Multipolar Fuzzy Hyperideals in Ordered Semihypergroups

Abstract

An multi-polar fuzzy set is a robust mathematical model to examine multipolar, multiattribute, and multi-index data. The multi-polar fuzzy sets was created as a useful mechanism to portray uncertainty in multiattribute decision making. In this article, we consider the theoretical applications of multi-polar fuzzy sets. We present the notion of multi-polar fuzzy sets in ordered semihypergroups and define multi-polar fuzzy hyperideals (bi-hyperideals, quasi hyperideals) in an ordered semihypergroup. Relations between multi-polar fuzzy hyperideals, multi-polar fuzzy bi-hyperideals and multi-polar fuzzy quasi hyperideals are discussed.

1. Introduction

Since the theory of fuzzy sets $\left(fSs\right)$, which began with Zadeh in [1], many generalizations of this theory have been researched. Zhang developed the concept of bipolar fuzzy sets $\left(bfSs\right)$, which is an extension of $\left(fSs\right)$ in [2]. The bipolar fuzzy set is used for dealing with the data which is bipolar in nature, and which depicts information with its counter property. Chen et al. [3] introduced an extension of $bfSs$, so called m-polar fuzzy sets $(m-pfSs)$. An $m-pfS$ solves multipolarity by assigning m degrees to each element of a crisp set. Later, the hybrid structures of $m-pfSs$ were presented in order to model uncertainties in multi-criteria decision-making $\left(MCDM\right)$ situations.

On the other hand, the theory of algebraic hyperstructures, introduced by Marty in 1934 [4], is a natural generalization of the theory of algebraic structures. In [5], Corsini and Leoreanu-Fotea collected numerous applications of algebraic hyperstructures. Hyperrings were discussed by Davvaz and others in e.g., [6,7]. Applications of hypergroups have appeared and were discussed by many researchers [8,9,10,11]. Following the terminology given by Zadeh, fuzzy sets in an ordered semigroup were defined by Kehayopulu and Tsingelis in [12,13,14].

Heidari and Davvaz [15] presented the idea of ordered semihypergroups, in fact a generalization of ordered semigroups. Changphas and Davvaz [16] worked on hyperideals of ordered semihypergroups. Also, the connection of the fuzzy sets and algebraic hyperstructures was studied by many authors, e.g., [17,18,19,20,21,22]. Recently, we have observed that $m-pfS$ theory has been extensively applied to algebraic structures: [23,24,25,26,27,28,29], graph theory [30], decision-making problems [31,32,33,34,35,36,37].

In this paper, we consider the theoretical applications of $m-pfSs$ and investigate the notion of $m-pfSs$ with an application to ordered semihypergroups. To facilitate our discussion, we first review some basic concepts on semihypergroup, fuzzy ordered semihypergroup, $m-pfS$ and $m-pf$ ordered semihypergroup in Section 2. In Section 3 and Section 4, the concepts of $m-pf$ left(right) hyperideals, $m-pf$ bi-hyperideals, and $m-pf$ quasi hyperideals are introduced, and numerous features are investigated. We give characterizations of $m-pf$ bi-hyperideals and $m-pf$ quasi hyperideals. In a regular ordered semihypergroup, we analyze relations between $m-pf$ bi-hyperideals and $m-pf$ quasi-hyperideals. Finally, in Section 5 we point to some conclusions.

2. Preliminaries

This section covers some of the fundamental concepts and definitions utilized in this work.

Let H be a non-empty set and let ${\mathcal{P}}^{*}\left(H\right)$ be the set of all non-empty subsets of H. A hyperoperation on H is a map $\circ :H\times H\to {\mathcal{P}}^{*}\left(H\right)$ and the pair $(H,\circ )$ is called a hypergroupoid. For any $x\in H$ and $A,B\in {\mathcal{P}}^{*}\left(H\right)$, we denote

$$A\circ B=\bigcup \{x\circ y\mid x\in A,y\in B\},x\circ A=\left\{x\right\}\circ A,A\circ x=A\circ \left\{x\right\}.$$

A hypergroupoid $(H,\circ )$ is called a semihypergroup if for all $x,y,z\in H$ we have $(x\circ y)\circ z=x\circ (y\circ z)$, which means that

$$\bigcup _{u\in x\circ y}u\circ z=\bigcup _{v\in y\circ z}x\circ v.$$

A non-empty subset A of a semihypergroup $(S,\circ )$ is called a sub-semihypergroup if $(A,\circ )$ is a semihypergroup, equivalently, $a\circ b\subseteq A$ for all $a,b\in A.$

Definition 1

([15]). Let S be a non-empty set and ≤ be an ordered relation on S. The tri-tuple $(S,\circ ,\le )$ is called an ordered semihypergroup if the following conditions are satisfied.

(i)

$(S,\circ )$ is a semihypergroup,

(ii)

$(S,\le )$ is a partially order set,

(iii)

For every $a,b,x\in S$, $a\le b$ implies $x\circ a\le x\circ b$ and $a\circ x\le b\circ x$.

For $K\subseteq S$, we denote $\left(K\right]:=\{a\in S\mid a\le kforsomek\in K\}.$

Let $(S,\circ ,\le )$ be an ordered semihypergroup $\varnothing \ne A\subseteq S$. Then

(i)

A is called a sub-semihypergroup of an ordered semihypergroup S if $(A,\circ ,\le )$ is an ordered semihypergroup.

(ii)

A is called a right (resp. left) hyperideal of an ordered semihypergroup S if $A\circ S\subseteq A$ (resp. $S\circ A\subseteq A$) and for every $a\in A$, $b\in S$ and $b\le a$ implies $b\in A$. If is both right and left hyperideal A of S is called a hyperideal (or two-side hyperideal) of S.

(iii)

A is called a bi-hyperideal of S if A is a subsemihypergroup of S, $A\circ S\circ A\subseteq A$ and for every $a\in A$, $b\in S$ and $b\le a$ implies $b\in A$.

