A New Interpretation of Multi-Polarity Fuzziness Subalgebras of BCK/BCI-Algebras

Abstract

In this paper, as a further generalization of m-polar fuzziness subalgebras, we present the conception of an m-polar $(\omega ,\theta )$-fuzzy subalgebra in BCK/BCI-algebras and discuss some interesting properties of it. We define an m-polar $({\in }_{\hat{\gamma }},{\in }_{\hat{\gamma }}\vee q_{\hat{\delta }})$-fuzzy subalgebras and explore some of its significant attributes. Certain features of m-polar $({\in }_{\hat{\gamma }},{\in }_{\hat{\gamma }}\vee q_{\hat{\delta }})$-fuzzy subalgebras are established.

Keywords:

• m-polar fuzzy subalgebra
• m-polar $(\omega ,\theta )$-fuzzy subalgebra
• m-polar $({\in }_{\hat{\gamma }},{\in }_{\hat{\gamma }}\vee q_{\hat{\delta }})$-fuzzy subalgebra

2010 Mathematics Subject Classifications:

• 06F35
• 03B52
• 03B05

1. Introduction

In 1966, Imai and Iséki created two algebraic structures known as ‘BCK/BCI algebras’ [1,2]. Since then, the concept and the generalizations have been researched in a variety of ways. Some classifications of BCK/BCI-algebras are considered in [3–6]. The theory of fuzzy sets, which was initiated by Zadeh in his pioneering paper [7] in 1965, was applied by many researchers to discuss the phenomena of uncertainty and vagueness in real life problems. In 1994, Zhang [8] gave a remarkable generalization of Zadeh's fuzzy set and presented the bipolar fuzzy sets. The various features of bipolar information in algebraic structures are considered in [9–12]. To deal with multi information in the fuzzy set theory, Chen et al. [13] gave the notation of multi polar valued function and constructed m-polar fuzzy (m-$\mathcal{F}$) sets. After the introduction of m-$\mathcal{F}$ sets by Chen et al., m-$\mathcal{F}$ set theory has become an active area of research in various fields such as lie algebras [14,15], ordered semihypergroups [16], subgroups [17], BCK/BCI-algebras [18–20]. For more studies related to BCK and BCI algebraic structures, see [21].

Fuzzy groups were introduced in 1971 due to Rosenfeld [22] while Bhakat and Das worked on fuzzy group of type $(\in ,\in \vee q)$ based on point fuzzy set in groups [23]. Jun, Muhiuddin and Al-Roqi worked on fuzzy subalgebra of type $(\alpha ,\text{β})$ based on point fuzzy set in BCK/BCI-algebras [24–26]. In this side, Ibrara et al. [27], Dudek et al. [28] and Narayanan et al. [29] extended [23] to semigroups, hemirings and near-rings, respectively. Al-Masarwah and Ahmad [30] worked on subalgebras of type $(\alpha ,\text{β})$ based on m-$\mathcal{F}$ points in BCK/BCI-algebras. Also, they worked on fuzzy ideals of type $(\in ,\in \vee q)$ based on point m-$\mathcal{F}$ in [31]. In [32], Ma et al. presented the idea of $({\in }_{\gamma },{\in }_{\gamma }\vee q_{\delta })$-fuzzy ideals in BCI-algebras. In [33], Jana and Pal presented $({\in }_{\gamma },{\in }_{\gamma }\vee q_{\delta })$ fuzzy soft BCI-algebras. Zulfiqar and Shabir [34], worked on fuzzy sub-commutative ideals of type $({\in }_{\gamma },{\in }_{\gamma }\vee q_{\delta })$ based on point fuzzy set in BCI-algebras. In Γ-hyperrings, Zhan [35] presented $({\in }_{\gamma },{\in }_{\gamma }\vee q_{\delta })$-fuzzy soft Γ-hyperideals while Zulfiqar worked on fuzzy fantastic ideal of type $({\overline{\in }}_{\gamma },{\overline{\in }}_{\gamma }\vee {\overline{q}}_{\delta })$ based on point fuzzy set in BCH–algebras [36].

Inspired by previous works in this direction, in this paper, we combine the m-$\mathcal{F}$ sets with BCK/BCI-subalgebras to broaden application fields of theory of fuzzy sets and provide more ways to study fuzzy algebras. We present a new kind of generalized m-$\mathcal{F}$ subalgebras of a BCK/BCI-algebra called, an m-polar $(\omega ,\theta )$-fuzzy subalgebra. We present some interesting properties of an m-polar $(\omega ,\theta )$-fuzzy subalgebra. Next, we define the notion of m-polar $({\in }_{\hat{\gamma }},{\in }_{\hat{\gamma }}\vee q_{\hat{\delta }})$-fuzzy subalgebras and investigate some related properties. Finally, some characterization theorems of m-polar $({\in }_{\hat{\gamma }},{\in }_{\hat{\gamma }}\vee q_{\hat{\delta }})$-fuzzy subalgebras are established.

2. Preliminaries

In this segment, we will go over some of the central tenets of BCK/BCI-algebras that will help us understand the paper better.

In each part of the paper,

• We use the ‘BCK/BCI-algebra ℧’ as the domain of discourse except where otherwise noted.

• We use the abbreviation m-$\mathcal{P}-$ $(\omega ,\theta )\mathcal{F}\mathcal{S}\left(s\right)$ instead of m-polar $(\omega ,\theta )$-fuzzy subalgebra(s), where ω and θ represent one of the symbols ${\in }_{\tilde{\gamma }},$ ${\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},$ $q_{\tilde{\delta }}$ or ${\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }}.$

Consider the following axioms, $\mathrm{\forall }\kappa ,\tau ,\upsilon \in \text{℧}:$

(K1)	$\left(\right(\kappa \divideontimes \tau )\divideontimes (\kappa \divideontimes \upsilon \left)\right)\divideontimes (\upsilon \divideontimes \tau )=0,$
(K2)	$(\kappa \divideontimes (\kappa \divideontimes \tau \left)\right)\divideontimes \tau =0,$
(K3)	$\kappa \divideontimes \kappa =0,$
(K4)	$0\divideontimes \kappa =0,$
(K5)	$\kappa \divideontimes \tau =0$ and $\tau \divideontimes \kappa =0$ imply $\kappa =\tau .$

Imai and co-workers [1,2], presented the following algebraic structure, and they called it a BCK-algebra:

Definition 2.1

An algebraic structure $(\text{℧};\divideontimes ,0)$ satisfying the above axioms (K1)–(K5) is called a BCK-algebra.

