An Innovative Approach towards Possibility Fuzzy Soft Ordered Semigroups for Ideals and Its Application

: The objective of this paper is put forward the novel concept of possibility fuzzy soft ideals and the possibility of fuzzy soft interior ideals. The various results in the form of the theorems with these notions are presented and further validated by suitable examples. In modern life decision-making problems, there is a wide applicability of the possibility fuzzy soft ordered semigroup which has also been constructed in the paper to solve the decision-making process. Elementary and fundamental concepts including regular, intra-regular and simple ordered semigroups in terms of possibility fuzzy soft ordered semigroup are presented. Later, the concept of left (resp. right) regular and left (resp. right) simple in terms of possibility fuzzy soft ordered semigroups are delivered. Finally, the notion of possibility fuzzy soft semiprime ideals in an ordered semigroup is defined and illustrated by theorems and example.


Introduction
As classical mathematical tools are unable to solve the modern uncertain problems, scientists are paying much attention to deriving tools for solving these uncertainties. Fuzzy mathematics is an approach put forwarded by Zadeh that opens up a new direction for scientists in solving such hard problems [1]. Fuzzy mathematics is a key tool to solve many problems from the field of control engineering, robotics, and artificial intelligence, medical diagnose, operational research, and neural networking. The algebraic structures act as a main tool in solving formal coding languages, machine learning, and in providing logical and mathematical basis for many engineering problems. The theory of semigroups has been used in various fields of applied sciences and engineering. Kuroki in [2] introduced fuzzy sets in terms of semigroups in 1991. He also studied fuzzy ideals in semigroups [3,4]. Several classes including simple, regular, intra-regular, weakly regular, and many more were introduced.
Later, Kehayopulu and Tsingelis in [5] studied some elementary properties of fuzzy sets in ordered groupoids, relating the concept of fuzzy ideals in terms of ordered semigroups for the first time. They further put forward the idea of fuzzy bi-ideals and fuzzy interior ideals in ordered semigroups [6,7]. Shabir and khan [8,9] studied some of the properties of fuzzy ideals in ordered semigroups and initiated the theory of fuzzy generalized bi-ideals. Khan et al. [10] presented generalized fuzzy interior ideals in ordered semigroups. Khan et al. [11] induced some new forms of fuzzy interior ideals of ordered semigroups.
An extension to fuzzy set theory, named as the soft set approach, was introduced by Moldstov [12,13] and Maji [14,15], which provided a parametrization tool. Alkalzaleh et al. [16] following Moldstov's approach and gave a new conception of the soft multi-set. An extension to fuzzy set theory, named as multi-fuzzy sets, is initiated by Sebastian in [17]. The fuzzy soft set approach provides a parameterization tool that reduces the uncertainties for theoretical problems and yielded more precise and better results. Well ahead, Yang et al. in [18] broached a new concept of interval-valued fuzzy set theory coupled with the soft set approach. Several other concepts, including intuitionistic fuzzy soft sets, expert sets, neutrosopic sets, vague sets, complex neutrosophic sets, and so on, were introduced recently for handling decision-making and other physical problems [19][20][21][22][23][24][25][26][27][28][29][30][31]. A possibility fuzzy soft set is also included in one of the aforementioned concepts introduced by Alkhalzaleh et al. in [32]. Authors have mentioned some applications of possibility fuzzy soft set theory in medical diagnoses and decision-making. Zhang and Shu [33] proposed a concept of possibility multi-fuzzy soft set theory and its application in decision sciences. Jun et al. [34] put forward a new side by coupling soft sets with ordered semigroups. Later, the ideas of fuzzy soft sets with ordered semigroups and different types of fuzzy soft ideals were further investigated in [35,36]. On basis of the aforementioned research, the concept of possibility fuzzy soft sets in ordered semigroup has been introduced by Habib et al. [37].
In the modern decision-making process, many researchers are studying different approaches of fuzzy soft sets coupling with semigroup theory and relating their possible applications in the field of decision sciences. This progress in the field of fuzzy mathematics motivates the Authors to develop a new direction to solve decision-making problems. The aim of this research is to initiate the concept possibility fuzzy soft ordered semigroups and its extension for interior ideals and left (resp. right) ideals. This new research had a massive scope in solving decision-making problems in the field of applied sciences. Thus, based on these observations, the primary objectives of the present work are summarized as follows: (1) To propose a novel approach of possibility fuzzy soft sets in terms of ordered semigroups and to present the characterizations of various classes of ordered semigroups coupling with possibility fuzzy soft sets. The rest of the manuscript is organized as follows: In Section 2, the authors reiterate some basic concepts and definitions that will help the readers to understand the concepts well. In Section 3, fundamental concepts of possibility fuzzy soft ordered semigroup along with suitable examples are introduced. Some basic definitions and theorems are included. Section 4 includes a new notion of possibility fuzzy soft interior ideals in ordered semigroups. Further, possibility fuzzy soft left (resp. right) ideals are also defined. Several examples, properties of possibility fuzzy soft left (resp. right) ideals and possibility fuzzy soft interior ideals are also investigated. In Section 5, the concept of possibility fuzzy soft simple ordered semigroup is introduced. Furthermore, important relations between regular and intra-regular ordered semigroups with possibility fuzzy soft interior ideals and possibility fuzzy soft left (resp. right) ideals are drawn out. Lastly, the notion of semi-prime possibility fuzzy soft ideals in ordered semigroups has been defined.

