**2. Preliminaries**

In this Section, we have studied the fundamental but essential definitions and preliminary findings based on semigroups that are important in their own right. These are necessary for later Sections.

If a groupoid (*<sup>P</sup>*, ·) satisfies the associative property, then it is called a semigroup. Throughout this paper, *P* will denote the semigroup, unless specified otherwise. A nonempty subset *H* of *P* is called a subsemigroup of *P* if *ab* ∈ *H* for every *a*, *b* ∈ *H*. In this paper, subsets mean non-empty subsets. A subset *H* of *P* is called a left ideal (resp. right ideal) of *P* if *PH* ⊆ *H* (resp. *HP* ⊆ *<sup>H</sup>*). If *H* is left and right ideal, then *H* is called a two-sided ideal or ideal of *P* [28] .

A subset *H* of *P* is called a generalized bi-ideal of *P* if *HPH* ⊆ *H*. The subsemigroup *H* of *P* is called a bi-ideal of *P* if *HPH* ⊆ *H*. A subset *H* of *P* is called a quasi-ideal of *P* if *HP* ∩ *PH* ⊆ *H*. The subsemigroup *H* of *P* is called an interior ideal of *P* if (*PH*)*P* ⊆ *H* [28].

A fuzzy subset *η* of *P* is a mapping from *P* to closed interval [0, 1], *that is η* : *P* → [0, 1] [10]. A bipolar fuzzy subset *η* of *P* is a mapping from *P* to closed interval [−1, 1] written as *η* = (*<sup>P</sup>*, *η*<sup>−</sup>, *<sup>η</sup>*+), where *η*<sup>−</sup> : *P* → [−1, 0] and *η*<sup>+</sup> : *P* → [0, 1]. It can differentiate between unrelated and contrary components of fuzzy problems. A natural one-to-one correspondence exists among the BFS and 2-polar fuzzy set ([0, 1]2-set). When data for real world complex situations come from *m* factors (*m* ≥ <sup>2</sup>), then *m*-PFS is used to deal with such problems. An *m*-PFS (or a [0, <sup>1</sup>]*<sup>m</sup>*-set) on *P* is a function *η* = *P* → [0, <sup>1</sup>]*<sup>m</sup>*. More generally, the *m*-PFS is the *m*-tuple of membership degree function of *P* that is *η* = (*η*1, *η*2, ... , *ηm*), where *ηκ* : *P* → [0, 1] is the mapping for every *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Here, **0** = (0, 0, ... , 0) is the smallest value in [0, 1]*m* and **1** = (1, 1, ... , 1) is the largest value in [0, 1]*m* [1].

The set of all *m*-PFSs of *P* is represented by *<sup>m</sup>*(*P*). We define relation ≤ on *m*(*P*) as follows: For any *m*-PFSs *η* = (*η*1, *η*2, ... , *ηm*) and *m* = (*<sup>m</sup>*1, *m*2, ... , *mm*) of *P*, *η* ≤ *m* means that *ηκ*(*a*) ≤ *<sup>m</sup>κ*(*a*) for every *a* ∈ *P* and *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. The symbols *η* ∧ *m* and *η* ∨ *m* mean the following *m*-PFSs of *P*. (*η* ∧ *m*)(*a*) = *η*(*a*) ∧ *m*(*a*) and (*η* ∨ *m*)(*a*) = *η*(*a*) ∨ *m*(*a*) that is (*ηκ* ∧ *<sup>m</sup>κ*)(*a*) = *ηκ*(*a*) ∧ *<sup>m</sup>κ*(*a*) for each *a* ∈ *P* and *κ* ∈ {1, 2, ... , *m*}; (*ηκ* ∨ *<sup>m</sup>κ*)(*a*) = *ηκ*(*a*) ∨ *<sup>m</sup>κ*(*a*) for each *a* ∈ *P* and *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. For two *m*-PFSs *η* = (*η*1, *η*2, ... , *ηm*) and *m* = (*<sup>m</sup>*1, *m*2, ... , *mm*), the product of *η* ◦ *m* = (*η*1 ◦ *m*1, *η*2 ◦ *m*2,..., *ηm* ◦ *mm*) is defined as

$$(\eta\_{\mathbf{x}} \circ m\_{\mathbf{x}}')(a) = \begin{cases} \bigvee\_{a=st} \{\eta\_{\mathbf{x}}(\mathbf{s}) \wedge m\_{\mathbf{x}}(t), \text{ if } a = st \text{ for some } \mathbf{s}, t \in P; \\\ 0, & \text{otherwise;} \end{cases}$$

for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. The next example shows the product of *m*-PFSs *η* and *m* of *P* for *m* = 4.

**Example 1.** *Consider the semigroup P* = {*ı*, *j*, , ¯*h*} *given in Table 3.*

**Table 3.** Table of multiplication of *P*.


We define 4-PFSs *η* = (*η*1, *η*2, *η*3, *η*4) and *m* = (*<sup>m</sup>*1, *m*2, *m*3, *<sup>m</sup>*4) as follows:

*η*(*ı*)=(0.2, 0.1, 0, 0.4), *η*(*j*)=(0.7, 0.5, 0.1, <sup>0</sup>), *η*()=(0.1, 0.3, 0.7, 0.4), *η*(*h*¯) = (0, 0, 0, 0.1) and

*m*(*ı*)=(0.7, 0.3, 0, 0.4), *m*(*j*)=(0.2, 0, 0, 0.1), *m*()=(0.2, 0.2, 0.4, <sup>0</sup>), *m*(*h*¯) = (0.2, 0.3, 0, <sup>0</sup>).

By defintion, we obtain

(*η*1 ◦ *<sup>m</sup>*1)(*ı*) = 0.7, (*η*1 ◦ *<sup>m</sup>*1)(*j*) = 0.1, (*η*1 ◦ *<sup>m</sup>*1)() = 0, (*η*1 ◦ *<sup>m</sup>*1)(*h*¯) = 0; (*η*2 ◦ *<sup>m</sup>*2)(*ı*) = 0.3, (*η*2 ◦ *<sup>m</sup>*2)(*j*) = 0.1, (*η*2 ◦ *<sup>m</sup>*2)() = 0, (*η*2 ◦ *<sup>m</sup>*2)(*h*¯) = 0; (*η*3 ◦ *<sup>m</sup>*3)(*ı*) = 0.1, (*η*3 ◦ *<sup>m</sup>*3)(*j*) = 0.4, (*η*3 ◦ *<sup>m</sup>*3)() = 0, (*η*3 ◦ *<sup>m</sup>*3)(*h*¯) = 0; (*η*4 ◦ *<sup>m</sup>*4)(*ı*) = 0.4, (*η*4 ◦ *<sup>m</sup>*4)(*j*) = 0.0, (*η*4 ◦ *<sup>m</sup>*4)() = 0, (*η*4 ◦ *<sup>m</sup>*4)(*h*¯) = 0. Hence, the product of *η* = (*η*1, *η*2, *η*3, *η*4) and *m* = (*<sup>m</sup>*1, *m*2, *m*3, *<sup>m</sup>*4) is defined by (*η* ◦ *m*)(*ı*)=(0.7, 0.3, 0.1, 0.4),(*η* ◦ *m*)(*j*)=(0.1, 0.1, 0.4, <sup>0</sup>), (*η* ◦ *m*)()=(0, 0, 0, <sup>0</sup>), (*η*◦ *m*)(*h*¯)=(0, 0, 0, <sup>0</sup>).

**Definition 1.** *Let η* = (*η*1, *η*2,..., *ηm*) *be an m-PFS of P.*


**Definition 2.** *An m-PFS η* = (*η*1, *η*2, ... , *ηm*) *of P is called an m-PFSS of P if, for all a*, *b* ∈ *P*, *η*(*ab*) ≥ min{*η*(*a*), *η*(*b*)}, *that is, ηκ*(*ab*) ≥ min{*ηκ*(*a*), *ηκ*(*b*)} *for all κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

**Definition 3.** *An m-PFS η* = (*η*1, *η*2, ... , *ηm*) *of P is called an m-PFI left* (*resp. right*) *of P for all a*, *b* ∈ *P*, *η*(*ab*) ≥ *η*(*b*) (*resp. η*(*ab*) ≥ *η*(*a*)), *that is ηκ*(*ab*) ≥ *ηκ*(*b*) (*resp. ηκ*(*ab*) ≥ *ηκ*(*a*)) *for all κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

An *m*-PFS *η* of *P* is called an *m*-PFI of *P* if *η* is both an *m*-PFI (left) and *m*-PFI (right) of *P*.