Definition 2.2

In a BCK/BCI-algebra ℧, for any $\kappa ,\tau \in \text{℧}:$

• $\kappa \divideontimes 0=\kappa$ holds $\mathrm{\forall }\kappa \in \text{℧},$

• $\kappa \divideontimes \tau =0$⇔$\kappa \le \tau$, where ‘≤’ is a partial ordering on ℧,

A subset $M\ne \varphi$ of ℧ is a subalgebra if $\kappa \divideontimes \tau \in M$ $\mathrm{\forall }\kappa ,\tau \in M.$

Definition 2.3

[13]

A mapping $\tilde{O}:\text{℧}\to [0,1{]}^m$ is a m-$\mathcal{F}$ set of ℧, where for any $\kappa \in \text{℧},$ $\hat{O}\left(\kappa \right)=\left({\hat{O}}^1\right(\kappa ),{\tilde{O}}^2(\kappa ),\dots ,{\tilde{O}}^m(\kappa \left)\right)$ and ${\tilde{O}}^j\left(\kappa \right)\in [0,1],$ for $j=1,2,\dots ,m.$

Definition 2.4

[18]

An m-$\mathcal{F}$ set $\tilde{O}$ of a BCK-algebra ℧ is an m-$\mathcal{F}$ subalgebra if, $\mathrm{\forall }\kappa ,\tau \in \text{℧}$ and $i=1,2,\dots ,m,$ $\tilde{O}(\kappa \divideontimes \tau )\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau \left)\right\}.$ That is, ${\tilde{O}}^j(\kappa \divideontimes \tau )\ge inf\left\{{\tilde{O}}^j\right(\kappa ),{\tilde{O}}^j(\tau \left)\right\}.$

Example 2.1

Let $\text{℧}=\{0,u,\nu ,ϵ\}$ be a set with the -operation given by the below Table:

Table

	0	u	ν	ε
0	0	0	0	0
u	u	0	0	u
ν	ν	ν	0	ν
ε	ε	ε	ε	0

Then, $(\text{℧};\divideontimes ,0)$ is a BCK-algebra [6]. Let $\tilde{O}$ be an m-$\mathcal{F}$ set on ℧ defined by: $\tilde{O}=\left\{\begin{array}{ll}〈0,& (0.6,0.67,0.77,0.56)〉,\\ 〈u,& (0.5,0.23,0.47,0.32)〉,\\ 〈\nu ,& (0.3,0.54,0.56,0.32)〉,\\ 〈ϵ,& (0.3,0.43,0.38,0.56)〉.\end{array}\right\}$ Then, $\tilde{O}$ is an m-$\mathcal{F}$ subalgebra of ℧.

Theorem 2.5

[18]

An m-$\mathcal{F}$ set $\tilde{O}$ of ℧ is an m-$\mathcal{F}$ subalgebra of ℧ ⇔ for any $\tilde{\pi }\in (0,1{]}^m,$ $\tilde{\pi }$-cut subset ${\tilde{O}}_{\tilde{\pi }}=\{\kappa \in \text{℧}\mid \tilde{O}(\kappa )\ge \tilde{\pi }\}$ is a subalgebra of ℧.

Proof.

The proof is obvious.

An m-$\mathcal{F}$ set $\tilde{O}$ of ℧ having the form $\tilde{O}\left(\tau \right)=\{\begin{array}{ll}\tilde{\pi }\in (0,1{]}^m,& \text{if }\tau =\kappa \\ \tilde{0},& \text{if }\tau \ne \kappa .\end{array}$ is an m-$\mathcal{F}$ point with support ℧ and value $\tilde{\pi }$ [18], and is symbolized by ${\kappa }_{\tilde{\pi }}.$

An m-$\mathcal{F}$ point ${\kappa }_{\tilde{\pi }}\in \tilde{O}$ if $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }.$ That is $p_i\circ \tilde{O}\left(\kappa \right)\ge {\pi }_i$ $\mathrm{\forall }i=1,2,\dots ,m.$ Also, ${\kappa }_{\tilde{\pi }}q\tilde{O}$ if $\tilde{O}\left(\kappa \right)+\tilde{\pi }>\hat{1}.$ That is, $p_i\circ \tilde{O}\left(\kappa \right)+{\pi }_i>\hat{1}$ $\mathrm{\forall }i=1,2,\dots ,m$).

By ${\kappa }_{\tilde{\pi }}\in \vee q\tilde{O}$ (resp., ${\kappa }_{\tilde{\pi }}\in \wedge q\tilde{O}$) we mean that ${\kappa }_{\tilde{\pi }}\in \tilde{O}$ or ${\kappa }_{\tilde{\pi }}q\tilde{O}$ (resp., ${\kappa }_{\tilde{\pi }}\in \tilde{O}$ and ${\kappa }_{\tilde{\pi }}q\tilde{O}).$ If $\varphi \ne C\subseteq \text{℧},$ then the m-polar characteristic function of C, say ${\hat{\chi }}_C,$ where ${\hat{\chi }}_C=\{\begin{array}{ll}\hat{1}=(1,1,\dots ,1),& \text{if }\kappa \in C\\ \tilde{0}=(0,0,\dots ,0),& \text{if }\kappa \notin C.\end{array}$ Clearly, the m-polar characteristic function is an m-$\mathcal{F}$ subset of ℧.

3. m-Polar $(\omega ,\theta )$-Fuzzy Subalgebras

In the section, we present the conception of an m-$\mathcal{P}-$ $(\omega ,\theta )\mathcal{F}\mathcal{S}$ in BCK/BCI-algebras and discuss some interesting properties of it. Let ω and θ represent one of the symbols ${\in }_{\tilde{\gamma }},$ ${\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},$ $q_{\tilde{\delta }}$ or ${\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }}$ except where otherwise noted.

