m-Polar Picture Fuzzy Bi-Ideals and Their Applications in Semigroups

Abstract

The concept of symmetry is fundamental to the study of algebra; it serves as the basis for a branch of group theory that is essential to abstract algebra. A semigroup is a structure that builds upon the concept of a group, similarly extending the idea of symmetry found within groups. In this study, we specifically focus on semigroups. The main objective of this research is to apply the notion of m-polar picture fuzzy sets (m-PPFSs), with m being a natural number, in investigations into semigroups, as this concept generalizes m-polar fuzzy sets (m-PFSs) and picture fuzzy sets (PFSs). This research introduces the concepts of m-polar picture fuzzy left ideals (m-PPFLs), m-polar picture fuzzy right ideals (m-PPFRs), m-polar picture fuzzy ideals (m-PPFIs), m-polar picture fuzzy bi-ideals (m-PPFBs), and m-polar picture fuzzy generalized bi-ideals (m-PPFGBs) in semigroups. This study examines the relationships between these concepts, showing that every m-PPFL (m-PPFR) in the semigroups is also an m-PPFB, and that every m-PPFB in the semigroups is an m-PPFGB. However, the opposite is not true. Additionally, we provide the characteristics of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in semigroups. We further discuss the connections between the m-PPFLs (m-PPFIs) and the m-PPFBs within the framework of regular semigroups, and most importantly, we show that, if the semigroup is regular, then the m-PPFBs and m-PPFGBs are equal. Finally, we utilize the properties of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs within semigroups to explore the classifications of regular semigroups.

1. Introduction

Symmetry is a fundamental concept in algebra; it forms the basis of a branch of group theory that is essential to abstract algebra. Additionally, the role of symmetry in decision-making is significant both theoretically and in practice as it aids with systematically and equitably understanding the “structure” of choices, the “bias” of perception, and the “fairness” of judgments made regarding each choice. Zabzina et al. [1] presented a decision-making model in which symmetry-breaking is followed by a symmetry-restoring bifurcation, whereby enormous systems return to an even distribution of exploitation among options. Zhou et al. [2] explored the symmetry in distances to both positive and negative ideal solutions, similar to the TOPSIS methodology. Kumar et al. [3] analyzed the elements that substantially influence breaches in healthcare information security using a hybrid fuzzy-based symmetrical technique called AHP-TOPSIS. The concept of PFSs is included within decision-making, which depends on the principle of symmetry to assist with making decisions. The concept of PFSs was introduced by Cuong and Kreinovich [4] in 2013 as a generalization of the notions of fuzzy sets (FSs) [5] and intuitionistic fuzzy sets (IFSs) [6]. The PFS incorporates a third dimension, the neutral membership degree, together with the membership degree used in traditional fuzzy sets and the non-membership degree introduced by intuitionistic fuzzy sets. There are “yes” and “no” options as well as “abstain” or “undecided” options. This is a useful tool that can be used in many areas that involve difficult problems. The PFS notion has been studied in a variety of fields. For example, Thao et al. [7] introduced a divergence measure of picture fuzzy sets, developed a multi-criteria decision-making algorithm, and applied it to medical diagnosis and classification problems. Furthermore, Verma and Rohtagi [8] introduced novel similarity measures between two picture fuzzy sets, defining distance measures and weighted versions. The authors applied these measures in pattern recognition and medical diagnosis, demonstrating improved performance. Moreover, its applicability extends to algebra; Yiarayong [9] applied the concept of PFSs to semigroup theory. He identified various categories of regular semigroups, including intra-regular and semisimple types, by examining their fuzzy left and right ideals, known as picture fuzzy ideals. Subsequently, Nakkhasen employed the concept of PFSs to further characterize semigroups by exploring various types of picture fuzzy ideals [10,11]. Afterward, Kankaew et al. [12] introduced eight new concepts related to picture fuzzy sets and also discussed the connections between these concepts in UP-algebras. The references offer additional research on the notion of PFSs in algebraic structures [13,14,15].

In 2014, Chen et al. [16] introduced the notion of m-PFSs as a generalization of bipolar fuzzy sets (BFSs) [17], while the BFSs generalize the FSs. The primary advantage of m-PFSs is their capacity to concurrently manage multi-dimensional data, a capability that conventional fuzzy sets lack. For instance, a 4-PFS can immediately represent a decision regarding “technical, economic, environmental, and social aspects”. In 2018, Akram and Shahzadi [18] investigated the application of m-PFSs in hypergraphs, introducing regular and totally regular m-polar fuzzy hypergraphs. They examined their characteristics, applications in decision-making, and the creations of effective algorithms for addressing such issues. In 2019, Al-Masarwah and Ahmad [19] introduced the notion of m-polar $(\alpha ,\beta )$-fuzzy ideals in BCK/BCI-algebras, which generalizes fuzzy ideals, bipolar fuzzy ideals, and bipolar $(\alpha ,\beta )$-fuzzy ideals within these algebras. They characterized m-polar $(\in ,\in \vee q)$-fuzzy ideals through level cut subsets and defined m-polar commutative ideals. In semigroups, Bashir et al. [20] came up with the idea of an m-PFS. Then, they considered generalized key results from BFSs to m-PFSs, focusing on m-polar fuzzy subsemigroups (ideals, generalized bi-ideals, bi-ideals, quasi-ideals, and interior ideals) in semigroups. They showed that every m-polar fuzzy bi-ideal of semigroups is the m-polar fuzzy generalized bi-ideal of semigroups, but this might not hold true the other way around. Thereafter, they explored the features of m-polar fuzzy ideals and used those features to describe regular and intra-regular semigroups. Additional investigations into m-PFSs have taken place, for instance [21,22,23]. Currently, there are still studies on the concept of different types of FSs to help address diverse data in different situations; for instance, El-Bably et al. [24] expanded traditional rough set theory through generalized rough set theory to improve decision-making under uncertainty. They introduced initial-minimal and initial-maximal neighborhoods, yielding eight new generalized rough approximations, which demonstrated accuracy rates up to $100\%$ in experiments on integrated nano-topological structures for continuous uncertainty in COVID-19 diagnosis. Moreover, Osman et al. [25] demonstrated the applicability of multi-pretopological models in medical information systems. They showed how multiset-based topology can effectively represent and analyze repetitive or uncertain medical data. Furthermore, Alshammari et al. [26] explored r-fuzzy soft $\delta$-open sets within the context of fuzzy soft topological spaces. They also examined related concepts, including fuzzy soft $\delta$-closure, $\delta$-interior, and various types of fuzzy soft continuity, such as semi-continuous, pre-continuous, virtually continuous, and weakly continuous functions.

Building on the previously mentioned concepts of PFSs and m-PFSs, Doggra and Pal [27] introduced the concept of m-PPFSs within BCK-algebras as an extension of both PFSs and m-PFSs. They proved that an m-polar image fuzzy implicative ideal of a BCK-algebra is an m-polar picture fuzzy ideal. However, this is only true in implicative BCK-algebras. Research into m-PFSs in algebraic structures is a relatively new and growing concept, particularly with semigroups, which are popular structures in algebraic studies. Therefore, we are presented with the opportunity to apply the concept of m-PFSs to the study of semigroups. The objective of this research is to use the concept of m-PPFSs in an investigation into semigroups. The concepts of m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs are introduced, and the connections among these ideas in semigroups are investigated. Subsequently, we provide characterizations of each of the concepts that were discussed in semigroups. Finally, we focus on the properties of the m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs within semigroups to help with the characterization of regular semigroups.

2. Preliminaries

In this section, we will examine the fundamental concepts and characteristics that will be used in subsequent sections. A semigroup is a structure $(S,·)$ consisting of a nonempty set S and a binary associative operation · on S. For each of the nonempty subsets A and B of semigroup S, we define $AB=\left\{ab\mid a\in A,b\in B\right\}$. Let S denote a semigroup. A nonempty subset T of S is defined as a subsemigroup of S if $TT\subseteq T$. A nonempty subset A of S is known as a left ideal (right ideal) of S if $SA\subseteq A$ ($AS\subseteq A$). A nonempty subset A of S is called an ideal of S if it is accepted as both a left ideal and a right ideal of S. A subsemigroup B of S is called a bi-ideal of S if $BSB\subseteq B$. A nonempty subset G of S is known as a generalized bi-ideal of S if $GSG\subseteq G$.

A fuzzy set (FS) [5] $\mu$ of a nonempty set X is a function that maps elements from X to the closed interval $[0,1]$, denoted as $\mu :X\to [0,1]$. A picture fuzzy set (PFS) [4] $\mathcal{A}$ on a universe X is defined as follows, $\mathcal{A}=\left\{\left(x,{\mu }_{\mathcal{A}}\left(x\right),{\eta }_{\mathcal{A}}\left(x\right),{\nu }_{\mathcal{A}}\left(x\right)\right)\mid x\in X\right\}$, where ${\mu }_{\mathcal{A}}\left(x\right),{\eta }_{\mathcal{A}}\left(x\right),{\nu }_{\mathcal{A}}\left(x\right)\in [0,1]$ denote the degree of positive membership, the degree of neutral membership, and the degree of negative membership, respectively, for each $x\in X$ to set $\mathcal{A}$, such that ${\mu }_{\mathcal{A}}$, ${\eta }_{\mathcal{A}}$, and ${\nu }_{\mathcal{A}}$ satisfy the following condition: the value of $0⩽{\mu }_{\mathcal{A}}\left(x\right)+{\eta }_{\mathcal{A}}\left(x\right)+{\nu }_{\mathcal{A}}\left(x\right)⩽1$ for all $x\in X$. An object of the form $\mathcal{A}=\left\{\left(x,{\mu }_{\mathcal{A}}^{+}\left(x\right),{\mu }_{\mathcal{A}}^{-}\left(x\right)\right)\mid x\in X\right\}$ is called a bipolar fuzzy set (BFS) [17] of a nonempty set X. Here, ${\mu }_{\mathcal{A}}^{+}:X\to [0,1]$ and ${\mu }_{\mathcal{A}}^{-}:X\to [-1,0]$.

An m-polar fuzzy set (m-PFS) [16] $\mathcal{A}$ over the universe set X is defined as $\mathcal{A}=\left\{\left(x,{\mu }_{\mathcal{A}}\left(x\right)\right)\mid x\in X\right\}$, where ${\mu }_{\mathcal{A}}:X\to {[0,1]}^m$ (with m representing a natural number) has been classified as an m-polar fuzzy set of X,

$${\mu }_{\mathcal{A}}\left(x\right)=\left(\left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(x\right),\left({\pi }_2\circ {\mu }_{\mathcal{A}}\right)\left(x\right),\dots ,\left({\pi }_m\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\right).$$

In this regard, ${\pi }_i:{[0,1]}^m\to [0,1]$ is the i-th projection mapping, such that $({\pi }_i\circ {\mu }_{\mathcal{A}})\left(x\right)$ represents the i-th component of ${\mu }_{\mathcal{A}}\left(x\right)$ for $i=1,2,\dots ,m$.

In addition, the poset with respect to partial order relation “≼” on ${[0,1]}^m$ is defined as for any $t=(t_1,t_2,\dots ,t_m),r=(r_1,r_2,\dots ,r_m)\in {[0,1]}^m$, $t\preccurlyeq r$ if and only if $t_i⩽r_i$ for all $i=1,2,\dots ,m$. In addition, $t\prec r$ if and only if $t_i<r_i$ for all $i=1,2,\dots ,m$. So, in the case of $t\succcurlyeq r$ and $t\succ r$, this results in $r\preccurlyeq t$ and $r\prec t$, respectively. Furthermore, we say that $\mathbf{0}=(0,0,\dots ,0)$ (m-tuple) is the smallest value and $\mathbf{1}=(1,1,\dots ,1)$ (m-tuple) is the largest value in ${[0,1]}^m$.