The example given below is of 4-PFI of *P*.

**Example 2.** *Let P* = {*ı*, *j*, , ¯*h*} *be a semigroup given in Table 4.*

**Table 4.** Table of multiplication of *P*.


We define a 4-PFS *η* = (*η*1, *η*2, *η*3, *η*4) of *P* as follows:

*η*(*ı*)=(0.7, 0.6, 0.6, 0.4), *η*(*j*)=(0.2, 0, 0, 0.1), *η*()=(0.5, 0.4, 0.3, 0.1), *η*(*h*¯) = (0.5, 0.4, 0.3, 0.1).

Clearly, *η* = (*η*1, *η*2, *η*3, *η*4) is both 4-PFIs (left and right) of *P*. Hence *η* is a 4-PFI of *P*.

**Definition 4.** *Let a subset H of P. Then, the m-polar characteristic function CH* : *H* → [0, 1]*m is defined as*

> *CH*(*h*) = (1, 1, . . . , <sup>1</sup>), *m-tuple if h* ∈ *H*; (0, 0, . . . , <sup>0</sup>), *m-tuple if h* ∈/ *H*.

#### **3. Characterization of Semigroups by** *m***-Polar Fuzzy Sets**

This is the most essential portion, because here we make our major contributions. With the help of several lemmas, theorems, and examples, the notions of *m*-PFSSs and *m*-PFIs of semigroups are explained in this section. We have proved that every *m*-PFBI of *P* is *m*-PFGBI, but the converse does not hold. For LA-semigroups, Shabir et al. [27] has proved this result. We have generalized the results in Shabir et al. [27] for semigroups. In whole paper, *δ* is an *m*-PFS of P that maps each element of P on (1, 1, . . . , 1).

**Lemma 1.** *Consider two subsets H and I of P. Then*

*1. CH* ∧ *CI* = *CH*∩*I*;

*2. CH* ∨ *CI* = *CH*∪*I*;

$$
\mathbf{3.} \qquad \mathbf{C}\_H \circ \mathbf{C}\_I = \mathbf{C}\_{HI}.
$$

**Proof.** The proof of (1) and (2) are obvious.

(3): Case 1: Let *a* ∈ *H I*. This implies that *a* = *hi* for some *h* ∈ *H* and *i* ∈ *I*. Therefore, *CH I*(*a*)=(1, 1, ... , <sup>1</sup>). Since *h* ∈ *H* and *i* ∈ *I*, we have *CH*(*h*)=(1, 1, ... , 1) or *CI*(*i*)=(1, 1, . . . , <sup>1</sup>). Now,

(*CH* ◦ *CI*)(*a*) = *<sup>a</sup>*=*bc* {*CH*(*b*) ∧ *CI*(*c*)} ≥ *CH*(*h*) ∧ *CI*(*i*) = (1, 1, . . . , <sup>1</sup>).

Therefore, *CH* ◦ *CI* = *CH I*. Case 2: If *a* ∈/ *H I*. This implies that *CH I*(*a*)=(0, 0, ... , <sup>0</sup>), since *a* ∈/ *hi* for every *h* ∈ *H* and *i* ∈ *I*. Therefore

$$\begin{aligned} (\mathsf{C}\_H \circ \mathsf{C}\_I)(a) &= \bigvee\_{a=hi} \{ \mathsf{C}\_H(h) \wedge \mathsf{C}\_I(i) \}, \\ &= (0, 0, \dots, 0). \end{aligned}$$

Hence *CH* ◦ *CI* = *CH I*.

**Lemma 2.** *Let H be a subset of P. Then, the given statements hold.*


**Proof.** (1) Consider *H* as the subsemigroup of *P*. We have to show that *CH*(*ab*) ≥ *CH*(*a*) ∧ *CH*(*b*) for all *a*, *b* ∈ *P*. Now, we consider some cases:

Case 1: Let *a*, *b* ∈ *H*. Then, *CH*(*a*) = *CH*(*b*)=(1, 1, ... , <sup>1</sup>). As *H* is a subsemigroup of *P*, so *ab* ∈ *H* implies that *CH*(*ab*)=(1, 1, . . . , <sup>1</sup>). Hence *CH*(*ab*) ≥ *CH*(*a*) ∧ *CH*(*b*).

Case 2: Let *a* ∈ *H*, *b* ∈/ *H*. Then, *CH*(*a*)=(1, 1, ... , <sup>1</sup>), *CH*(*b*)=(0, 0, ... , <sup>0</sup>). Hence, *CH*(*ab*) ≥ (0, 0, . . . , 0) = *CH*(*a*) ∧ *CH*(*b*).

Case 3: Let *a*, *b* ∈/ *H*. Then, *CH*(*a*) = *CH*(*b*)=(0, 0, ... , <sup>0</sup>). Clearly, *CH*(*ab*) ≥ (0, 0, . . . , 0) = *CH*(*a*) ∧ *CH*(*b*).

Case 4: Let *a* ∈/ *H*, *b* ∈ *H*. Then, *CH*(*a*)=(0, 0, ... , 0) and *CH*(*b*)=(1, 1, ... , <sup>1</sup>). Clearly, *CH*(*ab*) ≥ (0, 0, . . . , 0) = *CH*(*a*) ∧ *CH*(*b*).

Conversely, let *CH* be an *m*-PFSS of *P*. Let *a*, *b* ∈ *H*. Then, *CH*(*a*) = *CH*(*b*) = (1, 1, ... , <sup>1</sup>). By definition, *CH*(*ab*) ≥ *CH*(*a*) ∧ *CH*(*b*)=(1, 1, ... , 1) ∧ (1, 1, ... , 1) = (1, 1, ... , <sup>1</sup>), we have *CH*(*ab*)=(1, 1, ... , <sup>1</sup>). This implies that *ab* ∈ *H*, that is *H* is a subsemigroup of *P*.

(2) Suppose that *H* is the left ideal of *P*. We have to show that *CH*(*ab*) ≥ *CH*(*b*) for every *a*, *b* ∈ *P*. Now, consider the two cases:

Case 1: Let *b* ∈ *H* and *a* ∈ *P*. Then, *CH*(*b*)=(1, 1, ... , <sup>1</sup>). Since *H* is a left ideal of *P*, *ab* ∈ *H* implies that *CH*(*ab*)=(1, 1, . . . , <sup>1</sup>). Hence *CH*(*ab*) ≥ *CH*(*b*).

Case 2: Let *b* ∈/ *H* and *a* ∈ *P*. Then, *CH*(*b*)=(0, 0, . . . , <sup>0</sup>). Clearly, *CH*(*ab*) ≥ *CH*(*b*).

Conversely, let *CH* be an *m*-PFI (left) of *P*. Let *a* ∈ *P* and *b* ∈ *H*. Then, *CH*(*b*) = (1, 1, ... , <sup>1</sup>). By definition, *CH*(*ab*) ≥ *CH*(*b*)=(1, 1, ... , <sup>1</sup>), we have *CH*(*ab*)=(1, 1, ... , <sup>1</sup>). Thisimpliesthat*ab*∈*H*,thatis*H*isaleftidealof*P*.

In the same way, we can show that *H* is right ideal of *P* if, and only if, *CH* is an *m*-PFI (right) of *P*. Therefore, *H* is an ideal of *P* if, and only if, *CH* is an *m*-PFI of *P*.