(iv)

A is called a quasi-hyperideal of S if $(A\circ S]\cap (S\circ A]\subseteq A$ and for every $a\in A$, $b\in S$ and $b\le a$ implies $b\in A$.

We refer the reader to references [10,16] for further information regarding ordered semihypergroups.

Definition 2

([3]). An m-pfS $\widehat{\mathfrak{F}}$ on a nonempty set S is a mapping $\widehat{\mathfrak{F}}:S\to {[0,1]}^m$. The membership value of every element $x\in S$ is denoted by

$$\widehat{\mathfrak{F}}\left(x\right)=(({\pi }_1\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_2\circ \widehat{\mathfrak{F}})\left(x\right),\dots ,({\pi }_m\circ \widehat{\mathfrak{F}})\left(x\right))$$

where ${\pi }_i:{[0,1]}^m\to [0,1]$ is defined the $i-th$ projection mapping.

Let us recall that ${[0,1]}^m$ is considered as a poset with the pointwise order ≤ is defined by $x<y\iff {\pi }_i\left(x\right)\le {\pi }_i\left(y\right)$ for each $1\le i\le m,(x,y\in {[0,1]}^m)$. Moreover, $\widehat{0}=(0,0,\dots ,0)$ is the smallest value in ${[0,1]}^m$ and $\widehat{1}=(1,1,\dots ,1)$ is the largest value in ${[0,1]}^m$.

Definition 3.

Let $(S,\circ ,\le )$ be an ordered semihypergroup and $\varnothing \ne I\subseteq S$. Then the m-polar characteristic function ${\widehat{\chi }}_I:S\to {[0,1]}^m$ defined by

$$\begin{array}{c}\hfill {\widehat{\chi }}_I\left(y\right)=\left\{\begin{array}{cc}\widehat{1}\hfill & \mathrm{if}y\in I\hfill \\ \widehat{0},\hfill & otherwise\hfill \end{array}\right.\end{array}$$

for all $y\in S$.

Given an $m-pfS$ $\widehat{\mathfrak{F}}$ on an ordered semihypergroup S, we consider the set

$$U(\widehat{\mathfrak{F}},\widehat{t})=\{x\in S\mid \widehat{\mathfrak{F}}\left(x\right)\ge \widehat{t}\}$$

where $\widehat{t}=(t_1,t_2,\dots ,t_m)\in {[0,1]}^m$, that is,

$$U(\widehat{\mathfrak{F}},\widehat{t})=\{x\in S\mid ({\pi }_1\circ \widehat{\mathfrak{F}})\left(x\right)\ge t_1,({\pi }_2\circ \widehat{\mathfrak{F}})\left(x\right)\ge t_2,\dots ,({\pi }_m\circ \widehat{\mathfrak{F}})\left(x\right)\ge t_m\}$$

and

$$U({\widehat{\mathfrak{F}}}^{>},\widehat{t})=\{x\in S\mid \widehat{\mathfrak{F}}\left(x\right)>\widehat{t}\}$$

that is,

$$U(\widehat{\mathfrak{F}},\widehat{t})=\{x\in S\mid ({\pi }_1\circ \widehat{\mathfrak{F}})\left(x\right)>t_1,({\pi }_2\circ \widehat{\mathfrak{F}})\left(x\right)>t_2,\dots ,({\pi }_m\circ \widehat{\mathfrak{F}})\left(x\right)>t_m\}$$

which are called an m-polar level subset and strong level subset of $\widehat{\mathfrak{F}}$, respectively. It is clear that ${\displaystyle U(\widehat{\mathfrak{F}},\widehat{t})={\cap }_{1\le i\le m}{U}_i(\widehat{\mathfrak{F}},\widehat{t})}$, where $U_i(\widehat{\mathfrak{F}},\widehat{t})=\{x\in S\mid ({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\ge t_i\}$. For $z\in S$, we define $A_z=\{(x,y)\in S\times S\mid z\le x\circ y\}$. Let $\widehat{\mathfrak{F}}$ and $\widehat{\mathfrak{G}}$ be two $m-pfSs$ of an ordered semihypergroup S. Then, m-polar product $\widehat{\mathfrak{F}}\diamond \widehat{\mathfrak{G}}$ is defined by

$$\begin{array}{c}\hfill (\widehat{\mathfrak{F}}\diamond \widehat{\mathfrak{G}})\left(z\right)=\left\{\begin{array}{cc}{\displaystyle \underset{(x,y)\in A_z}{\bigvee }min\{\widehat{\mathfrak{F}}\left(x\right),\widehat{\mathfrak{G}}\left(y\right)\}}\hfill & \mathrm{if}{A}_z\ne \varnothing \hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.\end{array}$$

for all $z\in S$, that is,

$$\begin{array}{c}\hfill ({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{\mathfrak{G}})\left(z\right)=\left\{\begin{array}{cc}{\displaystyle \underset{(x,y)\in A_z}{\bigvee }min\{{\pi }_i\circ \widehat{\mathfrak{F}}\left(x\right),{\pi }_i\circ \widehat{\mathfrak{G}}\left(y\right)\}}\hfill & \mathrm{if}{A}_z\ne \varnothing \hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.\end{array}$$

for all $z\in S$ and $i=1,2,\dots ,m$.

3. m−pf Hyperideals

In this section, we introduce the notions of $m-pf$ hyperideals in ordered semihypergroups and investigate some of their connected properties.

Definition 4.

Let $(S,\circ ,\le )$ be an ordered semihypergroup. An $m-pfS$$\widehat{\mathfrak{F}}$ of S is called an m-pf sub-semihypergroup of S if and only if

$${\displaystyle min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}\le \underset{z\in x\circ y}{inf}\left\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\right\}}$$

for all $i=1,2,\dots ,m$ and $x,y\in S$.

Example 1.