Let $\left\{n_i|i\in \Lambda \right\}$ be a family of real numbers. Subsequently, we denote

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

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

If $\Lambda$ is finite, we write $n_1\wedge n_2\wedge \cdots \wedge n_m,$ instead of $\wedge \left\{n_1,n_2,\dots ,n_m\right\}$ and $n_1\vee n_2\vee \cdots \vee n_m,$ instead of $\vee \left\{n_1,n_2,\dots ,n_m\right\}$. Let $u=(u_1,u_2,\dots ,u_m),v=(v_1,v_2,\dots ,v_m)\in {[0,1]}^m$. We then denote the following:

(i)

$u=v$ iff $u_i=v_i$ for all $i=1,2,\dots ,m$;

(ii)

$u⋏v$ iff $u_i\wedge v_i$ for all $i=1,2,\dots ,m$;

(iii)

$u⋎v$ iff $u_i\vee v_i$ for all $i=1,2,\dots ,m$.

In 2020, Dogra and Pal [27] introduced the notion of m-PPFS as a generalization of FSs, PFSs, BFSs, and m-PFSs.

Definition 1

([27]). Let X be a nonempty set and m be a natural number. An m-polar picture fuzzy set (m-PPFS) $\mathcal{A}$ over X is defined as an object of the following type:

$$\mathcal{A}=\left\{\left(x,{\mu }_{\mathcal{A}}\left(x\right),{\eta }_{\mathcal{A}}\left(x\right),{\nu }_{\mathcal{A}}\left(x\right)\right)\mid x\in X\right\},$$

where ${\mu }_{\mathcal{A}}:X\to {[0,1]}^m,\eta :X\to {[0,1]}^m$, and ${\nu }_{\mathcal{A}}:X\to {[0,1]}^m$ with the condition $0⩽\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)+\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)+\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)⩽1$ for all $x\in X$ and for all $i=1,2,\dots ,m$.

In this study, we will denote the m-PPFS $\mathcal{A}=\left\{\left(x,{\mu }_{\mathcal{A}}\left(x\right),{\eta }_{\mathcal{A}}\left(x\right),{\nu }_{\mathcal{A}}\left(x\right)\right)\mid x\in X\right\}$ with the symbol $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$. Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ and $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ be any two m-PPFSs on universe X. Then, we denote the following:

(i)

$\mathcal{A}⊑\mathcal{B}$ iff $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)⩽\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)⩾\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)$, and

$\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)⩾\left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(x\right)$ for all $x\in X$ and for all $i=1,2,\dots ,m$;

(ii)

$\mathcal{A}=\mathcal{B}$ iff $\mathcal{A}⊑\mathcal{B}$ and $\mathcal{B}⊑\mathcal{A}$;

(iii)

$\mathcal{A}\sqcap \mathcal{B}=\left\{\left(x,\left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(x\right),\left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(x\right),\left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(x\right)\right)\mid x\in X\right\}$, that is,

$\mathcal{A}\sqcap \mathcal{B}$ iff $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)$, and

$\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(x\right)$ for all $x\in X$ and for all $i=1,2,\dots ,m$;

(iv)

$\mathcal{A}\bigsqcup \mathcal{B}=\left\{(x,\left({\mu }_{\mathcal{A}}⋎{\mu }_{\mathcal{B}}\right)\left(x\right),\left({\eta }_{\mathcal{A}}⋏{\eta }_{\mathcal{B}}\right)\left(x\right),\left({\nu }_{\mathcal{A}}⋏{\nu }_{\mathcal{B}}\right)\left(x\right))\mid x\in X\right\}$, that is,

$\mathcal{A}\bigsqcup \mathcal{B}$ iff $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)$, and

$\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(x\right)$ for all $x\in X$ and for all $i=1,2,\dots ,m$.

For any nonempty set X, we define an m-PPFS $\mathcal{S}=\left(\mathbf{1},\mathbf{0},\mathbf{0}\right)$ on X. It follows that $\mathcal{A}⊑\mathcal{S}$ for all m-PPFS $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ on X.

Next, we define the concepts of m-PPFSubs, m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in the semigroups as follows.

Definition 2.

Let S be a semigroup. An m-PPFS $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ over S is called an m-polar picture fuzzy subsemigroup (m-PPFSub) of S if for every $x,y\in S$,

(i) 

${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(y\right)$;

(ii) 

${\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(y\right)$;

(iii) 

${\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(y\right)$;

that is,

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)\hfill \end{array}$$

for all $i=1,2,\dots ,m$, respectively.

Definition 3.

Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ be an m-PPFS over semigroup S. Then,

(i) 

$\mathcal{A}$ is called an m-polar picture fuzzy left ideal (m-PPFL) of S if

$${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(y\right),{\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(y\right),\mathit{and}{\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(y\right),$$

that is, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right),\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)$, and

$\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)$ for all $x,y\in S$ and for all $i=1,2,\dots ,m$;

(ii) 

$\mathcal{A}$ is called an m-polar picture fuzzy right ideal (m-PPFR) of S if

$${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right),{\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right),\mathit{and}{\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right),$$

that is, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right),\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)$, and

$\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)$ for all $x,y\in S$ and for all $i=1,2,\dots ,m$;

(iii) 

An m-PPFS $\mathcal{A}$ is called an m-polar picture fuzzy ideal (m-PPFI) of S if it is both an m-PPFL and an m-PPFR of S.

Definition 4.

Let S be a semigroup. An m-PPFSub $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ of S is called an m-polar picture fuzzy bi-ideal (m-PPFB) of S if for any $x,y,z\in S$,

(i) 

${\mu }_{\mathcal{A}}\left(xyz\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(z\right)$;

(ii) 

${\eta }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(z\right)$;

(iii) 

${\nu }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(z\right)$;

that is,

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xyz\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)\hfill \end{array}$$

for all $i=1,2,\dots ,m$, respectively.

Definition 5.

An m-PPFS $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ over semigroup S is said to be an m-polar picture fuzzy generalized bi-ideal (m-PPFGB) of S if for each $x,y,z\in S$,

(i) 

${\mu }_{\mathcal{A}}\left(xyz\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(z\right)$;

(ii) 

${\eta }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(z\right)$;

(iii) 

${\nu }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(z\right)$;

that is,

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xyz\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)\hfill \end{array}$$

for all $i=1,2,\dots ,m$, respectively.

Proposition 1.

Every m-PPFL (m-PPFR) of semigroup S is also an m-PPFB of S.

Proof. 

Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ be any m-PPFL of semigroup S. Let $x,y,z\in S$ and $i=1,2,\dots ,m$. Then, we have the following:

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right).\hfill \end{array}$$

Similarly, we can prove that $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)$. Hence, ${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(y\right),{\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(y\right)$, and ${\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(y\right)$. It follows that $\mathcal{A}$ is an m-PPFSub of S. Moreover,

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xyz\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right).\hfill \end{array}$$

Also, $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xyz\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)$. Thus, ${\mu }_{\mathcal{A}}\left(xyz\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(z\right),{\eta }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(z\right)$, and ${\nu }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(z\right)$. Therefore, $\mathcal{A}$ is an m-PPFB of S. For the m-PPFR $\mathcal{A}$ of S, we can prove similarly.  □

The opposite of Proposition 1 is not necessarily true in general, as seen by the following example:

Example 1.

Consider semigroup $S=\left\{a,b,c,d\right\}$, as referenced in [28], presented in

Table 1. The operation “·” on S.

·	a	b	c	d
a	a	a	a	a
b	a	a	a	a
c	a	a	a	b
d	a	a	b	c

.

Define a 4-PPFS $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ on S as

Table 2. The representation of the value of a 4-PPFS $\mathcal{A}$ on S.

$\mathcal{A}$	${\mathit{\mu }}_{\mathcal{A}}$	${\mathit{\eta }}_{\mathcal{A}}$	${\mathit{\nu }}_{\mathcal{A}}$
a	$\left(0.6,0.5,0.4,0.7\right)$	$\left(0.1,0,0.1,0\right)$	$\left(0,0.1,0.1,0\right)$
b	$\left(0.4,0.2,0.2,0.5\right)$	$\left(0.3,0.2,0.4,0.2\right)$	$\left(0.3,0.5,0.4,0.3\right)$
c	$\left(0.5,0.4,0.3,0.6\right)$	$\left(0.2,0.1,0.3,0.1\right)$	$\left(0.1,0.3,0.3,0.2\right)$
d	$\left(0.1,0.1,0,0.2\right)$	$\left(0.4,0.3,0.5,0.3\right)$	$\left(0.5,0.6,0.5,0.5\right)$

.

Following on, straightforward computations indicate that the 4-PPFS $\mathcal{A}$ on S satisfies a 4-PPFB of S. Then, consider

$$\begin{array}{cc}\hfill ({\pi }_2\circ {\mu }_{\mathcal{A}})\left(dc\right)& =({\pi }_2\circ {\mu }_{\mathcal{A}})\left(b\right)=0.2\ngeqq 0.4=({\pi }_2\circ {\mu }_{\mathcal{A}})\left(c\right),\hfill \\ \hfill ({\pi }_4\circ {\mu }_{\mathcal{A}})\left(cd\right)& =({\pi }_4\circ {\mu }_{\mathcal{A}})\left(b\right)=0.5\ngeqq 0.6=({\pi }_4\circ {\mu }_{\mathcal{A}})\left(c\right).\hfill \end{array}$$

Here, we can see that ${\mu }_{\mathcal{A}}\left(dc\right)⋡{\mu }_{\mathcal{A}}\left(c\right)$ and ${\mu }_{\mathcal{A}}\left(cd\right)⋡{\mu }_{\mathcal{A}}\left(c\right)$. Therefore, the 4-PPFB $\mathcal{A}$ is neither a 4-PPFL nor a 4-PPFR of S.

Regarding Definition 4, it is not difficult to see that the m-PPFB $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ is also an m-PPFGB in semigroup S; however, the opposite is generally not valid, as seen in the following example:

Example 2.

Let $S=\left\{a,b,c,d\right\}$ denote a semigroup, as defined in [29], presented in

Table 3. The operation “·” on S.

·	a	b	c	d
a	a	a	a	a
b	a	a	a	a
c	a	a	b	a
d	a	a	b	b

.

Next, we define a 3-PPFS $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ on S as

Table 4. The representation of the value of a 3-PPFS $\mathcal{A}$ on S.

$\mathcal{A}$	${\mathit{\mu }}_{\mathcal{A}}$	${\mathit{\eta }}_{\mathcal{A}}$	${\mathit{\nu }}_{\mathcal{A}}$
a	$\left(0.5,0.4,0.6\right)$	$\left(0.1,0.2,0\right)$	$\left(0.2,0,0.1\right)$
b	$\left(0.1,0,0.2\right)$	$\left(0.3,0.4,0.3\right)$	$\left(0.6,0.5,0.5\right)$
c	$\left(0.5,0.3,0.4\right)$	$\left(0.2,0.2,0.1\right)$	$\left(0.3,0.1,0.2\right)$
d	$\left(0.4,0.2,0.3\right)$	$\left(0.2,0.3,0.3\right)$	$\left(0.4,0.3,0.4\right)$

.

Subsequent calculations verify that the 3-PPFS $\mathcal{A}$ on S achieves the conditions of a 3-PPFGB of S. Now, we obtain

$$\left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(dd\right)=\left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(b\right)=0.1\ngeqq 0.4=\left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(d\right)\wedge \left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(d\right).$$

Also, ${\mu }_{\mathcal{A}}\left(dd\right)⋡{\mu }_{\mathcal{A}}\left(d\right)⋏{\mu }_{\mathcal{A}}\left(d\right)$. Hence, $\mathcal{A}$ is not a 3-PPFSub of S. This implies that the 3-PPFGB $\mathcal{A}$ is also not a 3-PPFB of S.

Proposition 2.

Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ and $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ be any two m-PPFSs on semigroup S. Then, the following properties are true:

(i) 

If $\mathcal{A}$ and $\mathcal{B}$ are m-PPFSubs of S, then $\mathcal{A}\sqcap \mathcal{B}$ is also an m-PPFSub of S;

(ii) 

If $\mathcal{A}$ and $\mathcal{B}$ are m-PPFLs of S, then $\mathcal{A}\sqcap \mathcal{B}$ is also an m-PPFL of S;

(iii) 

If $\mathcal{A}$ and $\mathcal{B}$ are m-PPFRs of S, then $\mathcal{A}\sqcap \mathcal{B}$ is also an m-PPFR of S;

(iv) 

If $\mathcal{A}$ and $\mathcal{B}$ are m-PPFIs of S, then $\mathcal{A}\sqcap \mathcal{B}$ is also an m-PPFI of S;

(v) 

If $\mathcal{A}$ and $\mathcal{B}$ are m-PPFGBs of S, then $\mathcal{A}\sqcap \mathcal{B}$ is also an m-PPFGB of S;

(vi) 

If $\mathcal{A}$ and $\mathcal{B}$ are m-PPFBs of S, then $\mathcal{A}\sqcap \mathcal{B}$ is also an m-PPFB of S.

Proof. 

(i) Assume that $\mathcal{A}$ and $\mathcal{B}$ are m-PPFSubs of S. For every $x,y\in S$ and for every $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& \wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(xy\right)\hfill \\ \hfill & ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)\right]\wedge \left[\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(y\right)\right]\hfill \\ \hfill & =\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\right]\wedge \left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(y\right)\right],\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& \vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(xy\right)\hfill \\ \hfill & ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)\right]\vee \left[\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(y\right)\right]\hfill \\ \hfill & =\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\right]\vee \left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(y\right)\right].\hfill \end{array}$$

Similarly, it follows that $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(xy\right)⩽\left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(x\right)\right]\vee \left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(y\right)\right]$. This means that

$$\begin{array}{cc}\hfill \left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(xy\right)& \succcurlyeq \left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(x\right)⋏\left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(y\right),\hfill \\ \hfill \left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(xy\right)& \preccurlyeq \left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(x\right)⋎\left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(y\right),\hfill \\ \hfill \left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(xy\right)& \preccurlyeq \left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(x\right)⋎\left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(y\right).\hfill \end{array}$$

Hence, $\mathcal{A}\sqcap \mathcal{B}$ is an m-PPFSub of S.

(ii) Let $\mathcal{A}$ and $\mathcal{B}$ be m-PPFLs of S. For each $x,y\in S$ and for each $i=1,2,\dots ,m$, we obtain the following:

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(xy\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(y\right).\hfill \end{array}$$

It follows that $\left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(xy\right)\succcurlyeq \left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(y\right)$, $\left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(xy\right)\preccurlyeq \left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(y\right)$, and $\left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(xy\right)\preccurlyeq \left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(y\right)$. We find that $\mathcal{A}\sqcap \mathcal{B}$ is an m-PPFL of S.

(iii) The proof is similar to (ii).

(iv) It is followed by (ii) and (iii).

(v) Assume that $\mathcal{A}$ and $\mathcal{B}$ are m-PPFGBs of S. For any $x,y,z\in S$ and for any $i=1,2,\dots ,m$, we obtain

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xyz\right)& \wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(xyz\right)\hfill \\ \hfill & ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)\right]\wedge \left[\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(z\right)\right]\hfill \\ \hfill & =\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\right]\wedge \left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(z\right)\right],\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xyz\right)& \vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(xyz\right)\hfill \\ \hfill & ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)\right]\vee \left[\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(z\right)\right]\hfill \\ \hfill & =\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\right]\vee \left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(z\right)\right].\hfill \end{array}$$

Proving it in the same way as the previous case, we have that $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xyz\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(xyz\right)⩽\left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(x\right)\right]\vee \left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(z\right)\right]$. Thus,

$$\begin{array}{cc}\hfill \left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(xyz\right)& \succcurlyeq \left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(x\right)⋏\left({\mu }_{\mathcal{A}}⋏{\mu }_{\mathcal{B}}\right)\left(z\right),\hfill \\ \hfill \left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(xyz\right)& \preccurlyeq \left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(x\right)⋎\left({\eta }_{\mathcal{A}}⋎{\eta }_{\mathcal{B}}\right)\left(z\right),\hfill \\ \hfill \left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(xyz\right)& \preccurlyeq \left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(x\right)⋎\left({\nu }_{\mathcal{A}}⋎{\nu }_{\mathcal{B}}\right)\left(z\right).\hfill \end{array}$$

Therefore, $\mathcal{A}\sqcap \mathcal{B}$ is an m-PPFGB of S.

(vi) The proof is obtained by (i) and (v).  □

In general, the union of m-PPFSubs (m-PPFLs, m-PPFRs, m-PPFIs, m-PPFGBs, and m-PPFBs) in semigroups does not need to be the same, as in the following example.

Example 3.

Let semigroup $S=\left\{a,b,c,d\right\}$, as cited in [30], be given in

Table 5. The operation “·” on S.

·	a	b	c	d
a	a	a	a	a
b	a	b	c	b
c	a	a	a	a
d	a	a	b	d

.

Let us now explore two 3-PPFSs, $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ and $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$, on S as seen in

Table 6. The representation of the value of a 3-PPFS $\mathcal{A}$ on S.

$\mathcal{A}$	${\mathit{\mu }}_{\mathcal{A}}$	${\mathit{\eta }}_{\mathcal{A}}$	${\mathit{\nu }}_{\mathcal{A}}$
a	$\left(0.7,0.5,0.4\right)$	$\left(0.1,0.2,0.2\right)$	$\left(0,0.1,0\right)$
b	$\left(0.1,0.2,0.1\right)$	$\left(0.3,0.4,0.5\right)$	$\left(0.4,0.3,0.4\right)$
c	$\left(0.7,0.4,0.3\right)$	$\left(0.2,0.3,0.4\right)$	$\left(0.1,0.2,0.3\right)$
d	$\left(0.1,0.2,0.1\right)$	$\left(0.3,0.4,0.5\right)$	$\left(0.4,0.3,0.4\right)$

and

Table 7. The representation of the value of a 3-PPFS $\mathcal{B}$ on S.

$\mathcal{B}$	${\mathit{\mu }}_{\mathcal{B}}$	${\mathit{\eta }}_{\mathcal{B}}$	${\mathit{\nu }}_{\mathcal{B}}$
a	$\left(0.5,0.6,0.3\right)$	$\left(0.2,0.2,0.4\right)$	$\left(0.1,0,0\right)$
b	$\left(0.1,0.3,0.1\right)$	$\left(0.4,0.4,0.7\right)$	$\left(0.5,0.3,0.2\right)$
c	$\left(0.1,0.3,0.1\right)$	$\left(0.4,0.4,0.7\right)$	$\left(0.5,0.3,0.2\right)$
d	$\left(0.4,0.5,0.2\right)$	$\left(0.3,0.3,0.5\right)$	$\left(0.3,0.1,0.1\right)$

, respectively.

Through routine calculations, we determine that $\mathcal{A}$ and $\mathcal{B}$ can be considered as 3-PPFSubs of S. We now investigate

$$\begin{array}{cc}\hfill \left({\pi }_2\circ {\mu }_{\mathcal{A}}\right)\left(dc\right)& \vee \left({\pi }_2\circ {\mu }_{\mathcal{B}}\right)\left(dc\right)=\left({\pi }_2\circ {\mu }_{\mathcal{A}}\right)\left(b\right)\vee \left({\pi }_2\circ {\mu }_{\mathcal{B}}\right)\left(b\right)\hfill \\ \hfill & =0.2\vee 0.3=0.3\hfill \\ \hfill & \ngeqq 0.4=\left(0.2\vee 0.5\right)\wedge \left(0.4\vee 0.3\right)\hfill \\ \hfill & =\left[\left({\pi }_2\circ {\mu }_{\mathcal{A}}\right)\left(d\right)\vee \left({\pi }_2\circ {\mu }_{\mathcal{B}}\right)\left(d\right)\right]\wedge \left[\left({\pi }_2\circ {\mu }_{\mathcal{A}}\right)\left(c\right)\vee \left({\pi }_2\circ {\mu }_{\mathcal{B}}\right)\left(c\right)\right].\hfill \end{array}$$

It can be seen that $({\mu }_{\mathcal{A}}⋎{\mu }_{\mathcal{B}})\left(dc\right)⋡({\mu }_{\mathcal{A}}⋎{\mu }_{\mathcal{B}})\left(d\right)⋏({\mu }_{\mathcal{A}}⋎{\mu }_{\mathcal{B}})\left(c\right)$. Consequently, $\mathcal{A}\bigsqcup \mathcal{B}$ cannot satisfy the requirements to be identified as a 3-PPFSub of S.

3. Characterizations of Many Types of $\mathit{m}$-PPFIs

In this section, we will explore the characterizations of different types of m-PPFIs of semigroups using some newly developed concepts and some existing ones, the properties of which will guide this study into the next section.

For any subset A of a nonempty set S, we denote by ${\mathcal{C}}_A=\left({\mu }_{{\mathcal{C}}_A},{\eta }_{{\mathcal{C}}_A},{\nu }_{{\mathcal{C}}_A}\right)$ the m-polar picture characteristic function (m-PPCF) of S, where

$${\mathcal{C}}_A\left(x\right)=\left\{\begin{array}{cc}\left(\mathbf{1},\mathbf{0},\mathbf{0}\right)\hfill & \mathrm{if}x\in A,\hfill \\ \left(\mathbf{0},\mathbf{0},\mathbf{1}\right)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

This means that

$${\mu }_{{\mathcal{C}}_A}\left(x\right)=\left\{\begin{array}{cc}\mathbf{1}\hfill & \mathrm{if}x\in A,\hfill \\ \mathbf{0}\hfill & \mathrm{otherwise},\hfill \end{array}\right. {\eta }_{{\mathcal{C}}_A}\left(x\right)=\mathbf{0} \mathrm{for} \mathrm{all} x\in X, {\nu }_{{\mathcal{C}}_A}\left(x\right)=\left\{\begin{array}{cc}\mathbf{0}\hfill & \mathrm{if}x\in A,\hfill \\ \mathbf{1}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Observe that, if $A=S$, then ${\mathcal{C}}_A=\mathcal{S}$.

Let S be a semigroup and $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ and $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ be m-PPFSs over S. The product of $\mathcal{A}$ and $\mathcal{B}$ is defined by

$$\mathcal{A}\diamond \mathcal{B}=\left({\mu }_{\mathcal{A}}\diamond {\mu }_{\mathcal{B}},{\eta }_{\mathcal{A}}\diamond {\eta }_{\mathcal{B}},{\nu }_{\mathcal{A}}\diamond {\nu }_{\mathcal{B}}\right),$$

where

$$\begin{array}{cc}\hfill {\mu }_{\mathcal{A}}\diamond {\mu }_{\mathcal{B}}& =\left(\left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_1\circ {\mu }_{\mathcal{B}}\right),\dots ,\left({\pi }_m\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_m\circ {\mu }_{\mathcal{B}}\right)\right),\hfill \\ \hfill {\eta }_{\mathcal{A}}\diamond {\eta }_{\mathcal{B}}& =\left(\left({\pi }_1\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_1\circ {\eta }_{\mathcal{B}}\right),\dots ,\left({\pi }_m\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_m\circ {\eta }_{\mathcal{B}}\right)\right),\hfill \\ \hfill {\nu }_{\mathcal{A}}\diamond {\nu }_{\mathcal{B}}& =\left(\left({\pi }_1\circ {\nu }_{\mathcal{A}}\right)\diamond \left({\pi }_1\circ {\nu }_{\mathcal{B}}\right),\dots ,\left({\pi }_m\circ {\nu }_{\mathcal{A}}\right)\diamond \left({\pi }_m\circ {\nu }_{\mathcal{B}}\right)\right),\hfill \end{array}$$

and for any $i=1,2,\dots ,m$,

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\right]\left(x\right)& =\left\{\begin{array}{cc}{\displaystyle \underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(b\right)\right\}}\hfill & \mathrm{if}x\in S^2,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \\ \hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\right]\left(x\right)& =\left\{\begin{array}{cc}{\displaystyle \underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(b\right)\right\}}\hfill & \mathrm{if}x\in S^2,\hfill \\ 1\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \\ \hfill \left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\right]\left(x\right)& =\left\{\begin{array}{cc}{\displaystyle \underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(b\right)\right\}}\hfill & \mathrm{if}x\in S^2,\hfill \\ 1\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

Lemma 1.