**Lemma 3.** *For m-PFS η* = (*η*1, *η*2,..., *ηm*) *of P*, *the following properties hold.*

*1. η is an m-PFSS of P if, and only if, η* ◦ *η* ≤ *η*;

*2. η is an m-PFI (left) of P if, and only if, δ* ◦ *η* ≤ *η*;


**Proof.** (1) Assume that *η* = (*η*1, *η*2, ... , *ηm*) is an *m*-PFSS of *P*, that is, *ηκ*(*ab*) ≥ *ηκ*(*a*) ∧ *ηκ*(*b*) for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Let *p* ∈ *P*. If *p* is not expressible as *p* = *ab* for some *a*, *b* ∈ *P*; then, (*η* ◦ *η*)(*p*) = 0. Hence, (*η* ◦ *η*)(*p*) ≤ *η*(*p*). However, if *p* is expressible as *p* = *ab* for some *a*, *b* ∈ *P*, then

$$\begin{aligned} (\eta\_{\mathbb{K}} \circ \eta\_{\mathbb{K}})(p) &= \bigvee\_{p=ab} \{\eta\_{\mathbb{K}}(a) \wedge \eta\_{\mathbb{K}}(b)\} \\ &\le \bigvee\_{p=ab} \{\eta\_{\mathbb{K}}(ab)\} \\ &= \eta\_{\mathbb{K}}(p) \text{ for all } \kappa \in \{1, 2, \dots, m\}. \end{aligned}$$

Hence, *η* ◦ *η* ≤ *η*. Conversely, let *η* ◦ *η* ≤ *η* and *a*, *b* ∈ *P*. Then

$$\begin{array}{rcl} \eta\_{\mathbb{K}}(ab) & \geq & (\eta\_{\mathbb{K}} \circ \eta\_{\mathbb{K}})(ab) \\ & = & \bigvee\_{ab = uv} \{\eta\_{\mathbb{K}}(u) \wedge \eta\_{\mathbb{K}}(v)\} \\ & \geq & \eta\_{\mathbb{K}}(a) \wedge \eta\_{\mathbb{K}}(b) \text{ for all } \mathbb{K} \in \{1, 2, \dots, m\}. \end{array}$$

Hence, *ηκ*(*ab*) ≥ *ηκ*(*a*) ∧ *ηκ*(*b*). Thus, *η* is *m*-PFSS of *P*.

(2) Assume that *η* = (*η*1, *η*2, ... , *ηm*) is *m*-PFI (left) of *P*, that is, *ηκ*(*ab*) ≥ *ηκ*(*b*) for all *κ* ∈ {1, 2, ... , *m*} and *a*, *b* ∈ *P*. Let *p* ∈ *P*. If *p* is not expressible as *p* = *ab* for some *a*, *b* ∈ *P*, then (*δ* ◦ *η*)(*p*) = 0. Hence, *δ* ◦ *η* ≤ *η*. However, if *p* is expressible as *p* = *ab* for some *a*, *b* ∈ *P*, then

$$\begin{aligned} \left(\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}}\right)(p) &= \bigvee\_{p=ab} \{\delta\_{\mathbf{x}}(a) \wedge \eta\_{\mathbf{x}}(b)\} \\ &= \bigvee\_{p=ab} \{\eta\_{\mathbf{x}}(b)\} \\ &\leq \bigvee\_{p=ab} \eta\_{\mathbf{x}}(ab) \\ &= \eta\_{\mathbf{x}}(p) \text{ for all } \mathbf{x} \in \{1, 2, \dots, m\}. \end{aligned}$$

Hence *δ* ◦ *η* ≤ *η*. Conversely, let *δ* ◦ *η* ≤ *η* and *a*, *b* ∈ *P*. Then,

$$\begin{array}{rcl} \eta\_{\mathbb{K}}(ab) & \geq & (\delta\_{\mathbb{K}} \circ \eta\_{\mathbb{K}})(ab) \\ & = & \bigvee\_{ab=\iota\nu} \{\delta\_{\mathbb{K}}(\mu) \wedge \eta\_{\mathbb{K}}(\upsilon)\} \\ & \geq & \{\delta\_{\mathbb{K}}(a) \wedge \eta\_{\mathbb{K}}(b)\} \\ & = & \eta\_{\mathbb{K}}(b) \text{ for all } \kappa \in \{1, 2, \dots, m\}. \end{array}$$

Hence, *η*(*ab*) ≥ *η*(*b*). Thus, *η* is *m*-PFI (left) of *P*.


**Lemma 4.** *The given statements are true in P.*


**Proof.** Straightforward.

**Proposition 1.** *Let η* = (*η*1, *η*2, ... , *ηm*) *be an m-PFS of P. Then, η is an m-PFSS* (*resp. m-PFI*) *of P if, and only if, ηt* = {*a* ∈ *<sup>P</sup>*|*η*(*a*) ≥ *t*} = *φ is a subsemigroup* (*resp. ideal*) *of P for all t* ∈ (*<sup>t</sup>*1, *t*2,..., *tm*) ∈ (0, <sup>1</sup>]*<sup>m</sup>*.

**Proof.** Let *η* be an *m*-PFSS of *P*. Let *a*, *b* ∈ *η<sup>t</sup>*. Then, *ηκ*(*a*) ≥ *tκ* and *ηκ*(*b*) ≥ *tκ* for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. As *η* is an *m*-PFSS of *P*, this implies *ηκ*(*ab*) ≥ *ηκ*(*a*) ∧ *ηκ*(*b*) ≥ *tκ* ∧ *tκ* = *tκ* for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}. Therefore, *ab* ∈ *η<sup>t</sup>*. Then *ηt* is a subsemigroup of *P*.

Conversely, let *ηt* = *φ* be a subsemigroup of *P*. On the contrary, let us consider that *η* is not an *m*-PFSS of *P*. Suppose *a*, *b* ∈ *P* such that *ηκ*(*ab*) < *ηκ*(*a*) ∧ *ηκ*(*b*) for some *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Take *tκ* = *ηκ*(*a*) ∧ *ηκ*(*b*) for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Then, *a*, *b* ∈ *ηt* but *ab* ∈/ *η<sup>t</sup>*, there is a contradiction. Hence, *ηκ*(*ab*) ≥ *ηκ*(*a*) ∧ *ηκ*(*b*). Thus, *η* is an *m*-PFSS of *P*. Other cases can be proved on the same lines.

Now, we define the *m*-PFGBI of a semigroup.

**Definition 5.** *An m-PFS η* = (*η*1, *η*2, ... , *ηm*) *of P is called an m-PFGBI of P if for all a*, *b*, *c* ∈ *P*, *η*(*abc*) ≥ *η*(*a*) ∧ *η*(*c*), *that is ηκ*(*abc*) ≥ *ηκ*(*a*) ∧ *ηκ*(*c*) *for all κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

**Lemma 5.** *A subset H of P is generalized bi-ideal of P if and only if CH is an m-PFGBI of P.*

**Proof.** This Lemma 5 can be proved similarly to the proof of Lemma 2.

**Lemma 6.** *An m-PFS η of P is m-PFGBI of P if and only if, η* ◦ *δ* ◦ *η* ≤ *η, where δ is the m-PFS of P that maps each element of P on* (1, 1, . . . , 1)*.*

**Proof.** Suppose *η* = (*η*1, *η*2, ... , *ηm*) is the *m*-PFGBI of *P*, that is, *ηκ*(*abc*) ≥ *ηκ*(*a*) ∧ *ηκ*(*c*) for all *κ* ∈ {1, 2, ... , *m*} and *a*, *b*, *c* ∈ *P*. Let *p* ∈ *P*. If *p* is not expressible as *p* = *ab* for some *a*, *b* ∈ *P*, then (*η* ◦ *δ* ◦ *η*)(*p*) = 0. Hence, *η* ◦ *δ* ◦ *η* ≤ *η*. However, if *p* is expressible as *p* = *ab* for some *a*, *b* ∈ *P*. Then

$$\begin{aligned} (\eta\_{\mathbb{X}} \circ \delta\_{\mathbb{X}} \circ \eta\_{\mathbb{X}})(p) &= \bigvee\_{p=ab} \{ (\eta\_{\mathbb{X}} \circ \delta\_{\mathbb{X}})(a) \wedge \eta\_{\mathbb{X}}(b) \} \\ &= \bigvee\_{p=ab} \{ \bigvee\_{a=uv} \{ \eta\_{\mathbb{X}}(u) \wedge \delta\_{\mathbb{X}}(v) \} \wedge \eta\_{\mathbb{X}}(b) \} \\ &= \bigvee\_{p=ab} \{ \bigvee\_{a=uv} \{ \eta\_{\mathbb{X}}(u) \wedge \eta\_{\mathbb{X}}(b) \} \} \\ &\leq \bigvee\_{p=ab} \{ \bigvee\_{a=uv} \{ \eta\_{\mathbb{X}}(uv)(b) \} \} \\ &= \bigvee\_{p=ab} \{ \eta\_{\mathbb{X}}(ab) \} \text{ for all } \mathfrak{x} \in \{ 1, 2, \dots, m \}. \\ &= \eta\_{\mathbb{X}}(p) \text{ for all } \mathfrak{x} \in \{ 1, 2, \dots, m \}. \end{aligned}$$