Let $S=\{k,l,m,n\}$ with the following hyperoperations “∘” and the order “≤” on S be defined as follows:

$$\begin{array}{ccccc}\circ & k& l& m& n\\ k& \left\{k\right\}& \left\{k\right\}& \left\{k\right\}& \left\{k\right\}\\ l& \left\{k\right\}& \left\{k\right\}& \left\{k\right\}& \left\{k\right\}\\ m& \left\{k\right\}& \left\{k\right\}& \{k,l\}& \{k,l\}\\ n& \left\{k\right\}& \left\{k\right\}& \{k,l\}& \left\{k\right\}\end{array}$$

$$\le :=\left\{\right(k,k),(k,l),(k,m),(k,n),(l,l),(m,m),(n,l),(n,m),(n,n\left)\right\}.$$

The covering relation is given by: $\prec =\left\{\right(k,n),(l,n),(m,n\left)\right\}$. Then, $(S,\circ ,\le )$ is an ordered semihypergroup [21]. We define a $3-pfS$$\widehat{\mathfrak{F}}:S\to {[0,1]}^3$ as follows.

$$\begin{array}{c}\hfill \widehat{\mathfrak{F}}\left(x\right)=\left\{\begin{array}{cc}(0.4,0.5,0.7)\hfill & ifx=k,\hfill \\ (0.3,0.4,0.6)\hfill & ifx=l,\hfill \\ (0.2,0.3,0.1)\hfill & ifx\in \{m,n\}.\hfill \end{array}\right.\end{array}$$

It is easy to see that $\widehat{\mathfrak{F}}$ is a $3-pf$ sub-semihypergroup of S.

Definition 5.

Let $(S,\circ ,\le )$ be an ordered semihypergroup. An $m-pfS$$\widehat{\mathfrak{F}}$ of S is called an m-pf left (resp. right) hyperideal of S if

(i)

${\displaystyle ({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\le \underset{z\in x\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}(({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\le \underset{z\in x\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\})}$ for all $i=1,2,\dots ,m$ and $x,y\in S$.

(ii)

$x\le y$ implies $({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)$ for all $i=1,2,\dots ,m$ and $x,y\in S$.

If $\widehat{\mathfrak{F}}$ is both $m-pf$ left hyperideal and $m-pf$ right hyperideal of S, then $\widehat{\mathfrak{F}}$ is called an m-pf hyperideal of S.

Example 2.

Consider the ordered semihypergroup $(S,\circ ,\le )$ given in Example 1. We define the $3-pfS$$\widehat{\mathfrak{F}}:S\to {[0,1]}^3$ as follows.

$$\begin{array}{c}\hfill \widehat{\mathfrak{F}}\left(x\right)=\left\{\begin{array}{cc}(0.40,0.55,0.80),\hfill & ifx=k,\hfill \\ (0.19,0.45,0.78),\hfill & ifx=n,\hfill \\ (0.29,0.48,0.75),\hfill & ifx\in \{l,m\}.\hfill \end{array}\right.\end{array}$$

It is not difficult to see that $\widehat{\mathfrak{F}}$ is a $3-pf$ right hyperideal of S. But it is not a $3-pf$ left hyperideal of S. Since $z\in m\circ n$, ${\displaystyle \underset{z\in m\circ n}{inf}\{({\pi }_3\circ \widehat{\mathfrak{F}})\left(z\right)\}=0.75\neg \ge 0.78=({\pi }_3\circ \widehat{\mathfrak{F}})\left(n\right)}$.

We define the $4-pfS$$\widehat{\mathfrak{F}}:S\to {[0,1]}^4$ as follows.

$$\begin{array}{c}\hfill \widehat{\mathfrak{F}}\left(x\right)=\left\{\begin{array}{cc}(0.65,0.55,0.85,0.95)\hfill & ifx=k,\hfill \\ (0.35,0.45,0.75,0.80),\hfill & ifx\in \{l,m,n\}.\hfill \end{array}\right.\end{array}$$

Then $\widehat{\mathfrak{F}}$ is a $4-pf$ hyperideal of S.

Theorem 1.

Let $\widehat{\mathfrak{F}}$ be an $m-pfS$ of S. Then $\widehat{\mathfrak{F}}$ is an $m-pf$ left (resp. right) hyperideal of S if and only if m-polar level set $U(\widehat{\mathfrak{F}},\widehat{t})\ne \varnothing$ of $\widehat{\mathfrak{F}}$ is a left (resp. right) hyperideal of S for all $\widehat{t}\in {[0,1]}^m$.

Proof. 

Straightforward □

Corollary 1.

Let $\widehat{\mathfrak{F}}$ be an $m-pfS$ with the upper bound $\widehat{t_0}$ of an ordered semihypergroup S. Then the following are equivalent:

(i)

$\widehat{\mathfrak{F}}$ is an $m-pf$ left (resp. right) hyperideal of S.

(ii)

Each m-polar level subset $U(\widehat{\mathfrak{F}},\widehat{t})$, for $\widehat{t}\in {[0,\widehat{t_0}]}^m$ is a left (right) hyperideal of S.

(iii)

Each m-polar strong level subset $U({\widehat{\mathfrak{F}}}^{>},\widehat{t})$, for $\widehat{t}\in {[0,\widehat{t_0}]}^m$ is a left (right) hyperideal of S.

(iv)

Each m-polar level subset $U(\widehat{\mathfrak{F}},\widehat{t})$, for $\widehat{t}\in Im\left(\widehat{\mathfrak{F}}\right)$ is a left (right) hyperideal of S, where $Im\left(\widehat{\mathfrak{F}}\right)$ denotes the image of $\widehat{\mathfrak{F}}$.

(v)

Each m-polar strong level subset $U({\widehat{\mathfrak{F}}}^{>},\widehat{t})$, for $\widehat{t}\in Im(\widehat{\mathfrak{F}})\{\widehat{t_0}\}$ is a left (right) hyperideal of S.

(vi)

Each m-polar non-empty level subset of $U(\widehat{\mathfrak{F}},\widehat{t})$ is a left (right) hyperideal of S.