Let A and B be nonempty subsets of semigroup S. Then, the following properties achieve

(i) 

${\mathcal{C}}_A\sqcap {\mathcal{C}}_B={\mathcal{C}}_{A\cap B}$;

(ii) 

${\mathcal{C}}_A\diamond {\mathcal{C}}_B={\mathcal{C}}_{AB}$.

Proof. 

(i) Let $x\in S$. If $x\in A\cap B$, then $x\in A$ and $x\in B$. For every $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\left(x\right)& =1\wedge 1=1=({\pi }_i\circ {\mu }_{{\mathcal{C}}_{A\cap B}})\left(x\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\left(x\right)& =0\vee 0=0=\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_{A\cap B}}\right)\left(x\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\left(x\right)& =0\vee 0=0=\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_{A\cap B}}\right)\left(x\right).\hfill \end{array}$$

On the other hand, if $x\notin A\cap B$, then $x\notin A$ or $x\notin B$. For each $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\left(x\right)& =0=\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_{A\cap B}}\right)\left(x\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\left(x\right)& =0=\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_{A\cap B}}\right)\left(x\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\left(x\right)& =1=\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_{A\cap B}}\right)\left(x\right).\hfill \end{array}$$

We conclude that ${\mathcal{C}}_A\sqcap {\mathcal{C}}_B={\mathcal{C}}_{A\cap B}$.

(ii) Let $x\in S$. Case 1: $x\in AB$. Then, $x=ab$ for some $a\in A$ and $b\in B$. For any $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\left(u\right)\wedge \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\left(v\right)\right\}}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\left(b\right)\hfill \\ \hfill & =1=\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_{AB}}\right)\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\left(u\right)\vee \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\left(v\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\left(b\right)\hfill \\ \hfill & =0=\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_{AB}}\right)\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigwedge }\left\{\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\left(u\right)\vee \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\left(v\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\left(a\right)\vee \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\left(b\right)\hfill \\ \hfill & =0=\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_{AB}}\right)\left(x\right).\hfill \end{array}$$

Also,

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& =\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_{AB}}\right)\left(x\right),\hfill \\ \hfill \left[\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& =\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_{AB}}\right)\left(x\right),\hfill \\ \hfill \left[\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& =\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_{AB}}\right)\left(x\right)\hfill \end{array}$$

for all $i=1,2,\dots ,m$. This proves that ${\mathcal{C}}_A\diamond {\mathcal{C}}_B={\mathcal{C}}_{AB}$.

Case 2: $x\notin AB$. Then, $x\ne ab$ for all $a\notin A$ or $b\notin B$. For each $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_A}\right)\left(u\right)\wedge \left({\pi }_i\circ {\mu }_{{\mathcal{C}}_B}\right)\left(v\right)\right\}}\hfill \\ \hfill & =0=\left({\pi }_i\circ {\mu }_{{\mathcal{C}}_{AB}}\right)\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_A}\right)\left(u\right)\vee \left({\pi }_i\circ {\eta }_{{\mathcal{C}}_B}\right)\left(v\right)\right\}}\hfill \\ \hfill & =0=\left({\pi }_i\circ {\eta }_{{\mathcal{C}}_{AB}}\right)\left(x\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\diamond \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigwedge }\left\{\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_A}\right)\left(u\right)\vee \left({\pi }_i\circ {\nu }_{{\mathcal{C}}_B}\right)\left(v\right)\right\}}\hfill \\ \hfill & =1=\left({\pi }_i\circ {\nu }_{{\mathcal{C}}_{AB}}\right)\left(x\right).\hfill \end{array}$$

Therefore, ${\mathcal{C}}_A\diamond {\mathcal{C}}_B={\mathcal{C}}_{AB}$.  □

Lemma 2.

Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right),\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right),\mathcal{C}=\left({\mu }_{\mathcal{C}},{\eta }_{\mathcal{C}},{\nu }_{\mathcal{C}}\right),$ and $\mathcal{D}=\left({\mu }_{\mathcal{D}},{\eta }_{\mathcal{D}},{\nu }_{\mathcal{D}}\right)$ be m-PPFSs on semigroup S. If $\mathcal{A}⊑\mathcal{B}$ and $\mathcal{C}⊑\mathcal{D}$, then $\mathcal{A}\diamond \mathcal{C}⊑\mathcal{B}\diamond \mathcal{D}$.

Proof. 

Let $x\in S$. It is obvious in case $x\notin S^2$. If $x\in S^2$, then for each $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{C}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(u\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{C}}\right)\left(v\right)\right\}}\hfill \\ \hfill & {\displaystyle ⩽\underset{x=uv}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(u\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{D}}\right)\left(v\right)\right\}}\hfill \\ \hfill & =\left[\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{D}}\right)\right]\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{C}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=uv}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(u\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{C}}\right)\left(v\right)\right\}}\hfill \\ \hfill & {\displaystyle ⩾\underset{x=uv}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(u\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{D}}\right)\left(v\right)\right\}}\hfill \\ \hfill & =\left[\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{D}}\right)\right]\left(x\right).\hfill \end{array}$$

Similarly, we can show that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{C}}\right)\right]\left(x\right)⩾\left[\left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{D}}\right)\right]\left(x\right)$. This implies that $\mathcal{A}\diamond \mathcal{C}⊑\mathcal{B}\diamond \mathcal{D}$.  □

Theorem 1.

Let S be a semigroup, A be a nonempty subset of S, and ${\mathcal{C}}_A=\left({\mu }_{{\mathcal{C}}_A},{\eta }_{{\mathcal{C}}_A},{\nu }_{{\mathcal{C}}_A}\right)$ be an m-PPFS on S. Then, the following statements hold:

(i) 

A is a subsemigroup of S if and only if ${\mathcal{C}}_A$ is an m-PPFSub of S;

(ii) 

A is a left ideal of S if and only if ${\mathcal{C}}_A$ is an m-PPFL of S;

(iii) 

A is a right ideal of S if and only if ${\mathcal{C}}_A$ is an m-PPFR of S;

(iv) 

A is an ideal of S if and only if ${\mathcal{C}}_A$ is an m-PPFI of S;

(v) 

A is a bi-ideal of S if and only if ${\mathcal{C}}_A$ is an m-PPFB of S;

(vi) 

A is a generalized bi-ideal of S if and only if ${\mathcal{C}}_A$ is an m-PPFGB of S.

Proof. 

(i) Assume that A is a subsemigroup of S. Let $x,y\in S$. If $xy\in A$, then

$$\begin{array}{cc}\hfill {\mu }_{{\mathcal{C}}_A}\left(xy\right)& =\mathbf{1}\succcurlyeq {\mu }_{{\mathcal{C}}_A}\left(x\right)⋏{\mu }_{{\mathcal{C}}_A}\left(y\right),\hfill \\ \hfill {\eta }_{{\mathcal{C}}_A}\left(xy\right)& =\mathbf{0}\preccurlyeq {\eta }_{{\mathcal{C}}_A}\left(x\right)⋎{\eta }_{{\mathcal{C}}_A}\left(y\right),\hfill \\ \hfill {\nu }_{{\mathcal{C}}_A}\left(xy\right)& =\mathbf{0}\preccurlyeq {\nu }_{{\mathcal{C}}_A}\left(x\right)⋎{\nu }_{{\mathcal{C}}_A}\left(y\right).\hfill \end{array}$$

On the other hand, let $xy\notin A$. By assumption, $x\notin A$ or $y\notin A$. Thus, we have

$$\begin{array}{cc}\hfill {\mu }_{{\mathcal{C}}_A}\left(xy\right)& =\mathbf{0}={\mu }_{{\mathcal{C}}_A}\left(x\right)⋏{\mu }_{{\mathcal{C}}_A}\left(y\right),\hfill \\ \hfill {\eta }_{{\mathcal{C}}_A}\left(xy\right)& =\mathbf{0}={\eta }_{{\mathcal{C}}_A}\left(x\right)⋎{\eta }_{{\mathcal{C}}_A}\left(y\right),\hfill \\ \hfill {\nu }_{{\mathcal{C}}_A}\left(xy\right)& =\mathbf{1}={\nu }_{{\mathcal{C}}_A}\left(x\right)⋎{\nu }_{{\mathcal{C}}_A}\left(y\right).\hfill \end{array}$$

Hence, ${\mathcal{C}}_A$ is an m-PPFSub of S. Conversely, assume that ${\mathcal{C}}_A$ is an m-PPFSub of S. Let $a,b\in A$. Then, we have

$$\begin{array}{cc}\hfill {\mu }_{{\mathcal{C}}_A}\left(ab\right)& \succcurlyeq {\mu }_{{\mathcal{C}}_A}\left(a\right)⋏{\mu }_{{\mathcal{C}}_A}\left(b\right)=\mathbf{1},\hfill \\ \hfill {\eta }_{{\mathcal{C}}_A}\left(ab\right)& \preccurlyeq {\eta }_{{\mathcal{C}}_A}\left(a\right)⋎{\eta }_{{\mathcal{C}}_A}\left(b\right)=\mathbf{0},\hfill \\ \hfill {\nu }_{{\mathcal{C}}_A}\left(ab\right)& \preccurlyeq {\nu }_{{\mathcal{C}}_A}\left(a\right)⋎{\nu }_{{\mathcal{C}}_A}\left(b\right)=\mathbf{0}.\hfill \end{array}$$

This implies that ${\mu }_{{\mathcal{C}}_A}\left(ab\right)=\mathbf{1},{\eta }_{{\mathcal{C}}_A}\left(ab\right)=\mathbf{0}$, and ${\nu }_{{\mathcal{C}}_A}\left(ab\right)=\mathbf{0}$. So, ${\mathcal{C}}_A\left(ab\right)=(\mathbf{1},\mathbf{0},\mathbf{0})$, and it follows $ab\in A$. Therefore, A is a subsemigroup of S.

