{if}x=0,\hfill \\ (0.25,0.33,0.42,0.38)\hfill & \mathrm{if}x=a,\hfill \\ (0.22,0.30,0.40,0.20)\hfill & \mathrm{if}x=b,\hfill \\ (0.25,0.34,0.55,0.50)\hfill & \mathrm{if}x=c\hfill \end{array}\right.$$

It is routine to check that $\widehat{\alpha }$ is a 4-polar fuzzy subalgebra of $X$ .

(Al-Masarwah and Ahmad, 2019, Definition 3.7) An m-polar fuzzy set $\widehat{\alpha }$ on a BCK/BCI-algebra $X$ is called an m-polar fuzzy ideal of $X$ if the following conditions are valid.

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

Define a 4-polar fuzzy set $\widehat{\alpha }$ on $X$ as follows; $$\widehat{\alpha }:X\to {[0,1]}^4,x\mapsto \left\{\begin{array}{cc}(0.50,0.60,0.60,0.70)\hfill & \mathrm{if}x=0,\hfill \\ (0.40,0.50,0.50,0.70)\hfill & \mathrm{if}x=a,\hfill \\ (0.20,0.30,0.30,0.20)\hfill & \mathrm{if}x=b,d,\hfill \\ (0.30,0.40,0.40,0.50)\hfill & \mathrm{if}x=c\hfill \end{array}\right.$$

It is routine to check that $\widehat{\alpha }$ is a 4-polar fuzzy ideal of $X$ .

Lemma 2.6 Mohseni Takallo et al., 2019, Lemma 1

An m-polar fuzzy set $\widehat{\alpha }$ on a BCK/BCI-algebra $X$ is an m-polar fuzzy ideal of $X$ if and only if the following conditions are valid.

An m-polar fuzzy set $\widehat{\alpha }$ on a BCI-algebra $X$ is called an m-polar $(\in ,\in )$ -fuzzy p-ideal of $X$ if it satisfies (2.18) and

Let $X=\{0,1,2,3\}$ be a set with a binary operation $*$ which is given in Table 3.

Then $X$ is a BCI-algebra (see Huang, 2006). Define a 5-polar fuzzy set $\widehat{\alpha }$ on $X$ as follows: $$\widehat{\alpha }:X\to {[0,1]}^5,x\mapsto \left\{\begin{array}{cc}(0.7,0.6,0.8,0.5,0.9)\hfill & \mathrm{if}x=0,\hfill \\ (0.5,0.6,0.7,0.4,0.7)\hfill & \mathrm{if}x=1,\hfill \\ (0.3,0.4,0.6,0.2,0.5)\hfill & \mathrm{if}x=2,\hfill \\ (0.3,0.4,0.6,0.2,0.5)\hfill & \mathrm{if}x=3\hfill \end{array}\right.$$

It is routine to check that $\widehat{\alpha }$ is a 5-polar $(\in ,\in )$ -fuzzy p-ideal of $X$ .

An m-polar fuzzy set $\widehat{\alpha }$ on a BCI-algebra $X$ is called an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ if it satisfies (2.18) and

Let $X=\{0,1,a\}$ be a set with a binary operation $*$ which is given in Table 4.

Then $X$ is a BCI-algebra (see Huang, 2006). Define a 3-polar fuzzy set $\widehat{\alpha }$ on $X$ as follows: $$\widehat{\alpha }:X\to {[0,1]}^3,x\mapsto \left\{\begin{array}{cc}(0.75,0.63,0.82)\hfill & \mathrm{if}x=0,\hfill \\ (0.55,0.63,0.72)\hfill & \mathrm{if}x=1,\hfill \\ (0.35,0.43,0.62)\hfill & \mathrm{if}x=a\hfill \end{array}\right.$$

It is routine to check that $\widehat{\alpha }$ is a 3-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ .

Every m-polar $(\in ,\in )$ -fuzzy q-ideal of a BCI-algebra $X$ is an m-polar fuzzy ideal and an m-polar fuzzy subalgebra of $X$ .