(vii)

Each non-empty m-polar strong level subset of $U({\widehat{\mathfrak{F}}}^{>},\widehat{t})$ is a left (right) hyperideal of S.

Corollary 2.

Let $(S,\circ ,\le )$ be an ordered semihypergroup and ${\widehat{\chi }}_I$ is an $m-pfS$ of S. Then ${\widehat{\chi }}_I$ is an $m-pf$ left (right) hyperideal of S if and only if I is a left (right) hyperideal of S.

Theorem 2.

Let $(S,\circ ,\le )$ be an ordered semihypergroup and $\widehat{\mathfrak{F}}$ be an $m-pf$ hyperideal of S. Then $\widehat{\mathfrak{F}}$ is an $m-pf$ sub-semihypergroup of S.

Proof. 

Let $\widehat{\mathfrak{F}}$ be an $m-pf$ hyperideal of an ordered semihypergroup S. By Definition 4 $\widehat{\mathfrak{F}}$ is both $m-pf$ left hyperideal and $m-pf$ right hyperideal of S. Then, we have

$${\displaystyle ({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\le \underset{z\in x\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\le \underset{z\in x\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}}$$

for all $i=1,2,\dots ,m$ and $x,y\in S$.

$$\begin{array}{c}\hfill {\displaystyle \underset{z\in x\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\})\ge max\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}\ge min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}.}\end{array}$$

By Definition 4 $\widehat{\mathfrak{F}}$ is an $m-pf$ sub-semihypergroup of S. □

The converse of Theorem 2 is not true in general as seen in the Example 3.

Example 3.

Consider an ordered semihypergroup $S=\{k,l,m,n\}$ which is given in Example 1. Define a $4-pfS$$\widehat{\mathfrak{F}}:S\to {[0,1]}^4$ as follows.

$$\begin{array}{c}\hfill \widehat{\mathfrak{F}}\left(x\right)=\left\{\begin{array}{cc}(0.83,0.65,0.55,0.95),\hfill & ifx=k,\hfill \\ (0.55,0.23,0.18,0.71),\hfill & ifx\in \{l,m\},\hfill \\ (0.41,0.28,0.15,0.70),\hfill & ifx=n.\hfill \end{array}\right.\end{array}$$

It is easy to see that $\widehat{\mathfrak{F}}$ is a $4-pf$ sub-semihypergroup of S. However, $\widehat{\mathfrak{F}}$ is not a $4-pf$ hyperideal of S. Since $z\in m\circ n$, ${\displaystyle \underset{z\in m\circ n}{inf}\{({\pi }_2\circ \widehat{\mathfrak{F}})\left(z\right)\}=0.23\neg \ge 0.28=({\pi }_2\circ \widehat{\mathfrak{F}})\left(n\right)}$.

Let $\eta$ be a mapping from an ordered semihypergroup $S_1$ to an ordered semihypergroup $S_2$. Let $\widehat{\mathfrak{F}}$ be an $m-pfS$ of $S_1$ and $\widehat{\mathfrak{G}}$ be an $m-pfS$ of $S_2$. Then the inverse image ${\eta }^{-1}(\widehat{\mathfrak{G}})$ of $\widehat{\mathfrak{G}}$ is an $m-pfS$ of $S_1$ defined by ${\eta }^{-1}\left(\widehat{\mathfrak{G}}\right)\left(x\right)=\widehat{\mathfrak{G}}\left(\eta \left(x\right)\right)$ for all $x\in S_1.$ The image $\eta (\widehat{\mathfrak{F}})$ of $\widehat{\mathfrak{F}}$ is an $m-pfS$ of $S_2$ defined by

$$\begin{array}{cc}& \eta (\widehat{\mathfrak{F}})\left(y\right)=\left\{\begin{array}{cc}{\displaystyle \underset{x\in {\eta }^{-1}\left(y\right)}{sup}\{\widehat{\mathfrak{F}}\left(x\right)\}}\hfill & \mathrm{if}{\eta }^{-1}\left(y\right)\ne \varnothing ,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

for all $y\in S_2.$

Lemma 1.

Let $S_1$ and $S_2$ be two ordered semihypergroups and let $\eta :S_1\to S_2$ be a strong homomorphism.

(i)

If η is an epimorphism and $\widehat{\mathfrak{F}}$ is an $m-pf$ hyperideal of $S_1$, then $\eta (\widehat{\mathfrak{F}})$ is an $m-pf$ hyperideal of $S_2$.

(ii)

If $\widehat{\mathfrak{G}}$ is an $m-pf$ hyperideal of $S_2$, then ${\eta }^{-1}(\widehat{\mathfrak{G}})$ is an $m-pf$ hyperideal of $S_1$.

Proof. 

It is straightforward. □

4. m−pf bi-Hyperideals and m−pf Quasi-Hyperideals

In this section, we define the concepts of an $m-pf$ bi-hyperideal and an $m-pf$ quasi-hyperideal in ordered semihypergroup and give relationships between them.

Definition 6.

Let $(S,\circ ,\le )$ be an ordered semihypergroup. An m−pf subsemihypergroup $\widehat{\mathfrak{F}}$ of S is called an m−pf bi-hyperideal of S if and only if

(i)

${\displaystyle \underset{z\in x\circ a\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}\ge min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}}$ for all $i=1,2,\dots ,m$ and $x,a,y\in S$.

(ii)

for every $x,y\in S$, $x\le y$ implies $({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)$ for all $i=1,2,\dots ,m$.

Definition 7.

Let $(S,\circ ,\le )$ be an ordered semihypergroup. An $m-pfS$$\widehat{\mathfrak{F}}$ of S is called an m-pf quasi-hyperideal of S if and only if

(i)

$({\pi }_i\circ \widehat{\mathfrak{F}})\diamond ({\pi }_i\circ \widehat{1})\cap ({\pi }_i\circ \widehat{1})\diamond ({\pi }_i\circ \widehat{\mathfrak{F}})\subseteq ({\pi }_i\circ \widehat{\mathfrak{F}})$ for all $i=1,2,\dots ,m$, where $\widehat{1}:S\to {[0,1]}^m$ is an m-polar function defined $\widehat{1}\left(x\right)=(1,1,\dots ,1)$ for all $x\in S$.