Hence, *η* ◦ *δ* ◦ *η* ≤ *η*. Conversely, let *η* ◦ *δ* ◦ *η* ≤ *η* and *a*, *b*, *c* ∈ *P*. Then,

$$\begin{array}{rcl} \eta\_{\mathbb{K}}(abc) & \geq & (\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}}) \circ \eta\_{\mathbb{K}})((ab)c) \\ & = & \bigvee\_{(ab)\mathbb{x} = \mathsf{u}\mathsf{y}} \left\{ (\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}})(\mathsf{u}) \wedge \eta\_{\mathbb{K}}(\mathsf{v}) \right\} \\ & \geq & (\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}})(ab) \wedge \eta\_{\mathbb{K}}(\mathsf{c}) \\ & = & \bigvee\_{(ab) = \mathsf{x}\mathsf{y}} \left\{ (\eta\_{\mathbb{K}}(\mathsf{x}) \wedge \delta\_{\mathbb{K}})(\mathsf{y}) \right\} \wedge \eta\_{\mathbb{K}}(\mathsf{c}) \\ & = & \left\{ (\eta\_{\mathbb{K}}(a) \wedge \delta\_{\mathbb{K}})(b) \right\} \wedge \eta\_{\mathbb{K}}(\mathsf{c}) \\ & = & \eta\_{\mathbb{K}}(a) \wedge \eta\_{\mathbb{K}}(\mathsf{c}) \text{ for all } \mathsf{x} \in \{1, 2, \ldots, m\}. \end{array}$$

Hence, *η*(*abc*) ≥ *η*(*a*) ∧ *η*(*c*). Thus, *η* is *m*-PFGBI of *P*.

**Proposition 2.** *Assume that η* = (*η*1, *η*2, ... , *ηm*) *is an m-PFS of P. Then, η is an m-PFGBI of P if, and only if, ηt* = {*a* ∈ *<sup>P</sup>*|*η*(*a*) ≥ *t*} = *φ is a generalized bi-ideal of P for all t* = (*<sup>t</sup>*1, *t*2,..., *tm*) ∈ (0, <sup>1</sup>]*<sup>m</sup>*.

**Proof.** Let *η* be an *m*-PFGBI of *P*. Let *a*, *c* ∈ *ηt* and *b* ∈ *P*. Then, *ηκ*(*a*) ≥ *tκ* and *ηκ*(*c*) ≥ *tκ* for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Since *η* is *m*-PFGBI of *P*, we have *ηκ*(*abc*) ≥ *ηκ*(*a*) ∧ *ηκ*(*c*) ≥ *tκ* ∧ *tκ* = *tκ* for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}. Therefore, *abc* ∈ *η<sup>t</sup>*. That is *ηt* is a GBI of *P*.

Conversely, assuming that *ηt* = *φ* is a GBI of *P*. On the contrary, assume that *η* is not *m*-PFGBI of *P*. Suppose *a*, *b*, *c* ∈ *P*, such that *ηκ*(*abc*) < *ηκ*(*a*) ∧ *ηκ*(*c*) for some *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Take *tκ* = *ηκ*(*a*) ∧ *ηκ*(*c*) for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Then, *a*, *c* ∈ *ηt* but *abc* ∈/ *η<sup>t</sup>*, which is a contradiction. Hence, *ηκ*(*abc*) ≥ *ηκ*(*a*) ∧ *ηκ*(*c*), that is, *η* is *m*-PFGBI of *P*.

Next, we define the *m*-PFBI of a semigroup.

**Definition 6.** *A subsemigroup η* = (*η*1, *η*2, ... , *ηm*) *of P is called an m-PFBI of P if for all a*, *b*, *c* ∈ *P*, *η*(*abc*) ≥ *η*(*a*) ∧ *η*(*c*) *that is, ηκ*(*abc*) ≥ *ηκ*(*a*) ∧ *ηκ*(*c*) *for all κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

**Lemma 7.** *A subset H of P is a bi-ideal of P if, and only if, CH is an m-PFBI of P.*

**Proof.** Follows from Lemmas 2 and 5.

**Lemma 8.** *An m-PFSS η of P is an m-PFBI of P if and only if, η* ◦ *δ* ◦ *η* ≤ *η, where δ is the m-PFS of P, which maps each element of P on* (1, 1, . . . , 1)*.*

**Proof.** Follows from Lemma 6.

**Proposition 3.** *Let η* = (*η*1, *η*2, ... , *ηm*) *be a subsemigroup of P. Then η is an m-PFBI of P if and only if, ηt* = {*a* ∈ *<sup>P</sup>*|*η*(*a*) ≥ *t*} = *φ is a bi-ideal of P for all t* = (*<sup>t</sup>*1, *t*2,..., *tm*) ∈ (0, <sup>1</sup>]*<sup>m</sup>*.

**Proof.** Follows from Proposition 2.

**Remark 1.** *Every m-PFBI of P is an m-PFGBI of P.*

The Example 3 illustrates that the converse of above Remark may not be true.

**Example 3.** *Let P* = {*ı*, *j*, , ¯*h*} *be a semigroup given in Table 5.*

**Table 5.** Table of multiplication of *P.*


We define a 4-PFS *η* = (*η*1, *η*2, *η*3, *η*4) of *P* as follows: *η*(*ı*)=(0.1, 0.3, 0.3, 0.4), *η*(*j*)=(0, 0, 0, <sup>0</sup>), *η*()=(0, 0, 0, <sup>0</sup>), *η*(*h*¯)=(0.5, 0.6, 0.7, 0.8). Then, simple calculations show that the *η* is a 4-PFGBI of *P*.

Now, *η*(*j*) = *η*(*h*¯ *h*¯)=(0, 0, 0, 0) (0.5, 0.6, 0.7, 0.8) = *η*(*h*¯) ∧ *η*(*h*¯). Therefore, *η* is not a bi-ideal of *P*. Next, we define the *m*-PFQI of a semigroup.

**Definition 7.** *An m-PFS η* = (*η*1, *η*2, ... , *ηm*) *of P is called an m-PFQI of P if* (*η* ◦ *δ*) ∧ (*δ* ◦ *η*) ≤ *η*, *that is* (*ηκ* ◦ *δκ*) ∧ (*δκ* ◦ *ηκ*) ≤ *ηκ*, *for all κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

**Lemma 9.** *A subset H of P is a quasi ideal of P if and only if CH is an m-PFQI of P*.

**Proof.** Let *H* be a quasi ideal of *P*, that is *HP* ∩ *PH* ⊆ *H*. We show that (*CH* ◦ *δ*) ∧ (*δ* ◦ *CH*) ≤ *CH*, that is, ((*CH* ◦ *δ*) ∧ (*δ* ◦ *CH*))(*h*) ≤ *CH*(*h*) for every *h* ∈ *P*. We study the following cases:

Case 1: If *h* ∈ *H* then *CH*(*h*)=(1, 1, ... , 1) ≥ ((*CH* ◦ *δ*) ∧ (*δ* ◦ *CH*))(*h*). Hence (*CH* ◦ *δ*) ∧ (*δ* ◦ *CH*) ≤ *CH*.

Case 2: If *h* ∈/ *H* then *h* ∈/ *HP* ∩ *PH*. This implies that *h* = *bc* and *h* = *f e* for some *b* ∈ *H*, *c* ∈ *P*, *f* ∈ *P* and *e* ∈ *H*. Therefore, either (*CH* ◦ *δ*)(*h*)=(0, 0, ... , 0) or (*δ* ◦ *CH*)(*h*)=(0, 0, ... , 0) that is ((*CH* ◦ *δ*) ∧ (*δ* ◦ *CH*))(*h*)=(0, 0, ... , 0) ≤ *CH*(*h*). Hence (*CH* ◦ *δ*) ∧ (*δ* ◦ *CH*) ≤ *CH*.

Conversely, let *n* ∈ *HP* ∩ *PH*. Then *n* = *be* and *n* = *a f* , where *a*,*e* ∈ *P* and *b*, *f* ∈ *H*. Since *CH* is an *m*-PFQI of *P*, we have

$$\begin{array}{rcl} \mathsf{C}\_{H}(n) & \geq & \left( (\mathsf{C}\_{H} \circ \delta) \wedge (\delta \circ \mathsf{C}\_{H}) \right)(n) \\ & = & (\mathsf{C}\_{H} \circ \delta)(n) \wedge (\delta \circ \mathsf{C}\_{H})(n) \\ & = & \left\{ \bigvee\_{n=wv} \left\{ (\mathsf{C}\_{H}(w) \wedge \delta(v)) \right\} \wedge \left\{ \bigvee\_{n=pq} \{ \delta(p) \wedge \mathsf{C}\_{H}(q) \} \right\} \right. \\ & \geq & \left\{ \mathsf{C}\_{H}(b) \wedge \delta(e) \right\} \wedge \left\{ \delta(a) \wedge \mathsf{C}\_{H}(f) \right\} \text{ since } n=be \text{ and } n=af \\ & = & (1,1,\ldots,1). \end{array}$$

Therefore, *CH*(*n*)=(1, 1, . . . , <sup>1</sup>). Hence, *n* ∈ *H*.