Throughout this paper, Let $\tilde{\gamma },\tilde{\delta }\in [0,1{]}^m,$ where $\tilde{\gamma }=({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_m)<\tilde{\delta }=({\delta }_1,{\delta }_2,\dots ,{\delta }_m).$ For an m-$\mathcal{F}$ point ${\kappa }_{\tilde{\pi }}$ and an m-$\mathcal{F}$ set $\tilde{O}$ of ℧. We say that:

1. ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ if $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }.$

2. ${\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}$ if $\tilde{O}\left(\kappa \right)+\tilde{\pi }>2\tilde{\delta }.$

3. ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}$ if ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ or ${\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}.$

4. ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }}\tilde{O}$ if ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ and ${\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}.$

5. ${\kappa }_{\tilde{\pi }}\overline{\omega }\tilde{O}$ does not hold for $\omega =\{{\in }_{\tilde{\gamma }},q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},$ ${\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }}\}$

Definition 3.1

An m-$\mathcal{F}$ set $\tilde{O}$ of ℧ is called an m-$\mathcal{P}-$ $(\omega ,\theta )\mathcal{F}\mathcal{S}$ of ℧, where $\omega \ne {\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }},$ if $\mathrm{\forall }\tilde{\gamma }<\tilde{\pi },\tilde{\varpi }\le \hat{1}$ and $\kappa ,\tau \in \text{℧},$ ${\kappa }_{\tilde{\pi }}\omega \tilde{O},{\tau }_{\tilde{\varpi }}\omega \tilde{O}\Rightarrow (\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}\theta \tilde{O}$

Let $\tilde{O}$ be an m-$\mathcal{F}$ set of ℧ such that $\tilde{O}\left(\kappa \right)\le \tilde{\delta }$ $\mathrm{\forall }\kappa \in \text{℧}.$ Let $\kappa \in \text{℧}$ and $\tilde{\gamma }<\tilde{\pi }\le \hat{1}$ be such that ${\kappa }_{\tilde{\pi }}\in \wedge q_{\tilde{\delta }}\tilde{O}.$ Then, $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }$ and $\tilde{O}\left(\kappa \right)+\tilde{\pi }>2\tilde{\delta }.$ Thus, $2\tilde{\delta }<\tilde{O}\left(\kappa \right)+\tilde{\pi }\le \tilde{O}\left(\kappa \right)+\tilde{O}\left(\kappa \right)=2\tilde{O}\left(\kappa \right),$ so that $\tilde{O}\left(\kappa \right)>\tilde{\delta }.$ Hence, $\{{\kappa }_{\tilde{\pi }}\mid {\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }}\tilde{O}\}=\varphi .$ Therefore, the case $\omega ={\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }}$ in the above definition is omitted.

Example 3.1

Consider the BCK-algebra $(\text{℧};\divideontimes ,0)$ and an m-$\mathcal{F}$ set $\tilde{O}$ presented in Example 2.1. It is clear that by Definition 4.1 that $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{(0.2,0.1,\mathrm{0.3.},0.2)},{\in }_{(0.2,0.1,\mathrm{0.3.},0.2)}\vee q_{(0.61,0.68,0.78,0.57)})\mathcal{F}\mathcal{S}$ of ℧.

Theorem 3.2

Let $\tilde{\gamma }+\hat{1}=2\tilde{\delta }$ and $\tilde{O}$ be an m-$\mathcal{P}-$ $(\omega ,\theta )\mathcal{F}\mathcal{S}$ of ℧. Then, the set ${\tilde{O}}_{\tilde{\gamma }}=\{\kappa \in \text{℧}\mid \tilde{O}(\kappa )>\tilde{\gamma }\}$ is a subalgebra of ℧.

Proof.

Let $\kappa ,\tau \in \text{℧}$ be such that $\kappa ,\tau \in {\tilde{O}}_{\tilde{\gamma }}.$ Then, $\tilde{O}\left(\kappa \right)>\tilde{\gamma }$ and $\tilde{O}\left(\tau \right)>\tilde{\gamma }.$ Assume that $\tilde{O}(\kappa \divideontimes \tau )\le \tilde{\gamma }.$ If $\omega \in \{{\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\},$ then ${\kappa }_{\tilde{O}\left(\kappa \right)}\omega \tilde{O}$ and ${\tau }_{\tilde{O}\left(\tau \right)}\omega \tilde{O},$ but $\tilde{O}(\kappa \divideontimes \tau )\le \tilde{\gamma }<inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau \left)\right\}$ and $\tilde{O}(\kappa \divideontimes \tau )+inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau \left)\right\}\le \tilde{\gamma }+\hat{1}=2\tilde{\delta }.$ So, $(\kappa \divideontimes \tau {)}_{inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau \left)\right\}}\overline{\theta }\tilde{O}$ $\mathrm{\forall }\theta \in \{{\in }_{\tilde{\gamma }},q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\wedge q_{\tilde{\delta }}\},$ a contradiction. Hence, $\tilde{O}(\kappa \divideontimes \tau )>\tilde{\gamma },$ i.e. $\kappa \divideontimes \tau \in {\tilde{O}}_{\tilde{\gamma }}.$

Also, $\tilde{O}(\kappa \divideontimes \tau )+\hat{1}>\tilde{\gamma }+\hat{1}=2\tilde{\delta }.$ We get $(\kappa \divideontimes \tau {)}_{\hat{1}}{q}_{\tilde{\delta }}\tilde{O},$ but $\tilde{O}(\kappa \divideontimes \tau )\le \tilde{\gamma },$ so $(\kappa \divideontimes \tau {)}_{\hat{1}}\overline{{\in }_{\tilde{\gamma }}}\tilde{O}$ and $\tilde{O}(\kappa \divideontimes \tau )+\hat{1}\le \tilde{\gamma }+\hat{1}=2\tilde{\delta },$ so $(\kappa \divideontimes \tau {)}_{\hat{1}}\overline{q_{\tilde{\delta }}}\tilde{O},$ a contradiction. Thus, $\tilde{O}(\kappa \divideontimes \tau )>\tilde{\gamma },$ that is, $\kappa \divideontimes \tau \in {\tilde{O}}_{\tilde{\gamma }}.$ Therefore, ${\tilde{O}}_{\tilde{\gamma }}$ is a subalgebra of ℧.