Let $\widehat{\alpha }$ be an m-polar $(\in ,\in )$ -fuzzy q-ideal of a BCI-algebra $X$ . Putting $z=0$ in (3.3) and using (2.2) implies that $$({\pi }_i\circ \widehat{\alpha })(x)=({\pi }_i\circ \widehat{\alpha })(x*0)⩾\inf \{({\pi }_i\circ \widehat{\alpha })(x*(y*0)),({\pi }_i\circ \widehat{\alpha })(y)\}=\inf \{({\pi }_i\circ \widehat{\alpha })(x*y)),({\pi }_i\circ \widehat{\alpha })(y)\}$$ for all $x,y\in X$ and $i=1,2,\dots ,m$ . Hence $\widehat{\alpha }$ is an m-polar fuzzy ideal of $X$ . Putting $z=y$ in (3.3) and using (III) and (2.2) implies that $$({\pi }_i\circ \widehat{\alpha })(x*y)⩾\inf \{({\pi }_i\circ \widehat{\alpha })(x*(y*y)),({\pi }_i\circ \widehat{\alpha })(y)\}=\inf \{({\pi }_i\circ \widehat{\alpha })(x*0),({\pi }_i\circ \widehat{\alpha })(y)\}=\inf \{({\pi }_i\circ \widehat{\alpha })(x),({\pi }_i\circ \widehat{\alpha })(y)\}$$ for all $x,y\in X$ and $i=1,2,\dots ,m$ . Thus $\widehat{\alpha }$ is an m-polar fuzzy subalgebra of $X$ . □

Let $X=\{0,1,b,c\}$ be a set with a binary operation $*$ which is given in Table 5.

Then $X$ is a BCI-algebra (see Huang, 2006). Define a 4-polar fuzzy set $\widehat{\alpha }$ on $X$ as follows: $$\widehat{\alpha }:X\to {[0,1]}^4,x\mapsto \left\{\begin{array}{cc}(0.7,0.6,0.8,0.5)\hfill & \mathrm{if}x=0,\hfill \\ (0.3,0.4,0.5,0.2)\hfill & \mathrm{if}x=1,\hfill \\ (0.3,0.5,0.6,0.2)\hfill & \mathrm{if}x=b,\hfill \\ (0.3,0.4,0.5,0.2)\hfill & \mathrm{if}x=c\hfill \end{array}\right.$$

It is routine to check that $\widehat{\alpha }$ is an 4-polar fuzzy ideal and a 4-polar fuzzy subalgebra of $X$ . But it is not a 4-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ since $$({\pi }_2\circ \widehat{\alpha })(c*1)=({\pi }_2\circ \widehat{\alpha })(b)=0.5<0.6=\inf \{({\pi }_2\circ \widehat{\alpha })(c*(0*1)),({\pi }_2\circ \widehat{\alpha })(0)\}\text{.}$$

(Al-Masarwah and Ahmad, 2019, Proposition 3.9) If $\widehat{\alpha }$ is an m-polar fuzzy ideal of a BCI-algebra $X$ , then $$(\forall x,y\in X)(x⩽y\Rightarrow \widehat{\alpha }(x)⩾\widehat{\alpha }(y)),$$ that is, $({\pi }_i\circ \widehat{\alpha })(x)⩾({\pi }_i\circ \widehat{\alpha })(y)$ for all $x,y\in X$ with $x⩽y$ and $i=1,2,\dots ,m$ .

(Al-Masarwah and Ahmad, 2019, Proposition 3.14) If $\widehat{\alpha }$ is an m-polar fuzzy ideal of a BCI-algebra $X$ , then $$(\forall x,y,z\in X)(x*y⩽z\Rightarrow \widehat{\alpha }(x)⩾\inf \{\widehat{\alpha }(y),\widehat{\alpha }(z)\}),$$ that is, $({\pi }_i\circ \widehat{\alpha })(x)⩾\inf \{({\pi }_i\circ \widehat{\alpha })(y),({\pi }_i\circ \widehat{\alpha })(z)\}$ for all $x,y,z\in X$ with $x*y⩽z$ and $i=1,2,\dots ,m$ .

Given an m-polar fuzzy ideal $\widehat{\alpha }$ of a BCI-algebra $X$ , the following are equivalent.

• (1)  $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ .

• (2)  $\widehat{\alpha }$ satisfies $\widehat{\alpha }(x*y)⩾\widehat{\alpha }(x*(0*y))$ , that is, $$({\pi }_i\circ \widehat{\alpha })(x*y)⩾({\pi }_i\circ \widehat{\alpha })(x*(0*y))$$ for all $x,y\in X$ and $i=1,2,\dots ,m$ .

• (3)  $\widehat{\alpha }$ satisfies $\widehat{\alpha }((x*y)*z)⩾\widehat{\alpha }(x*(y*z))$ , that is, $$({\pi }_i\circ \widehat{\alpha })((x*y)*z)⩾({\pi }_i\circ \widehat{\alpha })(x*(y*z))$$ for all $x,y,z\in X$ and $i=1,2,\dots ,m$ .

(1) $\Rightarrow$ (2). If we replace $y$ and $z$ by 0 and $y$ , respectively, in (3.3) and use (2.16), then $$({\pi }_i\circ \widehat{\alpha })(x*y)⩾\inf \{({\pi }_i\circ \widehat{\alpha })(x*(0*y)),({\pi }_i\circ \widehat{\alpha })(0)\}=({\pi }_i\circ \widehat{\alpha })(x*(0*y)),$$ and so for $\widehat{\alpha }(x*y)⩾\widehat{\alpha }(x*(0*y))$ for all $x,y\in X$ .(2) $\Rightarrow$ (3). Note that $$((x*y)*(0*z))*(x*(y*z))=((x*y)*(x*(y*z)))*(0*z)⩽((y*z)*y)*(0*z)=(0*z)*(0*z)=0,$$ that is, $(x*y)*(0*z)⩽x*(y*z)$ for all $x,y,z\in X$ . It follows from (2) and Lemma 3.5 that $$({\pi }_i\circ \widehat{\alpha })((x*y)*z)⩾({\pi }_i\circ \widehat{\alpha })((x*y)*(0*z))⩾({\pi }_i\circ \widehat{\alpha })(x*(y*z)),$$ that is, $\widehat{\alpha }((x*y)*z)⩾\widehat{\alpha }(x*(y*z))$ for all $x,y,z\in X$ .

(3) $\Rightarrow$ (1). Note that $(x*z)*((x*y)*z)⩽x*(x*y)⩽y$ for all $x,y,z\in X$ . Using (3) and Lemma 3.6, we have $$({\pi }_i\circ \widehat{\alpha })(x*z)⩾\inf \{({\pi }_i\circ \widehat{\alpha })((x*y)*z),({\pi }_i\circ \widehat{\alpha })(y)\}⩾\inf \{({\pi }_i\circ \widehat{\alpha })(x*(y*z)),({\pi }_i\circ \widehat{\alpha })(y)\}$$ and so $\widehat{\alpha }(x*z)⩾\inf \{\widehat{\alpha }(x*(y*z)),\widehat{\alpha }(y)\}$ for all $x,y,z\in X$ . Therefore $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ . □

An m-polar fuzzy set $\widehat{\alpha }$ on a BCI-algebra $X$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ if and only if the m-polar level set $U(\widehat{\alpha }\text{;}\widehat{r})$ of $\widehat{\alpha }$ is a q-ideal of $X$ for all $\widehat{r}\in {[0,1]}^m$ .

Suppose that $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ and let $\widehat{r}=(r_1,r_2,\dots ,r_m)\in {\left(0,1\right]}^m$ . It is clear that $0\in U(\widehat{\alpha }\text{;}\widehat{r})$ . Let $x,y,z\in X$ be such that $x*(y*z)\in U(\widehat{\alpha }\text{;}\widehat{r})$ and $y\in U(\widehat{\alpha }\text{;}\widehat{r})$ . Then $({\pi }_i\circ \widehat{\alpha })(x*(y*z))⩾r_i$ and $({\pi }_i\circ \widehat{\alpha })(y)⩾r_i$ for all $i=1,2,\dots ,m$ . It follows from (3.3) that $$({\pi }_i\circ \widehat{\alpha })(x*z)⩾\inf \{({\pi }_i\circ \widehat{\alpha })(x*(y*z)),({\pi }_i\circ \widehat{\alpha })(y)\}⩾r_i$$ for $i=1,2,\dots ,m$ . Hence $x*z\in U(\widehat{\alpha }\text{;}\widehat{r})$ , and therefore $U(\widehat{\alpha }\text{;}\widehat{r})$ is a q-ideal of $X$ .