**Proposition 4.** *An m-PFS η* = (*η*1, *η*2, ... , *ηm*) *of P is an m-PFQI of P if, and only if, ηt* = {*a* ∈ *<sup>P</sup>*|*η*(*a*) ≥ *t*} = *φ is a quasi ideal of P for all t* = (*<sup>t</sup>*1, *t*2,..., *tm*) ∈ (0, <sup>1</sup>]*<sup>m</sup>*.

**Proof.** Let *η* is an *m*-PFQI of *P*. To show that *ηtP* ∩ *Pηt* ⊆ *η<sup>t</sup>*. Let *n* ∈ *ηtP* ∩ *Pη<sup>t</sup>*. Then, *n* ∈ *ηtP* and *n* ∈ *Pη<sup>t</sup>*. Therefore, *n* = *ba* and *n* = *sd* for some *a*,*s* ∈ *P* and *b*, *d* ∈ *η<sup>t</sup>*. Therefore, *ηκ* ≥ *tκ* for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

Now

$$\begin{array}{rcl} (\eta\_{\mathbf{x}} \circ \delta\_{\mathbf{x}})(n) &=& \bigvee\_{n=uv} \{\eta\_{\mathbf{x}}(u) \wedge \delta\_{\mathbf{x}}(v)\} \\ &\ge& \eta\_{\mathbf{x}}(b) \wedge \delta\_{\mathbf{x}}(a) \text{ because } n=ba \\ &=& \eta\_{\mathbf{x}}(b) \wedge 1 \\ &=& \eta\_{\mathbf{x}}(b) \\ &\ge& t\_{\mathbf{x}}. \end{array}$$

So

$$(\eta\_{\kappa} \circ \delta\_{\kappa})(n) \succeq t\_{\kappa} \text{ for all } \kappa \in \{1, 2, \dots, m\}.$$

Now

$$\begin{aligned} (\delta\_{\mathbb{K}} \circ \eta\_{\mathbb{K}})(n) &= \bigvee\_{n=\mathbb{n}\mathbb{P}} \{\delta\_{\mathbb{K}}(u) \wedge \eta\_{\mathbb{K}}(v)\} \\ &\ge \quad \delta\_{\mathbb{K}}(s) \wedge \eta\_{\mathbb{K}}(d) \text{ because } n=sd \\ &= \quad 1 \wedge \eta\_{\mathbb{K}}(d) \\ &= \quad \eta\_{\mathbb{K}}(d) \\ &\ge \quad t\_{\mathbb{K}}. \end{aligned}$$

So

$$(\delta\_{\kappa} \circ \eta\_{\kappa})(n) \succeq t\_{\kappa} \text{ for all } \kappa \in \{1, 2, \dots, m\}.$$

Therefore, ((*ηκ* ◦ *δκ*) ∧ (*δκ* ◦ *ηκ*))(*n*)=(*ηκ* ◦ *δκ*)(*n*) ∧ (*δκ* ◦ *ηκ*)(*n*) ≥ *tκ* ∧ *tκ* = *tκ* for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. So, ((*η* ◦ *δ*) ∧ (*δ* ◦ *η*))(*n*) ≥ *t*. Since *η*(*n*) ≥ ((*η* ◦ *δ*) ∧ (*δ* ◦ *η*))(*n*) ≥ *t*, so *n* ∈ *η<sup>t</sup>*. Hence, *ηt* is a quasi ideal of *P*.

Conversely, consider that *η* is not quasi ideal of *P*. Let *n* ∈ *P* be such that *ηκ*(*n*) < (*ηκ* ◦ *δκ*)(*n*) ∧ (*δκ* ◦ *ηκ*)(*n*) for some *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Choose *tκ* ∈ (0, 1], such that *tκ* = (*ηκ* ◦ *δκ*)(*n*) ∧ (*δκ* ◦ *ηκ*)(*n*) for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. This implies that *n* ∈ (*ηκ* ◦ *δκ*)*<sup>t</sup>κ* and *n* ∈ (*δκ* ◦ *ηκ*)*<sup>t</sup>κ* but *n* ∈/ (*ηκ*)*<sup>t</sup>κ* for some *κ*. Hence, *n* ∈ (*η* ◦ *<sup>P</sup>*)*t* and *n* ∈ (*P* ◦ *η*)*t* but *n* ∈/ (*η*)*<sup>t</sup>*, which is a contradiction. Hence, (*η* ◦ *δ*) ∧ (*δ* ◦ *η*) ≤ *η*.

**Lemma 10.** *Every m-PF one-sided ideal of P is an m-PFQI of P.*

**Proof.** This proof follows from Lemma 3.

In the next example, it is shown that the converse of the above Lemma may not be true.

**Example 4.** *Consider the semigroup P* = {*ı*, *j*, } *given in Table 6.*

**Table 6.** Table of multiplication of *P*.


Define a 3-PFS *η* = (*η*1, *η*2, *η*3) of *P* as follows: *η*(*ı*)=(0.3, 0.3, 0.4), *η*(*j*)=(0.7, 0.8, 0.9), *η*()=(0, 0, <sup>0</sup>).

Then, simple calculations show that *ηj* is QI of *P*. Therefore, by using Proposition 4, *η* is 3-PFQI of *P*. Now,

*η*() = *η*(*ı*)=(0, 0, 0) *η*(*ı*)=(0.3, 0.3, 0.4). So *η* is not 3-PFI (right) of *P*.

**Lemma 11.** *Let η* = (*η*1, *η*2, ... , *ηm*) *and m* = (*<sup>m</sup>*1, *m*2, ... , *mm*) *be two m-PFI*(*right*) *and m-PFI*(*left*) *of P*, *respectively. Then η* ∧ *m is m-PFQI of P.*

**Proof.** Let *n* ∈ *P*. If *n* = *bt* for some *b*, *t* ∈ *P*. Then, ((*η* ∧ *m*) ◦ *δ*) ∧ (*δ* ◦ (*η* ∧ *m*)) ≤ (*η* ∧ *<sup>m</sup>*). If *n* = *ab* for some *a*, *b* ∈ *P*, then

(((*ηκ* ∧ *<sup>m</sup>κ*) ◦ *δκ*) ∧ (*δκ* ◦ (*ηκ* ∧ *<sup>m</sup>κ*)))(*n*) = ((*ηκ* ∧ *<sup>m</sup>κ*) ◦ *δκ*)(*n*) ∧ (*δκ* ◦ (*ηκ* ∧ *<sup>m</sup>κ*))(*n*) = *<sup>n</sup>*=*ab* {(*ηκ* ∧ *<sup>m</sup>κ*)(*a*) ∧ *δκ*(*b*)} ∧ *<sup>n</sup>*=*ab* {*δκ*(*a*) ∧ (*ηκ* ∧ *<sup>m</sup>κ*)(*b*)} = *<sup>n</sup>*=*ab* {(*ηκ* ∧ *<sup>m</sup>κ*)(*a*)} ∧ *<sup>n</sup>*=*ab* {(*ηκ* ∧ *<sup>m</sup>κ*)(*b*)} = *<sup>n</sup>*=*ab* {(*ηκ* ∧ *<sup>m</sup>κ*)(*a*) ∧ (*ηκ* ∧ *<sup>m</sup>κ*)(*b*)} = *<sup>n</sup>*=*ab* {(*ηκ*(*a*) ∧ *<sup>m</sup>κ*(*a*)) ∧ (*ηκ*(*b*) ∧ *<sup>m</sup>κ*(*b*))} ≤ *<sup>n</sup>*=*ab* {*ηκ*(*a*) ∧ *<sup>m</sup>κ*(*b*)} ≤ *<sup>n</sup>*=*ab* {*ηκ*(*ab*) ∧ *<sup>m</sup>κ*(*ab*)} = *<sup>n</sup>*=*ab* {(*ηκ* ∧ *<sup>m</sup>κ*)(*ab*)} = (*ηκ* ∧ *<sup>m</sup>κ*)(*n*) for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

Hence, ((*η* ∧ *m*) ◦ *δ*) ∧ (*δ* ◦ (*η* ∧ *m*)) ≤ (*η* ∧ *<sup>m</sup>*), that is, *η* ∧ *m* is *m*-PFQI of *P*.