Theorem 3.3

Let $\tilde{\gamma }+\hat{1}=2\tilde{\delta }$ and $\varphi \ne C\subseteq \text{℧}.$ Then, C is a subalgebra of ℧ ⇔ the m-$\mathcal{F}$ subset $\tilde{O}$ of ℧ defined by

1. $\tilde{O}\left(\kappa \right)\ge \tilde{\delta },\mathrm{\forall }\kappa \in C,$

2. $\tilde{O}\left(\kappa \right)\le \tilde{\gamma },\mathrm{\forall }\kappa \notin C.$

is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Proof.

Let C be a subalgebra of ℧. Let $\kappa ,\tau \in \text{℧}$ and $\tilde{\gamma }<\tilde{\pi },\tilde{\varpi }\le \hat{1}$ such that ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{\in }_{\tilde{\gamma }}\tilde{O}.$ Then $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }$ and $\tilde{O}\left(\tau \right)\ge \tilde{\varpi }>\tilde{\gamma }.$ Thus, $\kappa ,\tau \in C$ and so $\kappa \divideontimes \tau \in C,$ that is, $\tilde{O}(\kappa \divideontimes \tau )\ge \tilde{\delta }.$ If $inf\{\tilde{\pi },\tilde{\varpi }\}\le \tilde{\delta },$ then $\tilde{O}(\kappa \divideontimes \tau )\ge \tilde{\delta }\ge inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\gamma }.$ Hence, $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\tilde{O}.$ If $inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\delta },$ then $\tilde{O}(\kappa \divideontimes \tau )+inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\delta }+\tilde{\delta }=2\tilde{\delta }$ and so $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{q}_{\tilde{\delta }}\tilde{O}.$ Thus, $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}.$ Hence, $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Conversely, assume that $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧. Then $C={\tilde{O}}_{\tilde{\gamma }}.$ Thus, by Theorem 3.2, C is a subalgebra of ℧.

Corollary 3.4

Let $\tilde{\gamma }+\hat{1}=2\tilde{\delta }$ and $\varphi \ne C\subseteq \text{℧}.$ Then, C is a subalgebra of ℧ ⇔${\hat{\chi }}_C$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Theorem 3.5

Let $\tilde{\gamma }+\hat{1}=2\tilde{\delta }$ and $\varphi \ne C\subseteq \text{℧}.$ Then, C is a subalgebra of ℧ ⇔ the m-$\mathcal{F}$ subset $\tilde{O}$ of ℧ defined by

1. $\tilde{O}\left(\kappa \right)\ge \tilde{\delta },\mathrm{\forall }\kappa \in C,$

2. $\tilde{O}\left(\kappa \right)\le \tilde{\gamma },\mathrm{\forall }\kappa \notin C.$

is an m-$\mathcal{P}-$ $(q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Proof.

Let C be a subalgebra of ℧. Let $\kappa ,\tau \in \text{℧}$ and $\tilde{\gamma }<\tilde{\pi },\tilde{\varpi }\le \hat{1}$ such that ${\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{q}_{\tilde{\delta }}\tilde{O}.$ Then $\tilde{O}\left(\kappa \right)+\tilde{\pi }>2\tilde{\delta }$ and $\tilde{O}\left(\tau \right)+\tilde{\varpi }>2\tilde{\delta },$ which implies that $\tilde{O}\left(\kappa \right)>2\tilde{\delta }-\tilde{\pi }\ge 2\tilde{\delta }-\hat{1}=\tilde{\gamma }$ and $\tilde{O}\left(\tau \right)>2\tilde{\delta }-\tilde{\varpi }\ge 2\tilde{\delta }-\hat{1}=\tilde{\gamma }.$ Thus, by definition $x,y\in C$ and so $\kappa \divideontimes \tau \in C,$ which implies that $\tilde{O}(\kappa \divideontimes \tau )\ge \tilde{\delta }.$ Now, if $inf\{\tilde{\pi },\tilde{\varpi }\}\le \tilde{\delta },$ then $\tilde{O}(\kappa \divideontimes \tau )\ge \tilde{\delta }\ge inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\gamma }.$ Hence, $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\tilde{O}.$ If $inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\delta },$ then $\tilde{O}(\kappa \divideontimes \tau )+inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\delta }+\tilde{\delta }=2\tilde{\delta }$ and so $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{q}_{\tilde{\delta }}\tilde{O}.$ Therefore, $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}\in \vee q_{\tilde{\delta }}\tilde{O}.$ Hence, $\tilde{O}$ is an m-$\mathcal{P}-$ $(q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Conversely, assume that $\tilde{O}$ m-polar m-$\mathcal{P}-$ $(q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧. Then, $C={\tilde{O}}_{\tilde{\gamma }}.$ Thus, by Theorem 3.2, C is a subalgebra of ℧.

Corollary 3.6

Let $\tilde{\gamma }+\hat{1}=2\tilde{\delta }$ and $\varphi \ne C\subseteq \text{℧}.$ Then, C is a subalgebra of ℧ ⇔${\hat{\chi }}_C$ is an m-$\mathcal{P}-$ $(q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Theorem 3.7

Let $\tilde{\gamma }+\hat{1}=2\tilde{\delta }$ and $\varphi \ne C\subseteq \text{℧}.$ Then, C is a subalgebra of ℧ ⇔ the m-$\mathcal{F}$ subset $\tilde{O}$ of ℧ defined by

1. $\tilde{O}\left(\kappa \right)\ge \tilde{\delta },\mathrm{\forall }\kappa \in C,$

2. $\tilde{O}\left(\kappa \right)\le \tilde{\gamma },\mathrm{\forall }\kappa \notin C.$

is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Proof.

Let C be a subalgebra of ℧. Let $\kappa ,\tau \in \text{℧}$ and $\tilde{\gamma }<\tilde{\pi },\tilde{\varpi }\le \hat{1}$ such that ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O},$ which implies that ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ or ${\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{\in }_{\tilde{\gamma }}\tilde{O}$ or ${\tau }_{\tilde{\varpi }}{q}_{\tilde{\delta }}\tilde{O}.$ If ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{q}_{\tilde{\delta }}\tilde{O},$ then $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }$ and $\tilde{O}\left(\tau \right)+\tilde{\varpi }>2\tilde{\delta }.$ This implies that $\tilde{O}\left(\tau \right)>2\tilde{\delta }-\tilde{\varpi }\ge 2\tilde{\delta }-\hat{1}=\tilde{\gamma }.$ Thus, $x,y\in C$ and so $\kappa \divideontimes \tau \in C.$ Analogous as in Theorems 3.3 and 3.5, we obtain $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}.$ Hence, $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧. The other cases can be considered similar to this case.