Conversely, suppose that the m-polar level set $U(\widehat{\alpha }\text{;}\widehat{r})$ of $\widehat{\alpha }$ is a q-ideal of $X$ for all $\widehat{r}\in {[0,1]}^m$ . If $\widehat{\alpha }(0)<\widehat{\alpha }(a)$ for some $a\in X$ and take $\widehat{r}≔\widehat{\alpha }(a)$ , then $a\in U(\widehat{\alpha }\text{;}\widehat{r})$ and $0\notin U(\widehat{\alpha }\text{;}\widehat{r})$ . This is a contradiction, and so $\widehat{\alpha }(0)⩾\widehat{\alpha }(x)$ for all $x\in X$ . Now, suppose that there exist $a,b,c\in X$ such that $\widehat{\alpha }(a*c)<\inf \{\widehat{\alpha }(a*(b*c)),\widehat{\alpha }(b)\}$ . If we take $$\widehat{r}≔\inf \{\widehat{\alpha }(a*(b*c)),\widehat{\alpha }(b)\},$$ then $a*(b*c)\in U(\widehat{\alpha }\text{;}\widehat{r})$ and $b\in U(\widehat{\alpha }\text{;}\widehat{r})$ . Since $U(\widehat{\alpha }\text{;}\widehat{r})$ is a q-ideal of $X$ , it follows that $a*c\in U(\widehat{\alpha }\text{;}\widehat{r})$ . Hence $\widehat{\alpha }(a*c)⩾\widehat{r}$ , which is a contradiction. Thus $\widehat{\alpha }(x*z)⩾\inf \{\widehat{\alpha }(x*(y*z)),\widehat{\alpha }(y)\}$ for all $x,y,z\in X$ . Therefore $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ . □

If $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of a BCI-algebra $X$ , then the set $$J≔\{x\in X|\widehat{\alpha }(x)=\widehat{\alpha }(0)\}$$ is a q-ideal of $X$ .

Let $\widehat{\alpha }$ and $\widehat{\beta }$ be m-polar fuzzy ideals of a BCI-algebra $X$ such that $\widehat{\alpha }⩽\widehat{\beta }$ and $\widehat{\alpha }(0)=\widehat{\beta }(0)$ . If $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ , then so is $\widehat{\beta }$ .

Assume that $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ . Since $$(x*(x*(0*y)))*(0*y)=(x*(0*y))*(x*(0*y))=0$$ for all $x,y\in X$ , we have $$({\pi }_i\circ \widehat{\alpha })((x*(x*(0*y)))*(0*y))=({\pi }_i\circ \widehat{\alpha })(0)=({\pi }_i\circ \widehat{\beta })(0)$$ for $i=1,2,\dots ,m$ . It follows from Theorem 3.7 that $$({\pi }_i\circ \widehat{\alpha })((x*(x*(0*y)))*y)⩾({\pi }_i\circ \widehat{\alpha })((x*(x*(0*y)))*(0*y))=({\pi }_i\circ \widehat{\beta })(0)\text{.}$$

Hence $$({\pi }_i\circ \widehat{\beta })((x*y)*(x*(0*y)))=({\pi }_i\circ \widehat{\beta })((x*(x*(0*y)))*y)⩾({\pi }_i\circ \widehat{\alpha })((x*(x*(0*y)))*y)⩾({\pi }_i\circ \widehat{\beta })(0)⩾({\pi }_i\circ \widehat{\beta })(x*(0*y)),$$ which implies from (2.17) that $$({\pi }_i\circ \widehat{\beta })(x*y)⩾\inf \{({\pi }_i\circ \widehat{\beta })((x*y)*(x*(0*y))),({\pi }_i\circ \widehat{\beta })(x*(0*y))\}=({\pi }_i\circ \widehat{\beta })(x*(0*y))$$ for all $x,y\in X$ and $i=1,2,\dots ,m$ . Therefore $\widehat{\beta }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ by Theorem 3.7. □

Let $f:X\to Y$ be an epimorphism of BCI-algebras. If $\widehat{\beta }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $Y$ , then the m-polar fuzzy set $\widehat{\alpha }$ on $X$ defined by $$\widehat{\alpha }:X\to {[0,1]}^m,x\mapsto \widehat{\beta }(f(x)),$$ that is, $({\pi }_i\circ \widehat{\alpha })(x)=({\pi }_i\circ \widehat{\beta })(f(x))$ for $x\in X$ and $i=1,2,\dots ,m$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ .

Let $\widehat{\beta }$ be an m-polar $(\in ,\in )$ -fuzzy q-ideal of $Y$ . For any $x\in X$ , we have $$({\pi }_i\circ \widehat{\alpha })(x)=({\pi }_i\circ \widehat{\beta })(f(x))⩽({\pi }_i\circ \widehat{\beta })(0)=({\pi }_i\circ \widehat{\beta })(f(0))=({\pi }_i\circ \widehat{\alpha })(0)$$ for all $i=1,2,\dots ,m$ . Let $x,y,z\in X$ . Then $$({\pi }_i\circ \widehat{\alpha })(x*z)=({\pi }_i\circ \widehat{\beta })(f(x*z))=({\pi }_i\circ \widehat{\beta })(f(x)*f(y))⩾\inf \{({\pi }_i\circ \widehat{\beta })(f(x)*(f(y)*f(z))),({\pi }_i\circ \widehat{\beta })(f(y))\}=\inf \{({\pi }_i\circ \widehat{\beta })(f(x*(y*z))),({\pi }_i\circ \widehat{\beta })(f(y))\}=\inf \{({\pi }_i\circ \widehat{\alpha })(x*(y*z)),({\pi }_i\circ \widehat{\alpha })(y)\}$$ for all $i=1,2,\dots ,m$ . Therefore $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ . □

Let $f:X\to Y$ be an epimorphism of BCI-algebras. If $\widehat{\alpha }$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $X$ such that $$(\forall T\subseteq X)(\exists x_0\in T)\left(\widehat{\alpha }(x_0)={\displaystyle \underset{a\in T}{\sup }}\widehat{\alpha }(a)\right),$$ then the image $\widehat{\beta }$ of $\widehat{\alpha }$ under f which is defined by $$\widehat{\beta }:Y\to {[0,1]}^m,y\mapsto {\displaystyle \underset{x\in f^{-1}(y)}{\sup }}\widehat{\alpha }(x)$$ is an m-polar $(\in ,\in )$ -fuzzy q-ideal of $Y$ .

Since $0\in f^{-1}(0)$ , we have $$\widehat{\beta }(0)={\displaystyle \underset{x\in f^{-1}(0)}{\sup }}\widehat{\alpha }(x)=\widehat{\alpha }(0)⩾\widehat{\alpha }(x)$$ for all $x\in X$ , and so $$\widehat{\beta }(0)={\displaystyle \underset{x\in f^{-1}(y)}{\sup }}\widehat{\alpha }(x)=\widehat{\beta }(y)$$ for all $y\in Y$ . For any $a,b,c\in Y$ , let $x_0\in f^{-1}(a),y_0\in f^{-1}(b)$ and $z_0\in f^{-1}(c)$ satisfying $\widehat{\alpha }(x_0*z_0)=\underset{u\in f^{-1}(a*c)}{\sup }\widehat{\alpha }(u)$ , $\widehat{\alpha }(y_0)=\underset{u\in f^{-1}(b)}{\sup }\widehat{\alpha }(u)$ and $\widehat{\alpha }(x_0*(y_0*z_0))=\underset{u\in f^{-1}(a*(b*c))}{\sup }\widehat{\alpha }(u)$ . Then $$\widehat{\beta }(a*c)={\displaystyle \underset{u\in f^{-1}(a*c)}{\sup }}\widehat{\alpha }(u)=\widehat{\alpha }(x_0*z_0)⩾\inf \{\widehat{\alpha }(x_0*(y_0*z_0)),\widehat{\alpha }(y_0)\}=\inf \left\{{\displaystyle \underset{u\in f^{-1}(a*(b*c))}{\sup }}\widehat{\alpha }(u),{\displaystyle \underset{u\in f^{-1}(b)}{\sup }}\widehat{\alpha }(u)\}\right\}=\inf \left\{\widehat{\beta }(a*(b*c)),\widehat{\beta }(b)\right\}\text{.}$$