(ii) Assume that A is a left ideal of S. Let $x,y\in S$. If $xy\in A$, then

$${\mu }_{{\mathcal{C}}_A}\left(xy\right)=\mathbf{1}\succcurlyeq {\mu }_{{\mathcal{C}}_A}\left(y\right), {\eta }_{{\mathcal{C}}_A}\left(xy\right)=\mathbf{0}={\eta }_{{\mathcal{C}}_A}\left(y\right), \mathrm{and} {\nu }_{{\mathcal{C}}_A}\left(xy\right)=\mathbf{0}\preccurlyeq {\nu }_{{\mathcal{C}}_A}\left(y\right).$$

Otherwise, let $xy\notin A$. So, $y\notin A$. Then, we have

$${\mu }_{{\mathcal{C}}_A}\left(xy\right)=\mathbf{0}={\mu }_{{\mathcal{C}}_A}\left(y\right),{\eta }_{{\mathcal{C}}_A}\left(xy\right)=\mathbf{0}={\eta }_{{\mathcal{C}}_A}\left(y\right), \mathrm{and} {\nu }_{{\mathcal{C}}_A}\left(xy\right)=\mathbf{1}={\nu }_{{\mathcal{C}}_A}\left(y\right).$$

It turns out that ${\mathcal{C}}_A$ is an m-PPFL of S. Conversely, let $x\in S$ and $a\in A$. Then, we have

$${\mu }_{{\mathcal{C}}_A}\left(xa\right)\succcurlyeq {\mu }_{{\mathcal{C}}_A}\left(a\right)=\mathbf{1}, {\eta }_{{\mathcal{C}}_A}\left(xa\right)\preccurlyeq {\eta }_{{\mathcal{C}}_A}\left(a\right)=\mathbf{0}, \mathrm{and} {\nu }_{{\mathcal{C}}_A}\left(xa\right)\preccurlyeq {\nu }_{{\mathcal{C}}_A}\left(a\right)=\mathbf{0}.$$

Also, ${\mu }_{{\mathcal{C}}_A}\left(xa\right)=\mathbf{1},{\eta }_{{\mathcal{C}}_A}\left(xa\right)=\mathbf{0}$, and ${\nu }_{{\mathcal{C}}_A}\left(xa\right)=\mathbf{0}$. This means that ${\mathcal{C}}_A\left(xa\right)=(\mathbf{1},\mathbf{0},\mathbf{0})$, and then $xa\in A$. Hence, A is a left ideal of S.

(iii) The proof can be demonstrated using an approach similar to the one shown in (ii).

(iv) The proof is derived from (ii) and (iii).

(v) Assume that A is a generalized bi-ideal of S. Let $x,y,z\in S$. If $xyz\in A$, then

$$\begin{array}{cc}\hfill {\mu }_{{\mathcal{C}}_A}\left(xyz\right)& =\mathbf{1}\succcurlyeq {\mu }_{{\mathcal{C}}_A}\left(x\right)⋏{\mu }_{{\mathcal{C}}_A}\left(z\right),\hfill \\ \hfill {\eta }_{{\mathcal{C}}_A}\left(xyz\right)& =\mathbf{0}\preccurlyeq {\eta }_{{\mathcal{C}}_A}\left(x\right)⋎{\eta }_{{\mathcal{C}}_A}\left(z\right),\hfill \\ \hfill {\nu }_{{\mathcal{C}}_A}\left(xyz\right)& =\mathbf{0}\preccurlyeq {\nu }_{{\mathcal{C}}_A}\left(x\right)⋎{\nu }_{{\mathcal{C}}_A}\left(z\right).\hfill \end{array}$$

Suppose that $xyz\notin A$. Then, $x\notin A$ or $z\notin A$. It follows that

$$\begin{array}{cc}\hfill {\mu }_{{\mathcal{C}}_A}\left(xyz\right)& =\mathbf{0}={\mu }_{{\mathcal{C}}_A}\left(x\right)⋏{\mu }_{{\mathcal{C}}_A}\left(z\right),\hfill \\ \hfill {\eta }_{{\mathcal{C}}_A}\left(xyz\right)& =\mathbf{0}={\eta }_{{\mathcal{C}}_A}\left(x\right)⋎{\eta }_{{\mathcal{C}}_A}\left(z\right),\hfill \\ \hfill {\nu }_{{\mathcal{C}}_A}\left(xyz\right)& =\mathbf{1}={\nu }_{{\mathcal{C}}_A}\left(x\right)⋎{\nu }_{{\mathcal{C}}_A}\left(z\right).\hfill \end{array}$$

Hence, ${\mathcal{C}}_A$ is an m-PPFGB of S. Conversely, let $x\in S$ and $a,b\in A$. Then, we have

$$\begin{array}{cc}\hfill {\mu }_{{\mathcal{C}}_A}\left(axb\right)& \succcurlyeq {\mu }_{{\mathcal{C}}_A}\left(a\right)⋏{\mu }_{{\mathcal{C}}_A}\left(b\right)=\mathbf{1},\hfill \\ \hfill {\eta }_{{\mathcal{C}}_A}\left(axb\right)& \preccurlyeq {\eta }_{{\mathcal{C}}_A}\left(a\right)⋎{\eta }_{{\mathcal{C}}_A}\left(b\right)=\mathbf{0},\hfill \\ \hfill {\nu }_{{\mathcal{C}}_A}\left(axb\right)& \preccurlyeq {\nu }_{{\mathcal{C}}_A}\left(a\right)⋎{\nu }_{{\mathcal{C}}_A}\left(b\right)=\mathbf{0}.\hfill \end{array}$$

So, ${\mu }_{{\mathcal{C}}_A}\left(axb\right)=\mathbf{1},{\eta }_{{\mathcal{C}}_A}\left(axb\right)=\mathbf{0}$, and ${\nu }_{{\mathcal{C}}_A}\left(axb\right)=\mathbf{0}$. This shows that ${\mathcal{C}}_A\left(axb\right)=(\mathbf{1},\mathbf{0},\mathbf{0})$. We find that $axb\in A$. Therefore, A is a generalized bi-ideal of S.

(vi) It follows from (i) and (v).  □

The following definitions have been adapted from [27].

Definition 6.

Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ be an m-PPFS on a nonempty set, X, and let $t=(t_1,t_2,\dots ,t_m),r=(r_1,r_2,\dots ,r_m),s=(s_1,s_2,\dots ,s_m)\in {[0,1]}^m$. Set ${\mathcal{A}}_{(t,r,s)}$ is called a $(t,r,s)$-cut or $(t,r,s)$-level set of $\mathcal{A}$ over X and is defined by

$${\mathcal{A}}_{(t,r,s)}=\left\{x\in X\mid \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)⩾t_i,\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)⩽r_i,\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)⩽s_i\forall i=1,2,\dots ,m\right\},$$

and it satisfies the property $0⩽t_i+r_i+s_i⩽1$ for all $i=1,2,\dots ,m$.

Theorem 2.

Let S be a semigroup and $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ be an m-PPFS on S. Then, the following properties hold:

(i) 

$\mathcal{A}$ is an m-PPFSub of S if and only if for every $t,r,s\in {[0,1]}^m$, ${\mathcal{A}}_{(t,r,s)}$ is a subsemigroup of S when it is nonempty;

(ii) 

$\mathcal{A}$ is an m-PPFL of S if and only if for every $t,r,s\in {[0,1]}^m$, ${\mathcal{A}}_{(t,r,s)}$ is a left ideal of S when it is nonempty;

(iii) 

$\mathcal{A}$ is an m-PPFR of S if and only if for every $t,r,s\in {[0,1]}^m$, ${\mathcal{A}}_{(t,r,s)}$ is a right ideal of S when it is nonempty;

(iv) 

$\mathcal{A}$ is an m-PPFI of S if and only if for every $t,r,s\in {[0,1]}^m$, ${\mathcal{A}}_{(t,r,s)}$ is an ideal of S when it is nonempty;

(v) 

$\mathcal{A}$ is an m-PPFGB of S if and only if for every $t,r,s\in {[0,1]}^m$, ${\mathcal{A}}_{(t,r,s)}$ is a generalized bi-ideal of S when it is nonempty;

(vi) 

$\mathcal{A}$ is an m-PPFB of S if and only if for every $t,r,s\in {[0,1]}^m$, ${\mathcal{A}}_{(t,r,s)}$ is a bi-ideal of S when it is nonempty.

Proof. 

(i) Assume that $\mathcal{A}$ is an m-PPFSub of S. Let $t=(t_1,t_2,\dots ,t_m),r=(r_1,r_2,\dots ,r_m)$, $s=(s_1,s_2,\dots ,s_m)\in {[0,1]}^m$ be such that ${\mathcal{A}}_{(t,r,s)}\ne \varnothing$. Let $x,y\in {\mathcal{A}}_{(t,r,s)}$. Then, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)⩾t_i$, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)⩾t_i$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)⩽r_i$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)⩽r_i$, $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)⩽s_i$, and $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)⩽s_i$ for all $i=1,2,\dots ,m$. By assumption, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)⩾t_i,\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)⩽r_i,\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)⩽s_i\hfill \end{array}$$

for all $i=1,2,\dots ,m$. It follows that $xy\in {\mathcal{A}}_{(t,r,s)}$. Hence, ${\mathcal{A}}_{(t,r,s)}$ is a subsemigroup of S.

Conversely, assume that nonempty $(t,r,s)$-level set ${\mathcal{A}}_{(t,r,s)}$ is a subsemigroup of S for all $t,r,s\in {[0,1]}^m$. Let $x,y\in S$. Take

$$\begin{array}{cc}\hfill t_i^{\prime }& =\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill r_i^{\prime }& =\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill s_i^{\prime }& =\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)\hfill \end{array}$$

for some $t_i^{\prime },r_i^{\prime },s_i^{\prime }\in [0,1]$, where $i=1,2,\dots ,m$. It turns out that $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)⩾t_i$, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)⩾t_i$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)⩽r_i$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)⩽r_i$, $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)⩽s_i$, and $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)⩽s_i$ for all $i=1,2,\dots ,m$. Now, let $t^{\prime }=(t_1^{\prime },t_2^{\prime },\dots ,t_m^{\prime })$, $r^{\prime }=(r_1^{\prime },r_2^{\prime },\dots ,r_m^{\prime })$, and $s^{\prime }=(s_1^{\prime },s_2^{\prime },\dots ,s_m^{\prime })$. We find that $x,y\in {\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$, and so ${\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}\ne \varnothing$. By the given assumption, we find that ${\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$ is a subsemigroup of S. Thus, $xy\in {\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$. This implies that

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾t_i^{\prime }=\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽r_i^{\prime }=\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)& ⩽s_i^{\prime }=\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)\hfill \end{array}$$

for all $i=1,2,\dots ,m$. That is, ${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(y\right)$, ${\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(y\right)$, and ${\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(y\right)$. Therefore, $\mathcal{A}$ is an m-PPFSub of S.

(ii) Assume that $\mathcal{A}$ is an m-PPFL of S. Let $t=(t_1,t_2,\dots ,t_m),r=(r_1,r_2,\dots ,r_m)$, $s=(s_1,s_2,\dots ,s_m)\in {[0,1]}^m$ be such that ${\mathcal{A}}_{(t,r,s)}\ne \varnothing$. Let $x\in S$ and $y\in {\mathcal{A}}_{(t,r,s)}$. Then, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)⩾t_i,\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)⩽r_i$, and $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)⩽s_i$ for all $i=1,2,\dots ,m$. By assumption, we have ${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(y\right)$, ${\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(y\right)$, and ${\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(y\right)$. We obtain that

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)⩾t_i,\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)⩽r_i,\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)⩽s_i\hfill \end{array}$$

for all $i=1,2,\dots ,m$. This implies that $xy\in {\mathcal{A}}_{(t,r,s)}$. Thus, ${\mathcal{A}}_{(t,r,s)}$ is a left ideal of S.

Conversely, assume that nonempty $(t,r,s)$-level set ${\mathcal{A}}_{(t,r,s)}$ is a left ideal of S for all $t,r,s\in {[0,1]}^m$. Let $x,y\in S$. Choose $t_i^{\prime }=\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right),r_i^{\prime }=\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)$, $s_i^{\prime }=\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)$ for some $t_i^{\prime },r_i^{\prime },s_i^{\prime }\in [0,1]$, where $i=1,2,\dots ,m$. Let $t^{\prime }=(t_1^{\prime },t_2^{\prime },\dots ,t_m^{\prime })$, $r^{\prime }=(r_1^{\prime },r_2^{\prime },\dots ,r_m^{\prime })$, and $s^{\prime }=(s_1^{\prime },s_2^{\prime },\dots ,s_m^{\prime })$. We have that $y\in {\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$. By assumption, we find that ${\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$ is a left ideal of S. Thus, $xy\in {\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$. This implies that

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾t_i^{\prime }=\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽r_i^{\prime }=\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)& ⩽s_i^{\prime }=\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)\hfill \end{array}$$

for all $i=1,2,\dots ,m$. Also, ${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(y\right)$, ${\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(y\right)$, and ${\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(y\right)$. Therefore, $\mathcal{A}$ is an m-PPFL of S.