(ii)

for every $x,y\in S$, $x\le y$ implies $({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)$ for all $i=1,2,\dots ,m$.

Theorem 3.

Let $\widehat{\mathfrak{F}}$ be an $m-pf$ sub-semihypergroup of S. Then $\widehat{\mathfrak{F}}$ is an $m-pf$ bi-hyperideal of S if and only if m-polar level set $U(\widehat{\mathfrak{F}},\widehat{t})\ne \varnothing$ of $\widehat{\mathfrak{F}}$ is a bi-hyperideal of S for all $\widehat{t}\in {[0,1]}^m$.

Proof. 

Assume that $\widehat{\mathfrak{F}}$ is an m−pf bi-hyperideal of S. Let $\widehat{t}\in {[0,1]}^m$ with $U(\widehat{\mathfrak{F}},\widehat{t})\ne \varnothing$. Let $z\in U(\widehat{\mathfrak{F}},\widehat{t})\circ U(\widehat{\mathfrak{F}},\widehat{t})$. Then $z\in x\circ y$ for some $x,y\in U_i(\widehat{\mathfrak{F}},\widehat{t})$ for all $i=1,2,\dots ,m$. Since $\widehat{\mathfrak{F}}$ is an $m-pf$ sub-semihypergroup of S, we get

$${\displaystyle t_i\le min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}\le \underset{z\in x\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}.}$$

which implies $({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge t_i$ for all $z\in x\circ y$ and $1\le i\le m$. Hence ${\displaystyle z\in {\cap }_{1\le i\le m}{U}_i(\widehat{\mathfrak{F}},\widehat{t})=U(\widehat{\mathfrak{F}},\widehat{t})}$, that is $U(\widehat{\mathfrak{F}},\widehat{t})$ sub-semihypergroup of S. Let $z\in U(\widehat{\mathfrak{F}},\widehat{t})\circ S\circ U(\widehat{\mathfrak{F}},\widehat{t})$. Then, we have $z\in x\circ a\circ y$ for some $a\in S$, $x,y\in U_i(\widehat{\mathfrak{F}},\widehat{t})$ for all $i=1,2,\dots ,m$. By assumption,

$${\displaystyle t_i\le min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}\le \underset{z\in x\circ a\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}.}$$

which implies $({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge t_i$ for all $z\in x\circ a\circ y$ and $1\le i\le m$. This implies that ${\displaystyle z\in {\cap }_{1\le i\le m}{U}_i(\widehat{\mathfrak{F}},\widehat{t})=U(\widehat{\mathfrak{F}},\widehat{t})}$, that is $U(\widehat{\mathfrak{F}},\widehat{t})\circ S\circ U(\widehat{\mathfrak{F}},\widehat{t})\subseteq U(\widehat{\mathfrak{F}},\widehat{t})$. Now, let $x\in U(\widehat{\mathfrak{F}},\widehat{t})$ and $y\in S$ such that $y\le x$. Then $({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\ge t_i$ for all $i=1,2,\dots ,m$. Since $y\le x$, it follows that $({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\ge t_i$. This implies that ${\displaystyle y\in {\cap }_{1\le i\le m}{U}_i(\widehat{\mathfrak{F}},\widehat{t})=U(\widehat{\mathfrak{F}},\widehat{t})}$ By Definition 1 the m-polar level set $U(\widehat{\mathfrak{F}},\widehat{t})$ is a bi-hyperideal of S for all $\widehat{t}\in {[0,1]}^m$. Conversely, suppose that the m-polar level set $U(\widehat{\mathfrak{F}},\widehat{t})$ of $\widehat{\mathfrak{F}}$ is a bi hyperideal of S for all $\widehat{t}\in {[0,1]}^m$. Since $\widehat{\mathfrak{F}}$ is an $m-pf$ sub-semihypergroup of S, we obtain

$${\displaystyle min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}\le \underset{z\in x\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}}$$

for all $x,y\in S$ and $1\le i\le m$. Let $t_{i_0}=min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}$ for some $x,y\in S$ and $1\le i\le m$. Then $x,y\in U(\widehat{\mathfrak{F}},\widehat{t_0})$. Since $U(\widehat{\mathfrak{F}},\widehat{t_0})$ is a bi hyperideal of S, we have $x\circ a\circ y\subseteq U(\widehat{\mathfrak{F}},\widehat{t_0})$ for all $a\in S$. Then for every $z\in x\circ a\circ y$, we obtain $({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge t_{i_0}$ for all $i=1,2,\dots ,m$. Thus

$${\displaystyle \{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}=t_{i_0}\le \underset{z\in x\circ a\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}.}$$

Finally, let $x,y\in S$ such that $x\le y$. Let $t_{i_1}=({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)$ for all $1\le i\le m$. Then $y\in U(\widehat{\mathfrak{F}},\widehat{t_1})$. Since $U(\widehat{\mathfrak{F}},\widehat{t_1})$ is an $m-pf$ bi-hyperideal of S, $x\in U(\widehat{\mathfrak{F}},\widehat{t_1})$. Thus $({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)\ge t_{i_1}=({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)$ for all $i=1,2,\dots ,m$. Therefore, $\widehat{\mathfrak{F}}$ is an $m-pf$ bi-hyperideal of S. □

Theorem 4.

Let $\widehat{\mathfrak{F}}$ be an $m-pf$ set of S. Then $\widehat{\mathfrak{F}}$ is an $m-pf$ quasi-hyperideal of S if and only if m-polar level set $U(\widehat{\mathfrak{F}},\widehat{t})\ne \varnothing$ of $\widehat{\mathfrak{F}}$ is a quasi-hyperideal of S for all $\widehat{t}\in {[0,1]}^m$.