Conversely, assume that $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧. Then, $C={\tilde{O}}_{\tilde{\gamma }}.$ Thus, by Theorem 3.2, C is a subalgebra of ℧.

Corollary 3.8

Let $\tilde{\gamma }+\hat{1}=2\tilde{\delta }$ and $\varphi \ne C\subseteq \text{℧}.$ Then, C is a subalgebra of ℧ ⇔${\hat{\chi }}_C$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Theorem 3.9

Every m-$\mathcal{P}-$ $(q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Proof.

Let $\tilde{O}$ be an m-$\mathcal{P}-$ $(q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧. Let $\kappa ,\tau \in \text{℧}$ and $\tilde{\gamma }<\tilde{\pi },\tilde{\varpi }\le \hat{1}$ such that ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{\in }_{\tilde{\delta }}\tilde{O}.$ Then, $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{a}\mathrm{n}\mathrm{d}\tilde{O}\left(\tau \right)\ge \tilde{\varpi }>\tilde{\gamma }.$ Suppose that $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}\overline{{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}}\tilde{O}.$ Then $\tilde{O}\left(\kappa \divideontimes \tau \right)<inf\{\tilde{\pi },\tilde{\varpi }\}$ and $\tilde{O}\left(\kappa \divideontimes \tau \right)+inf\{\tilde{\pi },\tilde{\varpi }\}\le 2\tilde{\delta }.$ This implies that $\tilde{O}(\kappa \divideontimes \tau )+\tilde{O}(\kappa \divideontimes \tau )<\tilde{O}(\kappa \divideontimes \tau )+inf\{\tilde{\pi },\tilde{\varpi }\}\le 2\tilde{\delta }.$ This implies that $\tilde{O}(\kappa \divideontimes \tau )<\tilde{\delta }.$ Now, $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}<inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}.$ Choose $\tilde{\gamma }<\hat{r}\le \hat{1}$ such that $2\tilde{\delta }-sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge \hat{r}>2\tilde{\delta }-inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\},$ that is, $inf\{2\tilde{\delta }-\tilde{O}(\kappa \divideontimes \tau ),2\tilde{\delta }-\tilde{\gamma }\}\ge sup\{2\tilde{\delta }-\tilde{O}(\kappa ),2\tilde{\delta }-\tilde{O}(\tau ),\tilde{\delta }\}.$ This implies that $\hat{r}>2\tilde{\delta }-\tilde{O}\left(\kappa \right),\hat{r}>2\tilde{\delta }-\tilde{O}\left(\tau \right)$ and $2\tilde{\delta }-\tilde{O}(\kappa \divideontimes \tau )>\hat{r},$ implies that $\tilde{O}\left(\kappa \right)+\hat{r}>2\tilde{\delta },\tilde{O}\left(\tau \right)+\hat{r}>2\tilde{\delta }$ and $\tilde{O}(\kappa \divideontimes \tau )+\hat{r}<2\tilde{\delta }.$ Thus, $x_{\hat{r}}{q}_{\tilde{\delta }}\tilde{O},y_{\hat{r}}{q}_{\tilde{\delta }}\tilde{O}$ but $(\kappa \divideontimes \tau {)}_{\hat{r}}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O},$ which is a contradiction. Hence, $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Theorem 3.10

Every m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Proof.

The proof follows from the fact that if ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O},$ then ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}.$

Theorem 3.11

Every m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }})\mathcal{F}\mathcal{S}$ of ℧ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Proof.

Let $\tilde{O}$ be an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }})\mathcal{F}\mathcal{S}$ of ℧. Let $\kappa ,\tau \in \text{℧}$ and $\tilde{\gamma }<\tilde{\pi },\tilde{\varpi }\le \hat{1},$ so that the m-$\mathcal{F}$ points ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{\in }_{\tilde{\gamma }}\tilde{O}$, respectively. Then since ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ and ${\tau }_{\tilde{\varpi }}{\in }_{\tilde{\gamma }}\tilde{O},$ $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}.$ This shows that $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

4. m-Polar $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})$-Fuzzy Subalgebras

This section introduces the concept of m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}s$ and delves into some of its key features.

Definition 4.1

An m-$\mathcal{F}$ set $\tilde{O}$ of ℧ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ if it meets the condition (A), where

1. ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O},{\tau }_{\tilde{\varpi }}{\in }_{\tilde{\gamma }}\tilde{O}\Rightarrow (\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O},$ $\mathrm{\forall }\tilde{\gamma }<\tilde{\pi },\tilde{\varpi }\le \hat{1}$ and $\kappa ,\tau \in \text{℧}.$

Theorem 4.2

For an m-$\mathcal{F}$ set $\tilde{O}$ of ℧, (A) in Definition 4.1 is an equivalent to (B), where

1. $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}$ $\mathrm{\forall }\kappa ,\tau \in \text{℧}.$

Proof.

(A) ⇒ (B). Assume that (B) does not hold. Then, $\mathrm{\exists }\kappa ,\tau \in \text{℧}$ such that $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}<inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}.$ Then, $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}<\tilde{\pi }\le inf\left\{\tilde{O}\right(\kappa ),$ $\tilde{O}\left(\tau \right),\tilde{\delta }\}$ for some $\tilde{\gamma }<\tilde{\pi }\le \tilde{\delta }.$ Thus ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}$ and ${\tau }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O},$ but $(\kappa \divideontimes \tau {)}_{\tilde{\pi }}\overline{{\in }_{\tilde{\gamma }}\vee {\in }_{\tilde{\gamma }}}\tilde{O}.$ A contradiction and therefore, $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}$ $\mathrm{\forall }\kappa ,\tau \in \text{℧}.$

(B) ⇒ (A). Let ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O},{\tau }_{\tilde{\varpi }}{\in }_{\tilde{\gamma }}\tilde{O}.$ Then, $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }$ and $\tilde{O}\left(\tau \right)\ge \tilde{\varpi }>\tilde{\gamma }.$ If $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\tilde{O},$ then (A) is hold.

