**1. Introduction**

In 2014, Chen et al. [1] presented the *m*-PFS as an expansion of the BFS. The mathematical theories of a 2-polar fuzzy set and BFS are equivalent, and we can see that one connected to the other. The BFS is expanded into an *m*-PFS by applying the notion of one-to-one correspondence. Sometimes, different things are monitored in different ways. The *m*-PFS is effective in assigning degrees of membership to various objects in multi-polar data. The *m*-PFS gives only a positive degree of membership to each element. The *m*-PFS has an extensive range of implementations in real world problems related to the multiagent, multi-objects, multi-polar information, multi-index and multi-attributes. This theory is applicable when a company decides to construct an item, a country elects its political leaders, or a group of friends wants to visit a country, with various options. It can be used in decision making, co-operative games, disease diagnosis, to discuss the confusions and conflicts of communication signals in wireless communications and as a model to define clusters or categorization and multi-relationships. In sum, an *m*-PFS on *H* is a mapping *I* : *H* → [0, <sup>1</sup>]*<sup>m</sup>*.

Here, we will make a model-based example on *m*-PFS, and use it to conveniently select an appropriate employee in a company. Here, the selection of an employee is based on 4-PFS with their four qualities, which are honesty, punctuality, communication skills, and being hardworking. Let *H* = {*<sup>a</sup>*1, *a*2, *a*3, *a*4, *<sup>a</sup>*5} be the set of five employees in a company. We shall characterize them according to four qualities in the form of 4-PFS, given in Table 1:

**Citation:** Bashir, S.; Shahzadi, S.; Al-Kenani, A.N.; Shabir, M. Regular and Intra-Regular Semigroups in Terms of *m*-Polar Fuzzy Environment. *Mathematics* **2021**, *9*, 2031. https:// doi.org/10.3390/math9172031

Academic Editor: Sorin Nadaban

Received: 30 July 2021 Accepted: 20 August 2021 Published: 24 August 2021

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**Table 1.** Table of qualities in persons with their membership values.

Therefore, we attain a 4-PFS *η* : *H* → [0, 1]4 of *H* such that


Figure 1 is the graphical representation of 4-PFS:

**Figure 1.** Graphical representation of 4-polar fuzzy subset.

Here, 1 represents good remarks, 0.5 represents average and 0 represents bad remarks. Similarly, we can solve any other problem with uncertainty in multiple directions.

Zhang [2] proposed that the function is mapped to the interval [−1, 1] rather than [0, 1] in BFS theory. Lee [3] coined the term bipolar fuzzy ideals. BFS is useful for solving uncertain problems with two poles of a situation: positive and negative pole. For more applications of BFS, see [4–9]. In medical science, environmental research, and engineering, we may find data or information that are ambiguous or complicated. All mathematical equations and techniques in classical mathematics are exact, they cannot deal with such problems. Many tools have been developed to deal with such issues. After extensive effort, Zadeh [10] was the first to propose fuzzy set theory as a solution to such complicated issues. This idea is used in a variety of areas, including logic, measure theory, topological space, ring theory, group theory and real analysis. The theory of fuzzy group was first intitated by Rosenfeld [11]. Kuroki [12] and Mordeson [13] have extensively explored fuzzy semigroups.

Semigroups are very useful in many applications containing dynamical systems, control problems, partial differential equations, sociology, stochastic differential equations, biology, etc. Some examples of semigroups are the collection of all mappings of a set, under the composition of functions (termed a full transformation monoid) and the set of natural numbers N under either addition or multiplication. The word "semigroup" was introduced to provide a title for some structures that were not groups but were created

through the expansion of consequences. The proper semigroup theory was initiated by the working of Russian mathematician Anton Kazimirovich Suschkewitsch [14]. Quasi-ideals in semigroups were introduced by Otto Steinfeld [15].

The study of *m*-PF algebraic structures began with the concept of *m*-PF Lie subalgebras [16]. After that, the *m*-PF Lie ideals were studied in Lie algebras [17]. In 2017, Sarwar and Akram worked on new applications of *m*-PF matroids [18]. In 2019, Ahmad and Al-Masarwah introduced the concept of *m*-PF (commutative) ideals and *m*-polar (*<sup>α</sup>*, *β*)-fuzzy ideals in BCK/BCI-algebras [19,20]. To continue their work, they introduced a new aspect of generalized *m*-PF ideals and studied the normalization of *m*-PF subalgebras in [21,22]. Recently, Muhiuddin and Al-Kadi presented interval-valued *m*-PF BCK/BCI-Algebras [23]. Shabir et al. [24] studied regular and intra-regular semirings in terms of BFIs. Then, Bashir et al. [25,26] studied regular ordered ternary semigroups and semirings in terms of BFIs. Shabir et al. extended the work of [24], initiated the concept of *m*-PFIs in LAsemigroups and characterized the regular LA-semigroups according to the properties of these *m*-PFIs [27]. By extending the work of [24,27], the concept of *m*-PFIs in semigroups was introduced and characterizations of regular and intra-regular semigroups according to the properties of *m*-PFIs are given in this paper.

This paper is charaterized as follows: We present some basic concepts related to *m*-PFS in Section 2. The major part of this paper is Section 3, the *m*-PFSSs, *m*-PFIs (left, right), *m*-PFBIs, *m*-PFGBIs, *m*-PFQIs, *m*-PFIIs of semigroups are discussed with examples. In Section 4, the regular and intra-regular semigroups are characterized by the properties of *m*-PFIs. A comparison between this research and previous work is shown in Section 5. In Section 6, we also talk about the conclusions and future work.

The list of acronyms used in the research article is given in Table 2.


#### **Table 2.** List of acronyms.