(iii) The proof is similar to (ii).

(iv) It obvious by (ii) and (iii).

(v) Assume that $\mathcal{A}$ is an m-PPFGB of S. Let $t=(t_1,t_2,\dots ,t_m),r=(r_1,r_2,\dots ,r_m)$, $s=(s_1,s_2,\dots ,s_m)\in {[0,1]}^m$ be such that ${\mathcal{A}}_{(t,r,s)}\ne \varnothing$. Let $x,z\in {\mathcal{A}}_{(t,r,s)}$ and $y\in S$. Then, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)⩾t_i,\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)⩾t_i$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)⩽r_i,\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)⩽r_i$, $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)⩽s_i,$ and $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)⩽s_i$, where $i=1,2,\dots ,m$. Given the hypothesis, we have ${\mu }_{\mathcal{A}}\left(xyz\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(z\right)$, ${\eta }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(z\right)$, and ${\nu }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(z\right)$. Thus, for any $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xyz\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)⩾t_i,\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)⩽r_i,\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)⩽s_i.\hfill \end{array}$$

This means that $xyz\in {\mathcal{A}}_{(t,r,s)}$. Hence, ${\mathcal{A}}_{(t,r,s)}$ is a generalized bi-ideal of S.

Conversely, let $x,y,z\in S$. Take

$$\begin{array}{cc}\hfill t_i^{\prime }& =\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill r_i^{\prime }& =\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill s_i^{\prime }& =\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)\hfill \end{array}$$

for some $t_i^{\prime },r_i^{\prime },s_i^{\prime }\in [0,1]$, where $i=1,2,\dots ,m$. It follows that $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)⩾t_i$, $\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)⩾t_i$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)⩽r_i$, $\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)⩽r_i$, $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)⩽s_i$, and $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)⩽s_i$ for any $i=1,2,\dots ,m$. Then, $xz\in {\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$. Given the assumption, we have $xyz\in {\mathcal{A}}_{(t^{\prime },r^{\prime },s^{\prime })}$. So, for any $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xyz\right)& ⩾t_i^{\prime }=\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xyz\right)& ⩽r_i^{\prime }=\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xyz\right)& ⩽s_i^{\prime }=\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right).\hfill \end{array}$$

That is, ${\mu }_{\mathcal{A}}\left(xyz\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(z\right)$, ${\eta }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(z\right)$, and ${\nu }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(z\right)$. Therefore, $\mathcal{A}$ is an m-PPFGB of S.

(vi) The proof is complete by (i) and (v).  □

Theorem 3.

Let S be a semigroup and $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ be an m-PPFS on S. Then, the following conditions are true:

(i) 

$\mathcal{A}$ is an m-PPFSub of S if and only if $\mathcal{A}\diamond \mathcal{A}⊑\mathcal{A}$;

(ii) 

$\mathcal{A}$ is an m-PPFL of S if and only if $\mathcal{S}\diamond \mathcal{A}⊑\mathcal{A}$;

(iii) 

$\mathcal{A}$ is an m-PPFR of S if and only if $\mathcal{A}\diamond \mathcal{S}⊑\mathcal{A}$;

(iv) 

$\mathcal{A}$ is an m-PPFI of S if and only if $\mathcal{S}\diamond \mathcal{A}⊑\mathcal{A}$ and $\mathcal{A}\diamond \mathcal{S}⊑\mathcal{A}$;

(v) 

$\mathcal{A}$ is an m-PPFGB of S if and only if $\mathcal{A}\diamond \mathcal{S}\diamond \mathcal{A}⊑\mathcal{A}$;

(vi) 

$\mathcal{A}$ is an m-PPFB of S if and only if $\mathcal{A}\diamond \mathcal{A}⊑\mathcal{A}$ and $\mathcal{A}\diamond \mathcal{S}\diamond \mathcal{A}⊑\mathcal{A}$.

Proof. 

(i) Assume that $\mathcal{A}$ is an m-PPFSub of S. Let $x\in S$. If $x\notin S^2$, then $\mathcal{A}\diamond \mathcal{A}⊑\mathcal{A}$. Suppose that $x=ab$ for some $a,b\in S$. Then, for any $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & {\displaystyle ⩽\underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(ab\right)\right\}=\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & {\displaystyle ⩾\underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(ab\right)\right\}=\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right).}\hfill \end{array}$$

Furthermore, we have that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\right]\left(x\right)⩾\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)$. This implies that $({\mu }_{\mathcal{A}}\diamond {\mu }_{\mathcal{A}})\left(x\right)\preccurlyeq {\mu }_{\mathcal{A}}\left(x\right)$, $({\eta }_{\mathcal{A}}\diamond {\eta }_{\mathcal{A}})\left(x\right)\succcurlyeq {\eta }_{\mathcal{A}}\left(x\right)$, and $({\nu }_{\mathcal{A}}\diamond {\nu }_{\mathcal{A}})\left(x\right)\succcurlyeq {\nu }_{\mathcal{A}}\left(x\right)$. Thus, $\mathcal{A}\diamond \mathcal{A}⊑\mathcal{A}$.

Conversely, let $x,y\in S$. By assumption, we have $\left({\mu }_{\mathcal{A}}\diamond {\mu }_{\mathcal{A}}\right)\left(xy\right)\preccurlyeq {\mu }_{\mathcal{A}}\left(xy\right)$, $\left({\eta }_{\mathcal{A}}\diamond {\eta }_{\mathcal{A}}\right)\left(xy\right)\succcurlyeq {\eta }_{\mathcal{A}}\left(xy\right)$, and $\left({\nu }_{\mathcal{A}}\diamond {\nu }_{\mathcal{A}}\right)\left(xy\right)\succcurlyeq {\nu }_{\mathcal{A}}\left(xy\right)$. So, for every $i=1,2,\dots ,m$, it follows

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(xy\right)\hfill \\ \hfill & {\displaystyle =\underset{xy=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(xy\right)\hfill \\ \hfill & {\displaystyle =\underset{xy=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right).\hfill \end{array}$$

Additionally, we find that $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)$. It turns out that ${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)\wedge {\mu }_{\mathcal{A}}\left(y\right)$, ${\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)\vee {\eta }_{\mathcal{A}}\left(y\right)$, and ${\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)\vee {\nu }_{\mathcal{A}}\left(y\right)$. Therefore, $\mathcal{A}$ is an m-PPFSub of S.

(ii) Assume that $\mathcal{A}$ is an m-PPFL of S. Let $x\in S$. It is clear that $\mathcal{S}\diamond \mathcal{A}⊑\mathcal{A}$ in the case of $x\notin S^2$. Suppose that $a,b\in S$ exist, such that $x=ab$. Then, for each $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & {\displaystyle ⩽\underset{x=ab}{\bigvee }\left\{1\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(ab\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(ab\right)\right\}}\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & {\displaystyle ⩾\underset{x=ab}{\bigwedge }\left\{0\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(ab\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(ab\right)\right\}}\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right).\hfill \end{array}$$

Similarly, $\left[\left({\pi }_i\circ {\nu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\right]\left(x\right)⩾\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)$. This shows that $\left({\mu }_{\mathcal{S}}\diamond {\mu }_{\mathcal{A}}\right)\left(x\right)\preccurlyeq {\mu }_{\mathcal{A}}\left(x\right)$, $\left({\eta }_{\mathcal{S}}\diamond {\eta }_{\mathcal{A}}\right)\left(x\right)\succcurlyeq {\eta }_{\mathcal{A}}\left(x\right)$, and $\left({\nu }_{\mathcal{S}}\diamond {\nu }_{\mathcal{A}}\right)\left(x\right)\succcurlyeq {\nu }_{\mathcal{A}}\left(x\right)$. We find that $\mathcal{S}\diamond \mathcal{A}⊑\mathcal{A}$.

Conversely, let $x,y\in S$. By assumption, we have $\left({\mu }_{\mathcal{S}}\diamond {\mu }_{\mathcal{A}}\right)\left(xy\right)\preccurlyeq {\mu }_{\mathcal{A}}\left(xy\right)$, $\left({\eta }_{\mathcal{S}}\diamond {\eta }_{\mathcal{A}}\right)\left(xy\right)\succcurlyeq {\eta }_{\mathcal{A}}\left(xy\right)$, and $\left({\nu }_{\mathcal{S}}\diamond {\nu }_{\mathcal{A}}\right)\left(xy\right)\succcurlyeq {\nu }_{\mathcal{A}}\left(xy\right)$. For any $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xy\right)& ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(xy\right)\hfill \\ \hfill & {\displaystyle =\underset{xy=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)\hfill \\ \hfill & =1\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(y\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xy\right)& ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(xy\right)\hfill \\ \hfill & {\displaystyle =\underset{xy=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)\hfill \\ \hfill & =0\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(y\right).\hfill \end{array}$$

Similarly, $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xy\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(y\right)$. This means that ${\mu }_{\mathcal{A}}\left(xy\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(y\right)$, ${\eta }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(y\right)$, and ${\nu }_{\mathcal{A}}\left(xy\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(y\right)$. Therefore, $\mathcal{A}$ is an m-PPFL of S.

(iii) The proof is similar to the one presented in (ii).

(iv) It is achieved by (ii) and (iii).

(v) Assume that $\mathcal{A}$ is an m-PPFGB of S. Let $x\in S$. If $x\ne uvw$ for all $u,v,w\in S$, then the proof is complete. Suppose that $x=abc$ for some $a,b,c\in S$. Also, there exists $m=ab$ such that $x=mc$. For any $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigvee }\left\{\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\right]\left(m\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(c\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigvee }\left\{\underset{m=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(b\right)\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(c\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigvee }\left\{\underset{m=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\right)\wedge 1\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(c\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigvee }\left\{\underset{m=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(c\right)\right\}\right\}}\hfill \\ \hfill & {\displaystyle ⩽\underset{x=mc}{\bigvee }\left\{\underset{m=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(abc\right)\right\}\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(mc\right)\right\}}\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigwedge }\left\{\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\right]\left(m\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(c\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigwedge }\left\{\underset{m=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(b\right)\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(c\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigwedge }\left\{\underset{m=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\right)\vee 0\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(c\right)\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigwedge }\left\{\underset{m=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(c\right)\right\}\right\}}\hfill \\ \hfill & {\displaystyle ⩾\underset{x=mc}{\bigwedge }\left\{\underset{m=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(abc\right)\right\}\right\}}\hfill \\ \hfill & {\displaystyle =\underset{x=mc}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(mc\right)\right\}}\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right).\hfill \end{array}$$

Also, we find that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\right]\left(x\right)⩾\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)$. Thus, $\left({\mu }_{\mathcal{A}}\diamond {\mu }_{\mathcal{S}}\diamond {\mu }_{\mathcal{A}}\right)\left(x\right)\preccurlyeq {\mu }_{\mathcal{A}}\left(x\right)$, $\left({\eta }_{\mathcal{A}}\diamond {\eta }_{\mathcal{S}}\diamond {\eta }_{\mathcal{A}}\right)\left(x\right)\succcurlyeq {\eta }_{\mathcal{A}}\left(x\right)$, and $\left({\nu }_{\mathcal{A}}\diamond {\nu }_{\mathcal{S}}\diamond {\nu }_{\mathcal{A}}\right)\left(x\right)\succcurlyeq {\nu }_{\mathcal{A}}\left(x\right)$. Hence, $\mathcal{A}\diamond \mathcal{S}\diamond \mathcal{A}⊑\mathcal{A}$.