Proof. 

The proof is similar to the proof of Theorem 3. □

Corollary 3.

Let $(S,\circ ,\le )$ be an ordered semihypergroup and ${\widehat{\chi }}_I$ is an $m-pf$ sub-semihypergroup of S. Then ${\widehat{\chi }}_I$ is an $m-pf$ bi-hyperideal of S if and only if I is a bi-hyperideal of S.

Corollary 4.

Let $(S,\circ ,\le )$ be an ordered semihypergroup and ${\widehat{\chi }}_I$ is an $m-pfS$ of S. Then ${\widehat{\chi }}_I$ is an $m-pf$ quasi-hyperideal of S if and only if I is a quasi-hyperideal of S.

Example 4.

Let $S=\{k,l,m,n,p\}$ with the following hyperoperations “∘” and the order “≤” on S be defined as follows:

$$\begin{array}{cccccc}\circ & k& l& m& n& p\\ k& \left\{k\right\}& \left\{k\right\}& \left\{k\right\}& \left\{k\right\}& \left\{k\right\}\\ l& \left\{k\right\}& \{k,l\}& \left\{k\right\}& \{k,n\}& \left\{k\right\}\\ m& \left\{k\right\}& \{k,p\}& \{k,m\}& \{k,m\}& \{k,p\}\\ n& \left\{k\right\}& \{k,l\}& \{k,n\}& \{k,n\}& \{k,l\}\\ p& \left\{k\right\}& \{k,p\}& \left\{k\right\}& \{k,m\}& \left\{k\right\}\end{array}$$

$$\le :=\left\{\right(k,k),(l,l),(m,m),(n,n),(p,p),(k,l),(k,m),(k,n),(k,p\left)\right\}.$$

The covering relation is given by: $\prec =\left\{\right(k,l),(k,m),(k,n),(k,p\left)\right\}$. Then, $(S,\circ ,\le )$ is an ordered semihypergroup [22]. Define a $3-pfS$$\widehat{\mathfrak{F}}:S\to {[0,1]}^3$ as follows.

$$\begin{array}{c}\hfill \widehat{\mathfrak{F}}\left(x\right)=\left\{\begin{array}{cc}(0.85,0.81,0.71),\hfill & ifx=k,\hfill \\ (0.73,0.70,0.61),\hfill & ifx=m,\hfill \\ (0.51,0.65,0.53),\hfill & ifx=n,\hfill \\ (0.35,0.38,0.41),\hfill & ifx\in \{l,p\}.\hfill \end{array}\right.\end{array}$$

We have

$$\begin{array}{c}\hfill U_1(\widehat{\mathfrak{F}},\widehat{t})=\left\{\begin{array}{cc}\varnothing ,\hfill & if{t}_1\in (0.85,1],\hfill \\ \left\{k\right\},\hfill & if{t}_1\in (0.73,0.85],\hfill \\ \{k,m\}\hfill & if{t}_1\in (0.51,0.73],\hfill \\ \{k,m,n\}\hfill & if{t}_1\in (0.35,0.51],\hfill \\ S,\hfill & if{t}_1\in (0,0.35].\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill U_2(\widehat{\mathfrak{F}},\widehat{t})=\left\{\begin{array}{cc}\varnothing ,\hfill & if{t}_2\in (0.81,1],\hfill \\ \left\{k\right\},\hfill & if{t}_2\in (0.70,0.81],\hfill \\ \{k,m\}\hfill & if{t}_2\in (0.65,0.70],\hfill \\ \{k,m,n\}\hfill & if{t}_2\in (0.38,0.65],\hfill \\ S,\hfill & if{t}_2\in (0,0.38].\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill U_3(\widehat{\mathfrak{F}},\widehat{t})=\left\{\begin{array}{cc}\varnothing ,\hfill & if{t}_3\in (0.71,1],\hfill \\ \left\{k\right\},\hfill & if{t}_3\in (0.61,0.71],\hfill \\ \{k,m\}\hfill & if{t}_3\in (0.53,0.61],\hfill \\ \{k,m,n\}\hfill & if{t}_3\in (0.41,0.53],\hfill \\ S,\hfill & if{t}_3\in (0,0.41].\hfill \end{array}\right.\end{array}$$

Since $\left\{k\right\}$, $\{k,m\}$, $\{k,m,n\}$ and S are all quasi-hyperideal of S, then $U_i(\widehat{\mathfrak{F}},\widehat{t})$ is a quasi-hyperideal of S for all $i=1,2,3$. Thus ${\displaystyle U(\widehat{\mathfrak{F}},\widehat{t})={\cap }_{1\le i\le 3}{U}_i(\widehat{\mathfrak{F}},\widehat{t})}$ is a quasi-hyperideal of S, which implies from Theorem 4 that $\widehat{\mathfrak{F}}$ is a $3-pf$ quasi-hyperideal of S.

Theorem 5.

Every $m-pf$ left(resp. right) hyperideal of S is an $m-pf$ quasi hyperideal of S.

Proof. 

Let $\widehat{\mathfrak{F}}$ is an $m-pf$ left hyperideal of S and $z\in S$. Then

$$\begin{array}{ccc}\hfill ({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)& =& {\displaystyle \underset{z\le x\circ y}{\bigvee }min\{({\pi }_i\circ \widehat{1})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}}\hfill \\ & =& {\displaystyle \underset{z\le x\circ y}{\bigvee }({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\le \underset{z\le x\circ y}{\bigvee }({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)=({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right).}\hfill \end{array}$$

for $i=1,2,\dots ,m$. Thus $({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\le ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$. Similarly $({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right)\le ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$. Therefore

$$({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge min\{({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right),({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right)\}.$$

for $i=1,2,\dots ,m$ i.e., ${\pi }_i\circ \widehat{\mathfrak{F}}\supseteq ({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\cap ({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})$. Thus $\widehat{\mathfrak{F}}$ is an $m-pf$ quasi hyperideal of S. The proof for the case that $\widehat{\mathfrak{F}}$ is an $m-pf$ right hyperideal is similar. □

Example 5.