If $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\overline{\in }}_{\tilde{\gamma }}\tilde{O},$ then $Q(\kappa \divideontimes \tau )<inf\{\tilde{\pi },\tilde{\varpi }\}$. Since $\begin{array}{rl}sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}& \ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}\\ & \ge inf\{\tilde{\pi },\tilde{\varpi },\tilde{\delta }\}.\end{array}$ Implies $\tilde{O}(\kappa \divideontimes \tau )\ge \tilde{\delta }$ and $inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\delta }.$ Thus, $\tilde{O}(\kappa \divideontimes \tau )+inf\{\tilde{\pi },\tilde{\varpi }\}>\tilde{\delta }+\tilde{\delta }=2\tilde{\delta },$ implies $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{q}_{\tilde{\delta }}\tilde{O}.$ Hence, $(\kappa \divideontimes \tau {)}_{inf\{\tilde{\pi },\tilde{\varpi }\}}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}.$

Remark 4.3

An m-$\mathcal{F}$ set $\tilde{O}$ of ℧ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ if it satisfies (B).

Theorem 4.4

The intersection of any family of m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}s$ of ℧ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

Proof.

Let $\{{\tilde{O}}_i{\}}_{i\in I}$ be a family of m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}s$ of ℧ and $\kappa ,\tau \in \text{℧}.$ Then, $sup\left\{{\tilde{O}}_i\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{{\tilde{O}}_i\right(\kappa ),{\tilde{O}}_i(\tau ),\tilde{\delta }\}$ $\mathrm{\forall }i\in I.$ Thus, $\begin{array}{rl}sup\left\{({\wedge }_{i\in I}{\tilde{O}}_i)\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}& =sup\left\{{\wedge }_{i\in I}{\tilde{O}}_i\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\\ & \ge {\wedge }_{i\in I}(inf\left\{{\tilde{O}}_i\right(\kappa ),{\tilde{O}}_i(\tau ),\tilde{\delta }\})\\ & =inf\left\{({\wedge }_{i\in I}{\tilde{O}}_i)\right(\kappa ),({\wedge }_{i\in I}{\tilde{O}}_i)(\tau ),\tilde{\delta }\}.\end{array}$ Therefore, $sup\left\{\right({\wedge }_{i\in I}{\tilde{O}}_i\left)\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\right({\wedge }_{i\in I}{\tilde{O}}_i\left)\right(\kappa ),$ $\left({\wedge }_{i\in I}{\tilde{O}}_i\right)\left(\tau \right),\tilde{\delta }\}.$ Hence, ${\wedge }_{i\in I}{\tilde{O}}_i$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

For any m-$\mathcal{F}$ set $\tilde{O}$ of ℧ and $\tilde{\pi }\in [0,1{]}^m,$ we denote $\begin{array}{rl}& {\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}=\{\kappa \in \text{℧}\mid {\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}\},\\ & 〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }}=\{\kappa \in \text{℧}\mid {\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}\},\end{array}$ and $[\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }}=\{\kappa \in \text{℧}\mid {\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O}\}.$ It is clear that $[\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }}={\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}\cup 〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }}.$ The following theorems describe the relationship between m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}s$ and the crisp subalgebras in ℧.

Theorem 4.5

Let $\tilde{O}$ be an m-$\mathcal{F}$ set of ℧. Then, $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ ⇔${\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}\ne \varphi$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\gamma }<\tilde{\pi }\le \tilde{\delta }.$

Proof.

Let $\tilde{O}$ be an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ and let $\kappa ,\tau \in {\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}$ for $\tilde{\gamma }<\tilde{\pi }\le \tilde{\delta }.$ Then $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{a}\mathrm{n}\mathrm{d}\tilde{O}\left(\tau \right)\ge \tilde{\pi }>\tilde{\gamma }.$ Thus, we have $\begin{array}{rl}sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}& \ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}\\ & \ge inf\{\tilde{\pi },\tilde{\pi },\tilde{\delta }\}\\ & =inf\{\tilde{\pi },\tilde{\delta }\}\\ & =\tilde{\pi }>\tilde{\gamma },\end{array}$ that is, $\tilde{O}(\kappa \divideontimes \tau )\ge \tilde{\pi },$ implies, $\kappa \divideontimes \tau \in {\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}.$ Thus, ${\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}$ is a subalgebra of ℧.

Conversely, suppose that ${\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\gamma }<\tilde{\pi }\le \tilde{\delta }.$ Assume $\kappa ,\tau \in \text{℧}$ such that $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}<inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}.$ select $\tilde{\gamma }<\hat{r}\le \tilde{\delta }$ such that $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}<\hat{r}=inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}.$ Then, ${\kappa }_{\hat{r}}{\in }_{\tilde{\gamma }}\tilde{O},{\tau }_{\hat{r}}{\in }_{\tilde{\gamma }}\tilde{O},$ but $(\kappa \divideontimes \tau {)}_{\hat{r}}\overline{{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}}\tilde{O}.$ Since ${\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }}$ is a subalgebra of ℧, so $\kappa \divideontimes \tau \in {\tilde{O}}_{\tilde{\pi }}^{\tilde{\gamma }},$ which is a contradiction. Hence, $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}$ $forall\kappa ,\tau \in \text{℧}.$ Therefore, $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

If we take $\tilde{\gamma }=\tilde{0}$ and $\tilde{\delta }=\widehat{0.5}$ in Theorem 4.5, we can deduce the following corollary:

Corollary 4.6

Let $\tilde{O}$ be an m-$\mathcal{F}$ set of ℧. Then $\tilde{O}$ is an m-$\mathcal{P}-$ $(\in ,\in \vee q)\mathcal{F}\mathcal{S}$ of ℧ ⇔${\tilde{O}}_{\tilde{\pi }}=\{\kappa \in \text{℧}\mid {\kappa }_{\tilde{\pi }}\in \tilde{O}\}\ne \varphi$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\pi }\in (0,0.5{]}^m.$

Theorem 4.7

Let $\tilde{O}$ be an m-$\mathcal{F}$ set of ℧. If $1+\tilde{\gamma }=2\tilde{\delta },$ then $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ ⇔$〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }}\ne \varphi$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\delta }<\tilde{\pi }\le \hat{1}.$

Proof.