Conversely, let $x,y,z\in S$. By assumption, we have $\left({\mu }_{\mathcal{A}}\diamond {\mu }_{\mathcal{S}}\diamond {\mu }_{\mathcal{A}}\right)\left(xyz\right)\preccurlyeq {\mu }_{\mathcal{A}}\left(xyz\right)$, $\left({\eta }_{\mathcal{A}}\diamond {\eta }_{\mathcal{S}}\diamond {\eta }_{\mathcal{A}}\right)\left(xyz\right)\succcurlyeq {\eta }_{\mathcal{A}}\left(xyz\right)$, and $\left({\nu }_{\mathcal{A}}\diamond {\nu }_{\mathcal{S}}\diamond {\nu }_{\mathcal{A}}\right)\left(xyz\right)\succcurlyeq {\nu }_{\mathcal{A}}\left(xyz\right)$. For each $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(xyz\right)& ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(xyz\right)\hfill \\ \hfill & {\displaystyle \underset{xyz=uv}{\bigvee }\left\{\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\right]\left(u\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(v\right)\right\}}\hfill \\ \hfill & ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\right]\left(xy\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)\hfill \\ \hfill & {\displaystyle =\underset{xy=mn}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(m\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(n\right)\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(y\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge 1\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(z\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(xyz\right)& ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(xyz\right)\hfill \\ \hfill & {\displaystyle =\underset{xyz=uv}{\bigwedge }\left\{\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\right]\left(u\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(v\right)\right\}}\hfill \\ \hfill & ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\right]\left(xy\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)\hfill \\ \hfill & {\displaystyle =\underset{xy=mn}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(m\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(n\right)\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(y\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee 0\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(z\right).\hfill \end{array}$$

Similarly, we can show that $\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(xyz\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(z\right)$. That is, ${\mu }_{\mathcal{A}}\left(xyz\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(x\right)⋏{\mu }_{\mathcal{A}}\left(z\right)$, ${\eta }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(x\right)⋎{\eta }_{\mathcal{A}}\left(z\right)$, and ${\nu }_{\mathcal{A}}\left(xyz\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(x\right)⋎{\nu }_{\mathcal{A}}\left(z\right)$. Consequently, $\mathcal{A}$ is an m-PPFGB of S.

(vi) This result is obtained from conditions (i) and (v).  □

4. Regular Semigroups Characterized by Their $\mathit{m}$-PPFBs

In this section, we examine the connections among m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in regular semigroups. Afterwards, we characterize regular semigroups from the viewpoint of m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs of the semigroups.

Semigroup S is called regular [31] if, for each element a in S, $x\in S$ exists, such that $a=axa$.

Proposition 3.

In any regular semigroup S, m-PPFGBs and m-PPFBs coincide.

Proof. 

Regarding Definition 4, it is sufficient to show that any m-PPFGB is an m-PPFB of S. Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ be an m-PPFGB of S and $a,b\in S$. Then, $x\in S$ exists, such that $b=bxb$. For each $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(ab\right)=\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\left(bx\right)b\right)& ⩾\left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(b\right),\hfill \\ \hfill \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(ab\right)=\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\left(bx\right)b\right)& ⩽\left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(b\right),\hfill \\ \hfill \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(ab\right)=\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(a\left(bx\right)b\right)& ⩽\left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(a\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(b\right).\hfill \end{array}$$

It follows that ${\mu }_{\mathcal{A}}\left(ab\right)\succcurlyeq {\mu }_{\mathcal{A}}\left(a\right)⋏{\mu }_{\mathcal{A}}\left(b\right)$, ${\eta }_{\mathcal{A}}\left(ab\right)\preccurlyeq {\eta }_{\mathcal{A}}\left(a\right)⋎{\eta }_{\mathcal{A}}\left(b\right)$, and ${\nu }_{\mathcal{A}}\left(ab\right)\preccurlyeq {\nu }_{\mathcal{A}}\left(a\right)⋎{\nu }_{\mathcal{A}}\left(b\right)$. It turns out that $\mathcal{A}$ is an m-PPFB of S.  □

Proposition 3 shows that, if semigroup S is regular, then m-PPBs and m-PPFGBs of S are the same. However, this relationship between m-PPFBs and both m-PPFLs and m-PPFRs in regular semigroups does not always hold. The following example illustrates that, even if semigroup S is regular, the m-PPFBs differ from either m-PPFLs or m-PPFRs.

Example 4.

Let $S=\{a,b,c,d,e\}$ represent a semigroup, according to [29], as seen in

Table 8. The operation “·” on S.

·	a	b	c	d	e
a	a	a	a	a	a
b	a	a	a	b	c
c	a	b	c	a	a
d	a	a	a	d	e
e	a	d	e	a	a

.

It is not difficult to verify that S is regular. Now, let us consider a 4-PPFS $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ on S defined by

Table 9. The representation of the value of a 4-PPFS $\mathcal{A}$ on S.

$\mathcal{A}$	${\mathit{\mu }}_{\mathcal{A}}$	${\mathit{\eta }}_{\mathcal{A}}$	${\mathit{\nu }}_{\mathcal{A}}$
a	$\left(0.4,0.5,0.6,0.5\right)$	$\left(0.1,0.1,0,0.2\right)$	$\left(0,0.1,0.1,0\right)$
b	$\left(0.4,0.3,0.5,0.3\right)$	$\left(0.2,0.2,0.1,0.3\right)$	$\left(0.1,0.3,0.2,0.1\right)$
c	$\left(0.3,0.2,0.4,0.3\right)$	$\left(0.3,0.3,0.2,0.4\right)$	$\left(0.2,0.4,0.4,0.2\right)$
d	$\left(0,0.1,0.2,0.2\right)$	$\left(0.4,0.4,0.3,0.5\right)$	$\left(0.6,0.5,0.5,0.3\right)$
e	$\left(0,0.1,0.2,0.2\right)$	$\left(0.4,0.4,0.3,0.5\right)$	$\left(0.6,0.5,0.5,0.3\right)$

.

Further calculations confirm that the 4-PPFS $\mathcal{A}$ on S satisfies the requirements of a 4-PPFB of S. Then, we consider

$$\begin{array}{cc}\hfill \left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(eb\right)& =\left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(d\right)=0\ngeqq 0.4=\left({\pi }_1\circ {\mu }_{\mathcal{A}}\right)\left(b\right),\hfill \\ \hfill \left({\pi }_3\circ {\mu }_{\mathcal{A}}\right)\left(be\right)& =\left({\pi }_3\circ {\mu }_{\mathcal{A}}\right)\left(c\right)=0.4\ngeqq 0.5=\left({\pi }_3\circ {\mu }_{\mathcal{A}}\right)\left(b\right).\hfill \end{array}$$

This means that ${\mu }_{\mathcal{A}}\left(eb\right)⋡{\mu }_{\mathcal{A}}\left(b\right)$ and ${\mu }_{\mathcal{A}}\left(be\right)⋡{\mu }_{\mathcal{A}}\left(b\right)$. Therefore, the 4-PPFB $\mathcal{A}$ is neither a 4-PPFL nor a 4-PPFR of S.

Now, we recall the basic features of regular semigroup characterizations that are important to the investigation performed in this section.

Lemma 3

([31]). Let S be a semigroup. Then, S is regular if and only if $R\cap L\subseteq RL$ for every left ideal L and right ideal R of S.

Lemma 4

([31]). Let S be a semigroup. Then, S is regular if and only if $B=BSB$ for every bi-ideal B of S.

Theorem 4.

Let S be a semigroup. Then, S is regular if and only if $\mathcal{R}\sqcap \mathcal{L}=\mathcal{R}\diamond \mathcal{L}$, for all m-PPFL $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$ and for all m-PPFR $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$ of S.

Proof. 

Assume that S is regular. Let $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$ and $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$ be an m-PPFL and an m-PPFR of S, respectively. Regarding Lemma 2 and Theorem 3, we have $\mathcal{R}\diamond \mathcal{L}⊑\mathcal{S}\diamond \mathcal{L}⊑\mathcal{L}$ and $\mathcal{R}\diamond \mathcal{L}⊑\mathcal{R}\diamond \mathcal{S}⊑\mathcal{R}$. This means that $\mathcal{R}\diamond \mathcal{L}⊑\mathcal{R}\sqcap \mathcal{L}$. Next, let $x\in S$. Then, $y\in S$ exists, such that $x=xyx$. For every $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\left(xy\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(x\right)\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\left(xy\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(x\right)\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(x\right).\hfill \end{array}$$

From a similar proof of the case above, it follows that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{L}}\right)\right]\left(x\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{R}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{L}}\right)\left(x\right)$. We find that $\mathcal{R}\sqcap \mathcal{L}⊑\mathcal{R}\diamond \mathcal{L}$. Hence, $\mathcal{R}\sqcap \mathcal{L}=\mathcal{R}\diamond \mathcal{L}$.

Conversely, let L be a left ideal and R be a right ideal of S. Then, $RL\subseteq R\cap L$. Let $x\in R\cap L$. Regarding Theorem 1, we have ${\mathcal{C}}_L$ and ${\mathcal{C}}_R$ that are an m-PPFL and an m-PPFR of S, respectively. Regarding the given assumption and Lemma 1, we obtain ${\mathcal{C}}_{RL}={\mathcal{C}}_R\diamond {\mathcal{C}}_L={\mathcal{C}}_R\sqcap {\mathcal{C}}_L={\mathcal{C}}_{R\cap L}$. So, ${\mathcal{C}}_{RL}\left(x\right)={\mathcal{C}}_{R\cap L}\left(x\right)=(\mathbf{1},\mathbf{0},\mathbf{0})$; that is, $x\in RL$. This shows that $R\cap L\subseteq RL$. Thus, $RL=R\cap L$. Regarding Lemma 3, we find that S is regular.  □

Theorem 5.

Let S be a semigroup. Then, S is regular if and only if $\mathcal{B}=\mathcal{B}\diamond \mathcal{S}\diamond \mathcal{B}$ for every m-PPFB $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ of S.

Proof. 

Assume that S is regular. Let $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ be an m-PPFB of S. Regarding Theorem 3, we have $\mathcal{B}\diamond \mathcal{S}\diamond \mathcal{B}⊑\mathcal{B}$. Next, let $x\in S$. Then, $y\in S$ exists, such that $x=xyx$. For any $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(a\right)\wedge \left[\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\right]\left(b\right)\right\}}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\wedge \left[\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\right]\left(yx\right)\hfill \\ \hfill & {\displaystyle =\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\wedge \underset{yx=cd}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(c\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(d\right)\right\}}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\wedge \left\{\left({\pi }_i\circ {\mu }_{\mathcal{S}}\right)\left(y\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\right\}\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{B}}\right)\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(a\right)\vee \left[\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\right]\left(b\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\vee \left[\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\right]\left(yx\right)\hfill \\ \hfill & {\displaystyle =\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\vee \underset{yx=cd}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(c\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(d\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\vee \left\{\left({\pi }_i\circ {\eta }_{\mathcal{S}}\right)\left(y\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\right\}\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{B}}\right)\left(x\right).\hfill \end{array}$$

Similarly, we find that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{S}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\right]\left(x\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{B}}\right)\left(x\right)$. This shows that $\mathcal{B}⊑\mathcal{B}\diamond \mathcal{S}\diamond \mathcal{B}$. Therefore, $\mathcal{B}=\mathcal{B}\diamond \mathcal{S}\diamond \mathcal{B}$.

Conversely, let A be a bi-ideal of S. Then, $A\subseteq ASA$ and also ${\mathcal{C}}_A$ is an m-PPFB of S. Next, let $x\in ASA$. Regarding Theorem 1 and the hypothesis, we have ${\mathcal{C}}_A={\mathcal{C}}_A\diamond \mathcal{S}\diamond {\mathcal{C}}_A={\mathcal{C}}_{ASA}$. Thus, ${\mathcal{C}}_A\left(x\right)={\mathcal{C}}_{ASA}\left(x\right)=\left(\mathbf{1},\mathbf{0},\mathbf{0}\right)$. Also, $x\in A$. Hence, $ASA\subseteq A$. It follows that $A=ASA$. Regarding Lemma 4, we find that S is regular.  □

The following theorem is obtained by Proposition 1 and Theorem 5.