Let $S=\{k,l,m,n\}$ with the following hyperoperations “∘” and the order “≤” on S be defined as follows:

$$\begin{array}{ccccc}\circ & k& l& m& n\\ k& \left\{k\right\}& \{k,l\}& \{k,m\}& \left\{k\right\}\\ l& \left\{k\right\}& \{k,l\}& \{k,m\}& \left\{k\right\}\\ m& \left\{k\right\}& \{k,l\}& \{k,m\}& \left\{k\right\}\\ n& \left\{k\right\}& \{k,l\}& \{k,m\}& \left\{k\right\}\end{array}$$

$$\le :=\left\{\right(k,k),(l,l),(m,m),(n,n),(l,k),(m,k\left)\right\}.$$

The covering relation is given by: $\prec =\left\{\right(l,k),(m,k\left)\right\}$. Then $(S,\circ ,\le )$ is an ordered semihypergroup [22]. Define a $3-pfS$$\widehat{\mathfrak{F}}:S\to {[0,1]}^3$ as follows.

$$\begin{array}{c}\hfill \widehat{\mathfrak{F}}\left(x\right)=\left\{\begin{array}{cc}(0.85,0.81,0.71),\hfill & ifx\in \{k,l,m\},\hfill \\ (0.51,0.65,0.53),\hfill & ifx=n.\hfill \end{array}\right.\end{array}$$

Then, $\widehat{\mathfrak{F}}$ is a $3-pf$ left hyperideal of S. By Theorem 5$\widehat{\mathfrak{F}}$ is a $3-pf$ bi-hyperideal of S.

Theorem 6.

Every $m-pf$ quasi hyperideal of S is an $m-pf$ bi-hyperideal of S.

Proof. 

Let $\widehat{\mathfrak{F}}$ is an $m-pf$ quasi hyperideal of S and $x,y\in S$ and $z\in x\circ y$. Then for all $i=1,2,\dots ,m$,

$$\begin{array}{ccc}\hfill ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)& \ge & min\left\{({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right),({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right)\right\}\hfill \\ & =& {\displaystyle min\left\{\underset{z\le a\circ b}{\bigvee }min\left\{({\pi }_i\circ \widehat{1})\left(a\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(b\right)\right\}\right\},}\hfill \\ & =& {\displaystyle \underset{z\le u\circ v}{\bigvee }min\left\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(u\right),({\pi }_i\circ \widehat{1})\left(v\right)\right\}}\hfill \\ & \ge & min\left\{min\{({\pi }_i\circ \widehat{1})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\right\},\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{1})\left(y\right)\}\}\hfill \\ & \ge & min\left\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\right\}.\hfill \end{array}$$

So $({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}$ for all $z\in x\circ y$ and $i=1,2,\dots ,m$. Hence $\widehat{\mathfrak{F}}$ is an m-polar fuzzy sub-semihypergroup of S. Now $x,a,y\in S$ and $z\in x\circ a\circ y$. Then there exists $\omega \in a\circ y,{\omega }^{*}\in x\circ a$ such that $z\le x\circ \omega$ and $z\le {\omega }^{*}\circ a$. Then for all $i=1,2,\dots ,m$,

$$\begin{array}{ccc}\hfill ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)& \ge & min\left\{({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right),({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\right\}\hfill \\ & =& {\displaystyle min\{\underset{z\le x\circ \omega }{\bigvee }min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{1})\left(\omega \right)\},}\hfill \\ & =& {\displaystyle \underset{z\le {\omega }^{*}\circ y}{\bigvee }min\{({\pi }_i\circ \widehat{1})({\omega }^{*}),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}}\hfill \\ & \ge & min\left\{min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{1})\left(\omega \right)\},min\{({\pi }_i\circ \widehat{1})({\omega }^{*}),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}\right\}\hfill \\ & \ge & min\left\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\right\}\hfill \end{array}$$

for all $z\in x\circ a\circ y$. Thus ${\displaystyle \underset{z\in x\circ a\circ y}{inf}\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}\ge min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(y\right)\}}$ for all $i=1,2,\dots ,m$. Therefore $\widehat{\mathfrak{F}}$ is an $m-pf$ bi-hyperideal of S. □

Example 6.

Lets take into consideration the ordered semihypergroup $(S,\circ ,\le )$ given in Example 4. We define the $4-pfS$$\widehat{\mathfrak{F}}:S\to {[0,1]}^4$ as follows.

$$\begin{array}{c}\hfill \widehat{\mathfrak{F}}\left(x\right)=\left\{\begin{array}{cc}(0.81,0.75,0.80,0.90),\hfill & ifx=k,\hfill \\ (0.75,0.65,0.78,0.78),\hfill & ifx=m,\hfill \\ (0.45,0.48,0.75,0.65),\hfill & ifx=n,\hfill \\ (0.29,0.30,0.65,0.35),\hfill & ifx\in \{l,p\}.\hfill \end{array}\right.\end{array}$$

It is easily checked that $\widehat{\mathfrak{F}}$ is a $4-pf$ quasi hyperideal of S. By Theorem 6 $\widehat{\mathfrak{F}}$ is a $4-pf$ bi-hyperideal of S.

Definition 8

([15]). An ordered semihypergroup $(S,\circ ,\le )$ is called regular, if for every $a\in S$, there exists $x\in S$ such that $a\le a\circ x\circ a$.

Theorem 7.

Let $(S,\circ ,\le )$ be a regular ordered semihypergroup and $\widehat{\mathfrak{F}}$ be an $m-pf$ set of S. Then, $\widehat{\mathfrak{F}}$ is an $m-pf$ quasi-hyperideal of S if and only if $\widehat{\mathfrak{F}}$ is an $m-pf$ bi-hyperideal of S.