Let $\tilde{O}$ be an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧. Let $\kappa ,\tau \in 〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }}.$ Then, ${\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}\mathrm{a}\mathrm{n}\mathrm{d}{\tau }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}.$ This implies that $\tilde{O}\left(\kappa \right)>2\tilde{\delta }-\tilde{\pi }\ge 2\tilde{\delta }-\hat{1}=\tilde{\gamma }$ and $\tilde{O}\left(\tau \right)>2\tilde{\delta }-\tilde{\pi }\ge 2\tilde{\delta }-\hat{1}=\tilde{\gamma }.$ By hypothesis $\begin{array}{rl}sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}& \ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}\\ & >inf\{2\tilde{\delta }-\tilde{\pi },2\tilde{\delta }-\tilde{\pi },\tilde{\delta }\}\\ & =inf\{2\tilde{\delta }-\tilde{\pi },\tilde{\delta }\}\\ & >2\tilde{\delta }-\tilde{\pi }.\end{array}$ So that $\tilde{O}(\kappa \divideontimes \tau )>2\tilde{\delta }-\tilde{\pi }.$ Thus, $\tilde{O}(\kappa \divideontimes \tau )+\tilde{\pi }>2\tilde{\delta }.$ This implies that $(\kappa \divideontimes \tau {)}_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O},$ i.e. $\kappa \divideontimes \tau \in 〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }}.$ Hence, $〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }}$ is a subalgebra of ℧.

Conversely, assume that $〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }}$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\delta }<\tilde{\pi }\le \hat{1}.$ Let $\kappa ,\tau \in \text{℧}$ such that $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\delta }\}<inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}.$ Then $2\tilde{\delta }-inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}<2\tilde{\delta }-sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}.$ This implies $sup\{2\tilde{\delta }-\tilde{O}(\kappa ),2\tilde{\delta }-\tilde{O}(\tau ),\tilde{\delta }\}<inf\{2\tilde{\delta }-\tilde{O}(\kappa \divideontimes \tau ),2\tilde{\delta }-\tilde{\gamma }\}.$ If we take $\tilde{\delta }<\tilde{\pi }\le \hat{1}$ such that $sup\{2\tilde{\delta }-\tilde{O}(\kappa ),2\tilde{\delta }-\tilde{O}(\tau ),\tilde{\delta }\}<\tilde{\pi }\le inf\{2\tilde{\delta }-\tilde{O}(\kappa \divideontimes \tau ),2\tilde{\delta }-\tilde{\gamma }\}.$ Then $2\tilde{\delta }-\tilde{O}\left(\kappa \right)<\tilde{\pi },2\tilde{\delta }-\tilde{O}\left(\tau \right)<\tilde{\pi }$ and $\tilde{\pi }\le 2\tilde{\delta }-\tilde{O}(\kappa \divideontimes \tau )$ implies that $\tilde{O}\left(\kappa \right)+\tilde{\pi }>2\tilde{\delta },\tilde{O}\left(\tau \right)+\tilde{\pi }>2\tilde{\delta }$ and $\tilde{O}(\kappa \divideontimes \tau )+\tilde{\pi }\le 2\tilde{\delta }.$ Thus, ${\kappa }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O},$ ${\tau }_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O},$ but $(\kappa \divideontimes \tau {)}_{\tilde{\pi }}{\overline{q}}_{\tilde{\delta }}\tilde{O},$ that is, ℧ and y are in $〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }},$ but $\kappa \divideontimes \tau \notin 〈\tilde{O}{〉}_{\tilde{\pi }}^{\tilde{\delta }},$ a contradiction. Hence, $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}$ $\mathrm{\forall }\kappa ,\tau \in \text{℧}.$ This shows that $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

If we take $\tilde{\gamma }=\tilde{0}$ and $\tilde{\delta }=\widehat{0.5}$ in Theorem 4.7, we can deduce the following corollary:

Corollary 4.8

Let $\tilde{O}$ be an m-$\mathcal{F}$ set of ℧. Then, $\tilde{O}$ is an m-$\mathcal{P}-$ $(\in ,\in \vee q)\mathcal{F}\mathcal{S}$ of ℧ ⇔$〈\tilde{O}{〉}_{\tilde{\pi }}=\{\kappa \in \text{℧}\mid {\kappa }_{\tilde{\pi }}q\tilde{O}\}\ne \varphi$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\pi }\in (0.5,1{]}^m.$

Theorem 4.9

Let $\tilde{O}$ be an m-$\mathcal{F}$ set of ℧. If $1+\tilde{\gamma }=2\tilde{\delta },$ then $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ ⇔$[\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }}\ne \varphi$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\gamma }<\tilde{\pi }\le \hat{1}.$

Proof.

Let $\tilde{O}$ be an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧ and $\tilde{\gamma }<\tilde{\pi }\le \hat{1}.$ Let $\kappa ,\tau \in [\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }},$ so ${\kappa }_{\tilde{\pi }},{\tau }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O},$ i.e. (1) $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{o}\mathrm{r}\tilde{O}\left(\kappa \right)>2\tilde{\delta }-\tilde{\pi }>2\tilde{\delta }-\hat{1}=\tilde{\gamma }$(1) and (2) $\tilde{O}\left(\tau \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{o}\mathrm{r}\tilde{O}\left(\tau \right)>2\tilde{\delta }-\tilde{\pi }>2\tilde{\delta }-\hat{1}=\tilde{\gamma }$(2) If $\tilde{\gamma }<\tilde{\pi }\le \tilde{\delta },$ then $2\tilde{\delta }-\tilde{\pi }\ge \tilde{\delta }\ge \tilde{\pi }.$ Thus, from (1(1) $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{o}\mathrm{r}\tilde{O}\left(\kappa \right)>2\tilde{\delta }-\tilde{\pi }>2\tilde{\delta }-\hat{1}=\tilde{\gamma }$(1) ) and (2(2) $\tilde{O}\left(\tau \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{o}\mathrm{r}\tilde{O}\left(\tau \right)>2\tilde{\delta }-\tilde{\pi }>2\tilde{\delta }-\hat{1}=\tilde{\gamma }$(2) ), we get $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{a}\mathrm{n}\mathrm{d}\tilde{O}\left(\tau \right)\ge \tilde{\pi }>\tilde{\gamma }.$ Since $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\},$ then $\begin{array}{rl}\tilde{O}(\kappa \divideontimes \tau )& \ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}\\ & \ge inf\{\tilde{\pi },\tilde{\pi },\tilde{\delta }\}\\ & =inf\{\tilde{\pi },\tilde{\delta }\}=\tilde{\pi }>\tilde{\gamma }.\end{array}$ Hence, $(\kappa \divideontimes \tau {)}_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O}.$