Now, we define the *m*-PFII of a semigroup. **Definition 8.** *An m-PFSS η* = (*η*1, *η*2, ... , *ηm*) *of P is called an m-PFII of P if for all a*, *b*, *c* ∈ *P*, *η*(*abc*) ≥ *η*(*b*)*, that is, ηκ*(*abc*) ≥ *ηκ*(*b*) *for all κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

**Lemma 12.** *A subset H of P is an interior ideal of P if, and only if, CH is an m-PFII of P.*

**Proof.** Let *H* be any interior ideal of *P*. From Lemma 2, *CH* is an *m*-PFSS of *P*. Now we show that *CH*(*abc*) ≥ *CH*(*b*) for every *a*, *b*, *c* ∈ *P*. We consider the following two cases:

Case 1: Let *b* ∈ *H* and *a*, *c* ∈ *P*. Then *CH*(*b*)=(1, 1, ... , <sup>1</sup>). Since *H* is an interior ideal of *P*, then *abc* ∈ *H*. Then, *CH*(*abc*)=(1, 1, . . . , <sup>1</sup>). Hence, *CH*(*abc*) ≥ *CH*(*b*).

Case 2: Let *b* ∈/ *H* and *a*, *c* ∈ *P*. Then, *CH*(*b*)=(0, 0, ... , <sup>0</sup>). Clearly, *CH*(*abc*) ≥ *CH*(*b*). Hence, *CH* of *H* is an *m*-PFII of *P*.

Conversely, consider *CH* of *H* is an *m*-PFII of *P*. Then by Lemma 2, *H* is a subsemigroup of *P*. Let *b* ∈ *H* and *a*, *c* ∈ *P*. Then *CH*(*b*)=(1, 1, ... , <sup>1</sup>). By hypothesis, *CH*(*abc*) ≥ *CH*(*b*)=(1, 1, ... , <sup>1</sup>). Hence *CH*(*abc*)=(1, 1, ... , <sup>1</sup>). This implies that *abc* ∈ *H*, that is *H* is an interior ideal of *P*.

**Lemma 13.** *An m-PFSS η of P is an m-PFII of P if, and only if, δ* ◦ *η* ◦ *δ* ≤ *η*.

**Proof.** Let *η* = (*η*1, *η*2, ... , *ηm*) be *m*-PFII of *P*. We show that *δ* ◦ *η* ◦ *δ* ≤ *η*. Let *n* ∈ *P*. Then, for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

$$\begin{aligned} \left(\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}} \circ \delta\_{\mathbf{x}}\right)(n) &= \bigvee\_{n=uv} \{ (\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}})(u) \wedge \delta\_{\mathbf{x}}(v) \} \\ &= \bigvee\_{n=uv} \{ (\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}})(u) \} \\ &= \bigvee\_{n=uv} \{ \bigvee\_{u=ab} (\delta\_{\mathbf{x}}(a) \wedge \eta\_{\mathbf{x}}(b)) \} \\ &= \bigvee\_{n=(ab)v} \{ \eta\_{\mathbf{x}}(b) \} \\ &\leq \bigvee\_{n=(ab)v} \{ \eta\_{\mathbf{x}}((ab)v) \} \text{ as } \eta \text{ is an } m\text{-PFII of } P. \\ &= \eta\_{\mathbf{x}}(n) \text{ for all } \mathbf{x} \in \{ 1, 2, \dots, m \}. \end{aligned}$$

Therefore *δ* ◦ *η* ◦ *δ* ≤ *η*.

Conversely, let *δ* ◦ *η* ◦ *δ* ≤ *η*. We only show that *ηκ*(*abc*) ≥ *ηκ*(*b*) for every *a*, *b*, *c* ∈ *P* and for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}. Let *n* = *abc*. Now, for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}.

$$\begin{split} \eta\_{\mathbf{x}}(abc) &\geq \quad \{ (\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}}) \circ \delta\_{\mathbf{x}} \} ((ab)c) \\ &= \quad \bigvee\_{(ab)c = n\nu} \{ ((\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}})(\mu) \wedge \delta\_{\mathbf{x}}(v)) \} \\ &\geq \quad \{ \delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}} \} (ab) \wedge \delta\_{\mathbf{x}}(c) \\ &= \quad \{ \delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}} \} (ab) \\ &= \quad \bigvee\_{ab = \eta \circ} \{ \delta\_{\mathbf{x}}(p) \wedge \eta\_{\mathbf{x}}(q) \} \\ &\geq \quad \delta\_{\mathbf{x}}(a) \wedge \eta\_{\mathbf{x}}(b) \\ &= \quad \eta\_{\mathbf{x}}(b) \text{ for all } \mathbf{x} \in \{ 1, 2, \dots, m \}. \end{split}$$

Therefore, *ηκ*(*abc*) ≥ *ηκ*(*b*) for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}. Hence, *η* is *m*-PFII of *P*.

**Proposition 5.** *A subset η* = (*η*1, *η*2, ... , *ηm*) *of P is m-PFII of P if, and only if, ηt* = {*a* ∈ *<sup>P</sup>*|*η*(*a*) ≥ *t*} = *φ is an interior ideal of P for all t* = (*<sup>t</sup>*1, *t*2,..., *tm*) ∈ (0, 1]*m.*

**Proof.** This is the same as the proof of Propositions 1 and 2.

#### **4. Characterization of Regular and Intra-Regular Semigroups by** *m***-Polar Fuzzy Ideals**

A semigroup *P* is called regular if for all *x* ∈ *P*, there exists an element *a* ∈ *P* such that *x* = *xax*. A semigroup *P* is called an intra-regular semigroup if for all *x* ∈ *P*, there exists elements *b*, *c* ∈ *P* such that *x* = *bx*2*c*. Regular and intra-regular semigroups have been studied by several authors, see [24,28]. The characterizations of the regular and intra-regular semigroups in terms of *m*-PF ideals and *m*-PFBI are discussed with the help of many theorems in this section.

**Theorem 1** ([28])**.** *The following results are equivalent in P.*


**Theorem 2.** *Each m-PFQI η of P is an m-PFBI of P.*

**Proof.** Suppose that *η* = (*η*1, *η*2,..., *ηm*) be *m*-PFQI of *P*. Let *a*, *b* ∈ *P*. Then,

$$\begin{array}{lcl} \eta\_{\mathbf{x}}(ab) & \geq & \{ (\eta\_{\mathbf{x}} \circ \delta\_{\mathbf{x}}) \land (\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}}) \}(ab) \\ & = & (\eta\_{\mathbf{x}} \circ \delta\_{\mathbf{x}})(ab) \land (\delta\_{\mathbf{x}} \circ \eta\_{\mathbf{x}})(ab) \\ & = & [\bigvee\_{ab = op} \{ (\eta\_{\mathbf{x}}(o) \land \delta\_{\mathbf{x}})(p) \}] \land [\bigvee\_{ab = uv} \{ (\delta\_{\mathbf{x}}(u) \land \eta\_{\mathbf{x}}(v)) \}] \\ & \geq & \{ (\eta\_{\mathbf{x}}(a) \land \delta\_{\mathbf{x}})(b) \} \land \{ (\delta\_{\mathbf{x}}(a) \land \eta\_{\mathbf{x}}(b)) \} \\ & = & \{ (\eta\_{\mathbf{x}}(a) \land 1) \land \{ 1 \land \eta\_{\mathbf{x}}(b) \} \\ & = & \eta\_{\mathbf{x}}(a) \land \eta\_{\mathbf{x}}(b) \text{ for all } \mathbf{x} \in \{ 1, 2, \dots, m \}. \end{array}$$

So, *ηκ*(*ab*) ≥ *ηκ*(*a*) ∧ *ηκ*(*b*). Now, let *a*, *b*, *c* ∈ *P*. Then,

$$\begin{aligned} (\delta\_{\mathbb{k}} \circ \eta\_{\mathbb{k}}))((ab)\mathfrak{c}) &= \bigvee\_{(ab)\mathfrak{c}=\mathfrak{n}\mathfrak{v}} \{ (\delta\_{\mathbb{k}}(\mathfrak{u}) \wedge \eta\_{\mathbb{k}}(\mathfrak{v})) \} \\ &\geq \quad \delta\_{\mathbb{k}}(ab) \wedge \eta\_{\mathbb{k}}(\mathfrak{c}) \\ &= \quad 1 \wedge \eta\_{\mathbb{k}}(\mathfrak{c}) \\ &= \quad \eta\_{\mathbb{k}}(\mathfrak{c}). \end{aligned}$$