Proof. 

“⇒” Assume that $\widehat{\mathfrak{F}}$ be an $m-pf$ quasi-hyperideal of S It is clear that $\widehat{\mathfrak{F}}$ is an $m-pf$ bi-hyperideal of S by Theorem 6

“⇐” Let $z\in S$. If $A_z=\varnothing$, then it is clear that $(({\pi }_i\circ \widehat{\mathfrak{F}})\diamond ({\pi }_i\circ \widehat{1})\wedge ({\pi }_i\circ \widehat{1})\diamond ({\pi }_i\circ \widehat{\mathfrak{F}}))\left(z\right)\le ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$ for all $i=1,2,\dots ,m$. Thus $({\pi }_i\circ \widehat{\mathfrak{F}})\diamond ({\pi }_i\circ \widehat{1})\cap ({\pi }_i\circ \widehat{1})\diamond ({\pi }_i\circ \widehat{\mathfrak{F}})\subseteq \widehat{\mathfrak{F}}$. Let $A_z\ne \varnothing$. If $({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right)\le ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$, then we have that $({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right)\ge min\{({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right),({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}$, that is $({\pi }_i\circ \widehat{\mathfrak{F}}\diamond ({\pi }_i\circ \widehat{1})\cap ({\pi }_i\circ \widehat{1}\diamond ({\pi }_i\circ \widehat{\mathfrak{F}})\subseteq ({\pi }_i\circ \widehat{\mathfrak{F}})$. If $({\pi }_i\circ \widehat{\mathfrak{F}}\diamond \widehat{1})\left(z\right)>({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$, then there exists $x,y\in S$ such that

$$min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{1})\left(y\right)\}=({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)>({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right).$$

That is $x,y\in S$, $z\le x\circ y$ and $({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)>({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$. In this case we will prove that $({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\le ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$ for all $i=1,2,\dots ,m$. Let $(u,v)\in A_z$ for some $u,v\in S$. Since S is regular and $z\in S$ there exists $a\in S$ such that $z\le z\circ a\circ z$. From $z\le z\circ a\circ z$, $z\le x\circ y$, $z\le u\circ v$ we get $z\le x\circ y\circ a\circ u\circ v$. Since $\widehat{\phi }$ is an m−pf bi-hyperideal S, we obtain for all $i=1,2,\dots ,m$

$$\begin{array}{c}({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}})(x\circ (y\circ a\circ u)\circ v)\ge \hfill \\ \hfill {\displaystyle \underset{\alpha \in x\circ (y\circ a\circ u)\circ v}{inf}({\pi }_i\circ \widehat{\mathfrak{F}})\left(\alpha \right)\ge min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\}.}\end{array}$$

If $min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\}=({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)$ for all $i=1,2,\dots ,m$ then, $({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)$. This contradicts with $({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right)>({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)$. Hence, $min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(x\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\}=({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)$ and so $({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge ({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)=min\{({\pi }_i\circ \widehat{1})\left(u\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\}$ for all $z\le u\circ v$ and $i=1,2,\dots ,m.$ Therefore ${\displaystyle ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\ge \underset{z\le u\circ v}{\bigvee }min\{({\pi }_i\circ \widehat{1})\left(u\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\}=({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)}$. As a result for all $i=1,2,\dots ,m$, we have

$$\begin{array}{ccc}\hfill (({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\cap ({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}}))\left(z\right)& =& min\{({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\left(z\right),({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right)\}\hfill \\ & =& {\displaystyle min\{\underset{z\le u\circ v}{\bigvee }min\{({\pi }_i\circ \widehat{1})\left(u\right),({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\},}\hfill \\ & & {\displaystyle \underset{z\le u\circ v}{\bigvee }min\{({\pi }_i\circ \widehat{\mathfrak{F}})\left(u\right),({\pi }_i\circ \widehat{1})\left(v\right)\}\}}\hfill \\ & =& {\displaystyle min\left\{\underset{z\le u\circ v}{\bigvee }({\pi }_i\circ \widehat{\mathfrak{F}})\left(u\right),\underset{z\le u\circ v}{\bigvee }({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\right\}}\hfill \\ & =& ({\pi }_i\circ \widehat{\mathfrak{F}})\left(v\right)\le ({\pi }_i\circ \widehat{\mathfrak{F}})\left(z\right).\hfill \end{array}$$

That is $({\pi }_i\circ \widehat{\mathfrak{F}}\diamond {\pi }_i\circ \widehat{1})\cap ({\pi }_i\circ \widehat{1}\diamond {\pi }_i\circ \widehat{\mathfrak{F}})\subseteq {\pi }_i\circ \widehat{\mathfrak{F}}.$ □

5. Conclusions

Many problems that we commonly encounter in everyday life are difficult to describe and solve using standard mathematical procedures. This is because these problems involve a certain kind of uncertainty. In the past, the issue of solving examples with uncertainty attracted the attention of many scientists, who gradually proposed various procedures and methods for solving decision-making processes with uncertainty. Currently, mathematical theories that solves the problem of uncertainty incorporate $fS$ theory, soft set theory, $m-pfS$ theory, probability theory and so on. An $m-pf$ model is a generalized form of a bipolar fuzzy model. This model grant more exactness, flexibility and compatibility to the system when more than one agreement has to be solved. In our paper, we apply the $m-pfS$ theory to algebraic hyperstructures and investigate three unique notions: $m-pf$ hyperideals, $m-pf$ quasi-hyperideals and $m-pf$ bi-hyperideals of an ordered semihypergroup. Regular ordered semihypergroups are described using the characteristics of these fuzzy hyperideals.

Due to the fact that algebraic hyperstructures are possiblle to use for modeling of biological phenomena [11], our paper can be extended to study the applications of $m-pfS$ in inheritance phenomena based on other algebraic hyperstructures as hypergroups, polygroups, and hyperrings.