If $\tilde{\gamma }<\tilde{\pi }\le \hat{1},$ then $2\tilde{\delta }-\tilde{\pi }<\tilde{\delta }<\tilde{\pi }.$ Thus, from (1(1) $\tilde{O}\left(\kappa \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{o}\mathrm{r}\tilde{O}\left(\kappa \right)>2\tilde{\delta }-\tilde{\pi }>2\tilde{\delta }-\hat{1}=\tilde{\gamma }$(1) ) and (2(2) $\tilde{O}\left(\tau \right)\ge \tilde{\pi }>\tilde{\gamma }\mathrm{o}\mathrm{r}\tilde{O}\left(\tau \right)>2\tilde{\delta }-\tilde{\pi }>2\tilde{\delta }-\hat{1}=\tilde{\gamma }$(2) ), we get $\tilde{O}\left(\kappa \right)>2\tilde{\delta }-\tilde{\pi }\mathrm{a}\mathrm{n}\mathrm{d}\tilde{O}\left(\tau \right)>2\tilde{\delta }-\tilde{\pi }.$ Since $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\},$ then $\begin{array}{rl}\tilde{O}(\kappa \divideontimes \tau )& \ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}\\ & \ge inf\{2\tilde{\delta }-\tilde{\pi },2\tilde{\delta }-\tilde{\pi },\tilde{\delta }\}\\ & =inf\{2\tilde{\delta }-\tilde{\pi },\tilde{\delta }\}=2\tilde{\delta }-\tilde{\pi }.\end{array}$ Thus, $(\kappa \divideontimes \tau {)}_{\tilde{\pi }}{q}_{\tilde{\delta }}\tilde{O}.$ Hence, $(\kappa \divideontimes \tau {)}_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}\tilde{O},$ i.e. $(\kappa \divideontimes \tau )\in [\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }}.$ Therefore, $[\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }}$ is a subalgebra of ℧.

Conversely, Assume that $[\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }}$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\gamma }<\tilde{\pi }\le \hat{1}.$ Suppose $\kappa ,\tau \in \text{℧}$ such that $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}<inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}.$ Select $\tilde{\gamma }<\tilde{\pi }\le \hat{1}$ such that $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}<\tilde{\pi }=inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}.$ Then ${\kappa }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O},{\tau }_{\tilde{\pi }}{\in }_{\tilde{\gamma }}\tilde{O},$ but $(\kappa \divideontimes \tau {)}_{\tilde{\pi }}\overline{{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }}}\tilde{O}.$ Since $[\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }}$ is a subalgebra of ℧, so $\kappa \divideontimes \tau \in [\tilde{O}{]}_{\tilde{\pi }}^{\tilde{\delta }},$ a contradiction. Hence, $sup\left\{\tilde{O}\right(\kappa \divideontimes \tau ),\tilde{\gamma }\}\ge inf\left\{\tilde{O}\right(\kappa ),\tilde{O}(\tau ),\tilde{\delta }\}$ $\mathrm{\forall }\kappa ,\tau \in \text{℧}.$ Thus, $\tilde{O}$ is an m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$ of ℧.

If we take $\tilde{\gamma }=\tilde{0}$ and $\tilde{\delta }=\widehat{0.5}$ in Theorem 4.9, we can deduce the following corollary:

Corollary 4.10

Let $\tilde{O}$ be an m-$\mathcal{F}$ set of ℧. Then $\tilde{O}$ is an m-$\mathcal{P}-$ $(\in ,\in \vee q)\mathcal{F}\mathcal{S}$ of ℧ ⇔$[\tilde{O}{]}_{\tilde{\pi }}=\{\kappa \in \text{℧}\mid {\kappa }_{\tilde{\pi }}\in \vee q\tilde{O}\}\ne \varphi$ is a subalgebra of ℧ $\mathrm{\forall }\tilde{\pi }\in (0,1{]}^m.$

5. Conclusions

In this study, we present a new sort of generalized m-$\mathcal{F}$ subalgebras of ℧ called, an m-polar $(\omega ,\theta )$-fuzzy subalgebra and examined some of its features. We also defined and analyzed the notion of m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$s. Finally, we demonstrated some characterizations of m-$\mathcal{P}-$ $({\in }_{\tilde{\gamma }},{\in }_{\tilde{\gamma }}\vee q_{\tilde{\delta }})\mathcal{F}\mathcal{S}$s.

We hope that the research in this area can be expanded, and some of the results in this study have already laid the groundwork for further investigation into the further development of m-$\mathcal{F}$ BCK/BCI-algebras and their applications in other branches of algebra. In the future, our proposed structure could be used in a variety of areas, such as semihypergroups, lattice implication algebras, UP-algebras, and so on.

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No potential conflict of interest was reported by the author(s).

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Notes on contributors

Anas Al-Masarwah

Anas Al-Masarwah received the B.Sc. and M.Sc. degrees in mathematics from Yarmouk University, Jordan, and the Ph.D. degree in mathematics from Universiti Kebangsaan Malaysia, Malaysia. He is currently an Assistant Professor at the Department of Mathematics, Faculty of Science, Ajloun National University, Jordan. His research interests include algebras, logical algebras, algebraic structures and the foundation of mathematics. He has been selected as a referee for several journals such as Mathematics, Axioms, Symmetry, Artificial Intelligence Review, Bulletin of the Section of Logic, Discussiones Mathematicae-General Algebra and Applications, IETE Journal of Research, European Journal of Pure and Applied Mathematics, Fractal and Fractional.

Abd Ghafur Ahmad

Abd Ghafur Ahmad is a professor at the Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Malaysia. His research interests include algebras, logical algebras and combinatorial algebra theory especially in geometric and functional group theory. He has been selected as a referee for several journals such as Bulletin of Malaysian Mathematical Society, Malaysian Journal of Mathmatical Sciences, Sains Malaysiana, Matematika, Fuzzy Set and System and Journal of Quality and Measurement.