Therefore, (*δκ* ◦ *ηκ*)(*abc*) ≥ *ηκ*(*c*) for all *κ* ∈ {1, 2, ... , *<sup>m</sup>*}. Since (*ab*)*c* = *a*(*bc*) ∈ *aP*, so (*ab*)*c* = *ap* for some *p* ∈ *P*. Therefore,

$$\begin{aligned} (\eta\_{\mathbf{k}} \circ \delta\_{\mathbf{k}})(abc) &= \bigvee\_{(ab)c=ab} \{ (\eta\_{\mathbf{k}}(a) \wedge \delta\_{\mathbf{k}})(b) \} \\ &\ge \quad \eta\_{\mathbf{k}}(a) \wedge \delta\_{\mathbf{k}}(p) \text{ since } (ab)c = ap \\ &= \quad \eta\_{\mathbf{k}}(a) \wedge 1 \\ &= \quad \eta\_{\mathbf{k}}(a) .\end{aligned}$$

Therefore, (*ηκ* ◦ *δκ*)(*abc*) ≥ *ηκ*(*a*) for all *κ* ∈ {1, 2, . . . , *<sup>m</sup>*}. Now, by our supposition

$$\begin{array}{rcl} \eta\_{\mathbb{K}}(abc) & \geq & ((\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}}) \wedge (\delta\_{\mathbb{K}} \circ \eta\_{\mathbb{K}}))(abc) \\ & = & (\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}})(abc) \wedge (\delta\_{\mathbb{K}} \circ \eta\_{\mathbb{K}})(abc) \\ & \geq & \eta\_{\mathbb{K}}(a) \wedge \eta\_{\mathbb{K}}(c) \text{ for all } \kappa \in \{1, 2, \dots, m\}. \end{array}$$

Therefore, *η*(*abc*) ≥ *η*(*a*) ∧ *η*(*c*). Hence, *η* is *m*-PFBI of *P*.

**Theorem 3.** *The given statements are equivalent in P.*

*1. P is regular;* *2. η* ∧ *m* = *η* ◦ *m for every m-PFI*(*right*) *η and m-PFI*(*left*) *m of P.*

**Proof.** (1) =⇒ (2) : Let *η* = (*η*1, *η*2, ... , *ηm*) and *m* = (*<sup>m</sup>*1, *m*2, ... , *mm*) be two *m*-PFI(right) and *m*-PFI(left) of *P*. Let *o* ∈ *P*, we have

$$\begin{aligned} (\eta\_{\mathbf{x}} \diamond m\_{\mathbf{x}}')(o) &= \bigvee\_{o=bc} \{ (\eta\_{\mathbf{x}}(b) \land m\_{\mathbf{x}}'(c)) \} \\ &\le \bigvee\_{o=bc} \{ (\eta\_{\mathbf{x}}(bc) \land m\_{\mathbf{x}}'(bc)) \} \\ &= \eta\_{\mathbf{x}}(o) \land m\_{\mathbf{x}}'(o) \\ &= (\eta\_{\mathbf{x}} \land m\_{\mathbf{x}}')(o) \text{ for all } \mathbf{x} \in \{1, 2, \dots, m\}. \end{aligned}$$

Therefore, *η* ◦ *m* ≤ *η* ∧ *<sup>m</sup>*. As *P* is regular, then for every, *o* ∈ *P*, there exists *a* ∈ *P*, such that *o* = (*oa*)*<sup>o</sup>*.

$$\begin{array}{rcll} \left(\eta\_{\mathbb{X}} \wedge m\_{\mathbb{X}}^{\prime}\right)(o) &=& \eta\_{\mathbb{X}}(o) \wedge m\_{\mathbb{x}}^{\prime}(o) \\ &\leq& \eta\_{\mathbb{x}}(oa) \wedge m\_{\mathbb{x}}^{\prime}(o) \text{ as } \eta \text{ is } m\text{-PFRI of } P. \\ &\leq& \bigvee\_{o=bc} \{ (\eta\_{\mathbb{x}}(b) \wedge m\_{\mathbb{x}}^{\prime}(c)) \} \\ &=& (\eta\_{\mathbb{x}} \circ m\_{\mathbb{x}}^{\prime})(o) \text{ for all } \mathbb{x} \in \{1, 2, \ldots, m\}. \end{array}$$

So, *η* ∧ *m* ≤ *η* ◦ *<sup>m</sup>*. Therefore, *η* ∧ *m* = *η* ◦ *<sup>m</sup>*.

(2) =⇒ (1) : Let *o* ∈ *P*. Then, *η* = *oP* is a left ideal of *P* and *m* = *oP* ∪ *Po* is a right ideal of *P* generated by *o*. Then, by using Lemma 2**,** *Cη* and *Cm* the *m*-polar fuzzy characteristic fuctions of *η* and *m* are *m*-PFI(left) and *m*-PFI(right) of *P*, respectively. Then, we have

$$\begin{array}{rcl} \mathsf{C}\_{\mathsf{m}'\eta} & = & (\mathsf{C}\_{\mathsf{m}'} \circ \mathsf{C}\_{\eta}) \text{ by Lemma 1} \\ & = & (\mathsf{C}\_{\mathsf{m}'} \land \mathsf{C}\_{\mathsf{N}}) \text{ by 2} \\ & = & \mathsf{C}\_{\mathsf{m}' \cap \eta} \text{ by Lemma 1} \end{array}$$

Therefore, *m* ∩ *η* = *<sup>m</sup>η*. As a result, Theorem 1 shows that *P* is regular.

**Theorem 4.** *The following statements are equivalent in P.*


**Proof.** (1) =⇒ (2) : Let *η* = (*η*1, *η*2, ... , *ηm*) be an *m*-PFGBI of *P* and *o* ∈ *P*. Since *P* is regular, there exists *a* ∈ *P such that o* = (*oa*)*<sup>o</sup>*. Therefore, we have

$$\begin{array}{rcll} \left(\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}} \circ \eta\_{\mathbb{K}}\right)(o) & = & \bigvee\_{o=\mathbb{K}} \left\{ (\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}})(b) \wedge \eta\_{\mathbb{K}}(c) \right\} \text{ for some } b, c \in P \\ & & \geq & \left(\eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}}\right)(oa) \wedge \eta\_{\mathbb{K}}(o) \text{ since } o = (oa)o \\ & = & \bigvee\_{\text{or} = pq} \left\{ \eta\_{\mathbb{K}}(p) \wedge \delta\_{\mathbb{K}}\right)(q) \right\} \wedge \eta\_{\mathbb{K}}(o) \\ & \geq & \left\{ \eta\_{\mathbb{K}}(o) \wedge \delta\_{\mathbb{K}}\right\}(a) \right\} \wedge \eta\_{\mathbb{K}}(o) \\ & = & \eta\_{\mathbb{K}}(o) \text{ for all } \mathbb{K} \in \{1, 2, \dots, m\}. \end{array}$$

Hence, *η* ◦ *δ* ◦ *η* ≥ *η*. Since *η* is an *m*-PFGBI of *P*. Therefore, we have

$$\begin{aligned} \left(\eta\_{\mathbf{x}} \diamond \delta\_{\mathbf{x}} \diamond \eta\_{\mathbf{x}}\right)(o) &= \bigvee\_{o=rs} \{ (\eta\_{\mathbf{x}} \diamond \delta\_{\mathbf{x}})(r) \land \eta\_{\mathbf{x}}(s) \} \text{ for some } r, s \in P \\ &= \bigvee\_{o=rs} \{ \bigvee\_{r=uv} \{ \eta\_{\mathbf{x}}(u) \land \delta\_{\mathbf{x}}(v) \} \land \eta\_{\mathbf{x}}(s) \} \text{ for some } r, s \in P \\ &= \bigvee\_{o=rs} \{ \bigvee\_{r=uv} \{ \eta\_{\mathbf{x}}(u) \land \eta\_{\mathbf{x}}(s) \} \} \\ &\leq \bigvee\_{o=rs} \{ \bigvee\_{r=uv} \{ \eta\_{\mathbf{x}}((uv)(s)) \} \} \\ &= \bigvee\_{o=rs} \eta\_{\mathbf{x}}(rs) \\ &= \eta\_{\mathbf{x}}(o) \text{ for all } \mathbf{x} \in \{ 1, 2, \dots, m \}. \end{aligned}$$

So, *η* ◦ *δ* ◦ *η* ≤ *η*. Therefore, *η* = *η* ◦ *δ* ◦ *η*.

(2) =⇒ (3) : It is obvious.