Theorem 6.

Let S be a semigroup. Then, S is regular if and only if $\mathcal{G}=\mathcal{G}\diamond \mathcal{S}\diamond \mathcal{G}$ for every m-PPFGB $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ of S.

Theorem 7.

In semigroup S, the following conditions are equivalent:

(i) 

S is regular;

(ii) 

$\mathcal{G}\sqcap \mathcal{A}=\mathcal{G}\diamond \mathcal{A}\diamond \mathcal{G}$ for each m-PPFI $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ and each m-PPFGB $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ of S;

(iii) 

$\mathcal{B}\sqcap \mathcal{A}=\mathcal{B}\diamond \mathcal{A}\diamond \mathcal{B}$ for each m-PPFI $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ and each m-PPFB $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ of S.

Proof. 

(i) ⇒ (ii) Assume that S is regular. Let $\mathcal{A}=\left({\mu }_{\mathcal{A}},{\eta }_{\mathcal{A}},{\nu }_{\mathcal{A}}\right)$ and $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ be an m-PPFI and an m-PPFGB of S, respectively. Regarding Lemma 2 and Theorem 3, we have

$$\mathcal{G}\diamond \mathcal{A}\diamond \mathcal{G}⊑\mathcal{G}\diamond \mathcal{S}\diamond \mathcal{G}⊑\mathcal{G} \mathrm{and} \mathcal{G}\diamond \mathcal{A}\diamond \mathcal{G}⊑(\mathcal{S}\diamond \mathcal{A})\diamond \mathcal{S}⊑\mathcal{A}\diamond \mathcal{S}⊑\mathcal{A}.$$

That is, $\mathcal{G}\diamond \mathcal{A}\diamond \mathcal{G}⊑\mathcal{G}\sqcap \mathcal{A}$. Now, let $x\in S$. Then, $y\in S$ exists, such that $x=xyx$, and so $x=xyxyx$. For any $i=1,2,\dots ,n$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left[\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\right]\left(xyxy\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{xyxy=cd}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(c\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(d\right)\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)}\hfill \\ \hfill & =\left\{\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(yxy\right)\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\hfill \\ \hfill & ⩾\left\{\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{A}}\right)\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left[\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\right]\left(xyxy\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{xyxy=cd}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(c\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(d\right)\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)}\hfill \\ \hfill & =\left\{\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(yxy\right)\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\hfill \\ \hfill & ⩽\left\{\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right)\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\hfill \\ \hfill & =\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{A}}\right)\left(x\right).\hfill \end{array}$$

Similarly, we can show that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{G}}\right)\right]\left(x\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{A}}\right)\left(x\right)$. This shows that $\mathcal{G}\sqcap \mathcal{A}⊑\mathcal{G}\diamond \mathcal{A}\diamond \mathcal{G}$. Therefore, $\mathcal{G}\sqcap \mathcal{A}=\mathcal{G}\diamond \mathcal{A}\diamond \mathcal{G}$.

(ii) ⇒ (iii) Since every m-PPFB is also an m-PPFGB of S, we can conclude that condition (iii) has been completed.

(iii) ⇒ (i) Let $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ be any m-PPFB of S. The m-PPFS $\mathcal{S}$ is an m-PPFI of S, and, by assumption, we have $\mathcal{B}=\mathcal{B}\sqcap \mathcal{S}=\mathcal{B}\diamond \mathcal{S}\diamond \mathcal{B}$. Consequently, for Theorem 5, S is regular.  □

Theorem 8.

The following statements are equivalent in semigroup S:

(i) 

S is regular;

(ii) 

$\mathcal{G}\sqcap \mathcal{L}⊑\mathcal{G}\diamond \mathcal{L}$ for every m-PPFL $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$ and every m-PPFGB $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ of S;

(iii) 

$\mathcal{B}\sqcap \mathcal{L}⊑\mathcal{B}\diamond \mathcal{L}$ for every m-PPFL $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$ and every m-PPFB $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ of S.

Proof. 

(i) ⇒ (ii) Let $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$ and $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ be an m-PPFL and an m-PPFGB of S, respectively. Let $x\in S$. Then, $y\in S$ exists, such that $x=xyx$. For each $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(yx\right)\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\right]\left(x\right)& {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(yx\right)\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(x\right).\hfill \end{array}$$

Given a similar proof of the case above, we can show that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{L}}\right)\right]\left(x\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{L}}\right)\left(x\right)$. Thus, $\mathcal{G}\sqcap \mathcal{L}⊑\mathcal{G}\diamond \mathcal{L}$.

(ii) ⇒ (iii) It is clear.

(iii) ⇒ (i) Let $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$ and $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$ be an m-PPFL and an m-PPFR of S, respectively. Next, the m-PPFR $\mathcal{R}$ is also an m-PPFB of S. Regarding the hypothesis, we have $\mathcal{R}\sqcap \mathcal{L}⊑\mathcal{R}\diamond \mathcal{L}$. On the other hand, $\mathcal{R}\diamond \mathcal{L}⊑\mathcal{R}\sqcap \mathcal{L}$ always. This implies that $\mathcal{R}\sqcap \mathcal{L}=\mathcal{R}\diamond \mathcal{L}$. Consequently, for Theorem 4, S is regular.  □

The following theorem can be presented similarly to Theorem 8.

Theorem 9.

Let S be a semigroup. Then, the following statements are equivalent:

(i) 

S is regular;

(ii) 

$\mathcal{R}\sqcap \mathcal{G}⊑\mathcal{R}\diamond \mathcal{G}$ for every m-PPFR $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$ and every m-PPFGB $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ of S;

(iii) 

$\mathcal{R}\sqcap \mathcal{B}⊑\mathcal{R}\diamond \mathcal{B}$ for every m-PPFR $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$ and every m-PPFB $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{B}}\right)$ of S.

Theorem 10.

In semigroup S, the following conditions are equivalent:

(i) 

S is regular;

(ii) 

$\mathcal{R}\sqcap \mathcal{G}\sqcap \mathcal{L}⊑\mathcal{R}\diamond \mathcal{G}\diamond \mathcal{L}$ for each m-PPFL $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$, each m-PPFR $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$, and each m-PPFGB $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ of S;

(iii) 

$\mathcal{R}\sqcap \mathcal{B}\sqcap \mathcal{L}⊑\mathcal{R}\diamond \mathcal{B}\diamond \mathcal{L}$ for each m-PPFL $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$, each m-PPFR $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$, and each m-PPFB $\mathcal{B}=\left({\mu }_{\mathcal{B}},{\eta }_{\mathcal{B}},{\nu }_{\mathcal{G}}\right)$ of S.

Proof. 

(i) ⇒ (ii) Assume that S is regular. Let $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$, $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$, and $\mathcal{G}=\left({\mu }_{\mathcal{G}},{\eta }_{\mathcal{G}},{\nu }_{\mathcal{G}}\right)$ be an m-PPFL, an m-PPFR, and an m-PPFGB of S, respectively. For any $x\in S$, we have $x=xyx$ for some $y\in S$. Also, $x=xyxyxyx$. For every $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigvee }\left\{\left[\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\right]\left(a\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩾\left[\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\right]\left(xyxyx\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(yx\right)\hfill \\ \hfill & {\displaystyle ⩾\underset{xyxyx=cd}{\bigvee }\left\{\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\left(c\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(d\right)\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(x\right)}\hfill \\ \hfill & ⩾\left\{\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\left(xy\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(xyx\right)\right\}\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(x\right)\hfill \\ \hfill & ⩾\left({\pi }_i\circ {\mu }_{\mathcal{R}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{G}}\right)\left(x\right)\wedge \left({\pi }_i\circ {\mu }_{\mathcal{L}}\right)\left(x\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left[\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\right.& \left.\diamond \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\right]\left(x\right)\hfill \\ \hfill & {\displaystyle =\underset{x=ab}{\bigwedge }\left\{\left[\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\right]\left(a\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(b\right)\right\}}\hfill \\ \hfill & ⩽\left[\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\right]\left(xyxyx\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(yx\right)\hfill \\ \hfill & {\displaystyle ⩽\underset{xyxyx=cd}{\bigvee }\left\{\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\left(c\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(d\right)\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(x\right)}\hfill \\ \hfill & ⩽\left\{\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\left(xy\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(xyx\right)\right\}\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(x\right)\hfill \\ \hfill & ⩽\left({\pi }_i\circ {\eta }_{\mathcal{R}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\eta }_{\mathcal{L}}\right)\left(x\right).\hfill \end{array}$$

Similarly, we can show that $\left[\left({\pi }_i\circ {\nu }_{\mathcal{R}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{G}}\right)\diamond \left({\pi }_i\circ {\nu }_{\mathcal{L}}\right)\right]\left(x\right)⩽\left({\pi }_i\circ {\nu }_{\mathcal{R}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{G}}\right)\left(x\right)\vee \left({\pi }_i\circ {\nu }_{\mathcal{L}}\right)\left(x\right)$. It follows that $\mathcal{R}\sqcap \mathcal{G}\sqcap \mathcal{L}⊑\mathcal{R}\diamond \mathcal{G}\diamond \mathcal{L}$.

(ii) ⇒ (iii) Straightforward.

(iii) ⇒ (i) Let $\mathcal{L}=\left({\mu }_{\mathcal{L}},{\eta }_{\mathcal{L}},{\nu }_{\mathcal{L}}\right)$ and $\mathcal{R}=\left({\mu }_{\mathcal{R}},{\eta }_{\mathcal{R}},{\nu }_{\mathcal{R}}\right)$ be an m-PPFL and an m-PPFR of S, respectively. Since the m-PPFS $\mathcal{S}$ itself is an m-PPFB of S and regarding the hypothesis, we have $\mathcal{R}\sqcap \mathcal{L}=\mathcal{R}\sqcap \mathcal{S}\sqcap \mathcal{L}⊑\mathcal{R}\diamond \mathcal{S}\diamond \mathcal{L}⊑\mathcal{R}\diamond \mathcal{L}$. On the other hand, $\mathcal{R}\diamond \mathcal{L}⊑\mathcal{R}\sqcap \mathcal{L}$. Hence, $\mathcal{R}\diamond \mathcal{L}=\mathcal{R}\sqcap \mathcal{L}$. We conclude that for Theorem 4, S is regular.  □

5. Conclusions

In this study, we successfully applied the concept of m-PPFSs to the study of semigroups, introducing and investigating fundamental algebraic structures such as m-PPFSubs, m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs in semigroups. Our findings demonstrate that every m-PPFL (m-PPFR) of semigroups is also an m-PPFB, though the opposite does not always hold. Moreover, we established that every m-PPFB is also an m-PPFGB in semigroups, but the reverse is not necessarily true, as shown in Examples 1 and 2, respectively. Additionally, we investigated the characterizations of different kinds of m-PPFIs in semigroups related to particular m-PPFS features. Given the fact that the concept of m-PPFSs is an extension of the concept of PFSs, it can be seen that the results obtained from this research are comprehensive in terms of the results of the classification of regular semigroups by PFSs in reference [9]. Furthermore, we studied the connections between various m-PPFIs in regular semigroups. Ultimately, the regular semigroups are characterized by their m-PPFSubs, m-PPFLs, m-PPFRs, m-PPFIs, m-PPFBs, and m-PPFGBs. In future work, we intend to indicate various types of regularities in semigroups, particularly focusing on intra-regular and weakly regular semigroups. We aim to achieve these objectives by utilizing different forms of m-PPFIs or by investigating these concepts within other algebraic structures, such as ordered semigroups, $\Gamma$-semigroups, or semihypergroups.