(3) =⇒ (1) : Let *η*, *ρ* be *m*-PFI (right) and *m*-PFI (left) of *P*, respectively. Then *η* ∧ *ρ* is an *m*-PFQI of *P*. According to hypothesis

$$\begin{aligned} \eta\_{\mathbb{K}} \wedge \rho\_{\mathbb{K}} & \leq & (\eta\_{\mathbb{K}} \wedge \rho\_{\mathbb{K}}) \circ \delta\_{\mathbb{K}} \circ (\eta\_{\mathbb{K}} \wedge \rho\_{\mathbb{K}}) \\ & \leq & \eta\_{\mathbb{K}} \circ \delta\_{\mathbb{K}} \circ \rho\_{\mathbb{K}} \\ & \leq & \eta\_{\mathbb{K}} \circ \rho\_{\mathbb{K}}. \end{aligned}$$

However, *ηκ* ◦ *ρκ* ≤ *ηκ* ∧ *ρκ* always hold. Hence, *ηκ* ◦ *ρκ* = *ηκ* ∧ *ρκ*, that is *η* ◦ *ρ* ≤ *η* ∧ *ρ*. Therefore by Theorem 3, *P* is a regular semigroup. Hence, proved.

**Theorem 5.** *The following statements are equivalent in P.*


**Proof.** (1) =⇒ (2) : Consider *b* is any element of *P*. As *P* is regular, there exists *a* ∈ *P such that b* = *bab*. It follows that *b* = (*ba*)*b* = *b*(*ab*) for each *a* ∈ *P* and *P* is semigroup. Hence, we have

$$\begin{array}{rcll} (\rho \circ m' \circ \eta)(b) &=& \bigvee\_{b \multimap \text{ac}} \{ (\rho \circ m')(a) \wedge \eta(c) \} \\ &\geq& (\rho \circ m')(b) \wedge \eta(ab) \text{ since } b = b(ab) \\ &\geq& \bigvee\_{b \multimap \text{q}} \{ \rho(p) \wedge m'(q) \} \wedge \eta(b) \text{ as } \eta \text{ is an } m\text{-PFI}(\text{left}) \text{ of } P. \\ &\geq& (\rho(ba) \wedge m'(b)) \wedge \eta(b) \text{ since } b = (ba)b \\ &\geq& (\rho(b) \wedge m'(b)) \wedge \eta(b) \\ &=& ((\rho \wedge m')(b)) \wedge \eta(b) \\ &=& (\rho \wedge m' \wedge \eta)(b). \end{array}$$

Therefore, *ρ* ∧ *m* ∧ *η* ≤ *ρ* ◦ *m* ◦ *η*. So (1) implies (2).

(2) =⇒ (3) =⇒ (4) : Straightforward.

(4) =⇒ (1) : As *δ* is an *m*-PFQI of *P*, by the supposition, we have

$$\begin{array}{rcl} (\rho \wedge \eta)(b) &=& ((\rho \wedge \delta) \wedge \eta)(b) \\ & \leq & ((\rho \circ \delta) \circ \eta(b) \\ & = & \bigvee\_{b \multimap r s} \{ (\rho \circ \delta)(r) \circ \eta(s) \} \\ & = & \bigvee\_{b \multimap r s} \{ (\bigvee\_{r \multimap r} \{ \rho(c) \wedge \delta(d) \}) \wedge \eta(s) \} \\ & = & \bigvee\_{b \multimap r s} \{ (\bigvee\_{r \multimap r} \{ \rho(c) \wedge 1 \}) \wedge \eta(s) \} \\ & = & \bigvee\_{b \multimap r s} \{ (\bigvee\_{r \multimap r} \rho(c)) \wedge \eta(s) \} \\ & = & \bigvee\_{b \multimap r s} \{ (\bigvee\_{r \multimap r} \rho(c)) \wedge \eta(s) \} \\ & \leq & \bigvee\_{b \multimap r s} \{ (\bigvee\_{r \multimap r} \rho(c)) \} \wedge \eta(s) \} \\ & = & \bigvee\_{b \multimap s} \{ \rho(r) \wedge \eta(s) \} \\ & = & (\rho \circ \eta)(b). \end{array}$$

Therefore *ρ* ∧ *η* ≤ *ρ* ◦ *η*. But *ρ* ◦ *η* ≤ *ρ* ∧ *η* always. So, *ρ* ◦ *η* = *ρ* ∧ *η*. Hence, by using Theorem 3, *P* is regular.

**Theorem 6** ([28])**.** *The following conditions are equivalent in P.*


**Definition 9** ([24])**.** *A semigroup P is both regular and intra-regular if and only if H* = *H*<sup>2</sup> *for every bi-ideal H of P.*

**Theorem 7.** *A semigroup P is intra-regular if, and only if η* ∧ *ρ* ≤ *η* ◦ *ρ for every m-PFI*(*left*) *η and, for every, m-PFI*(*right*) *ρ of P.*

**Proof.** Consider *a* is any element of *P*. As *P* is intra-regular, there exists *x*, *y* ∈ *P such that a* = *xa*<sup>2</sup>*y*. Hence, we have

$$\begin{aligned} (\eta \circ \rho)(a) &= \bigvee\_{a=bc} \{\eta(b) \wedge \rho(c)\} \\ &\ge \quad \eta(xa) \wedge \rho(ay) \\ &\ge \quad \eta(a) \wedge \rho(a) \\ &= \quad (\eta \wedge \rho)(a) .\end{aligned}$$

This implies *η* ◦ *ρ* ≥ *η* ∧ *ρ*.

Conversely, assume that *η* ∧ *ρ* ≤ *η* ◦ *ρ* for all *m*-PFI(left) *η* and *m*-PFI(right) *ρ* of *P*. Let *H* be a right ideal and *I* be a left ideal of *P*, then *CH* is an *m*-PFI(right) and *CI* is an *m*-PFI(left) of *P*. By Lemma 1, *CH*∩*<sup>I</sup>* = *CH* ∧ *CI* ≤ *CH* ◦ *CI* = *CH I* which implies that *H* ∩ *I* ⊆ *H I*. Therefore, by Theorem 6, *P* is intra-regular.

**Theorem 8.** *For every m-PFBI η of P, η* ◦ *η* = *η if and only if, P is both regular and intra-regular.*

**Proof.** Let *P* be both regular and intra-regular semigroup. Let *η* be an *m*-PFBI of *P*. Thus, for *x* ∈ *P*, there exists *a*, *b*, *c* ∈ *P* such that *x* = *xax* and *x* = *bx*2*c*. Therefore, *x* = *xax* = *xaxax* = *xa*(*bx*2*c*)*ax* = (*xabx*)(*xcax*). Hence, we have

(*η* ◦ *η*)(*x*) = *<sup>x</sup>*=*bc* {*η*(*b*) ∧ *η*(*c*)} ≥ *η*(*xabx*) ∧ *η*(*xcax*) ≥ *η*(*x*) ∧ *η*(*x*) = *η*(*x*).

This implies *η* ◦ *η* ≥ *η*. By Lemma 3, *η* ◦ *η* ≤ *η* holds always. Therefore, *η* ◦ *η* = *η*.

Conversely, let *H* be a bi-ideal of *P*. Since every *m*-PFBI is *m*-PFSS of *P*. Then, Lemma 2, implies that *C H* is *m*-PFBI of *P*. Hence, by our supposition, *C H* = *C H* ◦ *C H*. Thus, *H* = *H*2. Therefore, by Theorem 9, *P* is both regular and intra-regular.

#### **5. Comparative Study and Discussion**

This section explains how this paper and the previous one are related to Shabir et al. [27]. Shabir et al. [24] studied regular and intra-regular semiring in terms of BFIs. Shabir et al. extended the work of [24] and initiated the concept of *m*-PFIs in LA-semigroups and characterized the regular LA-semigroups by the properties of these *m*-PFIs [27]. By extending the work of [24,27], the concept of *m*-PFIs in semigroups is introduced, and characterizations of regular and intra-regular semigroups by the properties of *m*-PFIs are given in this paper. Our approach is superior to that of Shabir et al. [27] because the associative property in LA-semigroups does not hold. There are also numerous structures that are handled by semigroups but not by LA-semigroups. If we take any non-empty set and define the operation on it as *a* ∗ *b* = *a*, then it is a semigroup, but not an LA-semigroup. To overcome this problem, we used a semigroup to generalize the whole results of Shabir et al. [27] and, as a result, our methodology offers a broader variety of applications than Shabir et al. [27].