Preliminaries
In this section, we recapitulate some basic definitions and lemmas that will further be used to derive some new and fundamental concepts.
For any non-empty subset L of Z is called a left (resp. right) ideal of Z if ⊆ (resp. ⊆ ) and for all ∈ , ∈ if ≤ ⇒ ∈ . Left (resp. right) ideals can be represented as ⊲ (resp. ⊲ ) [39]. If L is both left and right ideal of Z then L is known as an ideal of Z and is represented as ⊲ .
For an ordered semigroup Z, X is a non-empty subset of Z then X is known as an interior ideal of Z if For an ordered semigroup Z, P is a non-empty subset of Z then P is known as semiprime [39] if ∈ ⇒ ∈ (∀ ∈ ) or ⊆ (∀ ⊆ ). An ordered semigroup Z is regular [40]  If for all ⊲ (resp. ⊲ , = then Z is called left (resp. right) simple [41]. An ordered semigroup is simple if it is both right and left simple.
The possibility fuzzy soft set is among the generalized techniques used to resolve decision-making problems.

Possibility Fuzzy Soft Ordered Semigroup
Definition 13.  Table 1 and ordered relation is represented in Figure 1: By using Definition 13, is called PFSS over Z.
where ̅ and ̅ be two fuzzy subsets of Z. Based on possibility fuzzy soft ordered semigroups, the following theorems are stated and since the proofs are straightforward the proofs are omitted.
is also a possibility fuzzy soft ordered semigroup over Z.

Application of Possibility Fuzzy Soft Ordered Semigroups
Let = { , , } be a set of three players of cricket and we have defined the following multiplication table in Table 2 and ordered relation in Figure 2, on the basis of their average performance in last ten matches.  where = 1,2 and = 1,2,3. Here we will define a possibility fuzzy soft set ( ̅ , ) for the first observer or the committee member to choose the best player for the team: Hence, from the above matrix we found that ̅ ( ) ≠ ∅ and ̅ ( ) are fuzzy subsemigroup of Z. Thus, we concluded that, ( ̅ , ) is a possibility fuzzy soft ordered semigroup.
Similarly, let us define a mapping g : → ( ) × ( ), where (g , ) is the second observer or the committee member: .
After satisfying the above condition, (g , ) is a possibility fuzzy soft ordered semigroup. Now we will evaluate the combine results of the committee members, an AND operation must be applied. Let us define (ℎ , ) by using Theorem 4, keeping in mind that (ℎ , ) = ( ̅ , ) ∧ (g , ), The above matrix clearly shows that the membership value as well as the possible membership value for the third player is the highest. Hence, committee choose third player. Additionally, (ℎ , ) is a possibility fuzzy soft ordered semigroup of Z.

Possibility Fuzzy Soft Interior Ideals
This section elaborates some innovative concepts of possibility fuzzy soft left (resp. right) ideals and possibility fuzzy soft interior ideals. Some relatable classes of ordered semigroup including regular, intra-regular, and simple ordered relations are defined along with their suitable examples.  Table 3 and ordered relation as represented by Figure 3:   Figure 3. Hesse diagram.
As in Equation (7  Proof. Intersection of any two possibility fuzzy soft sets is defined as g | , L ∩ g | , L = (g , L), where L ∩ L = L then ∀n ∈ L implies n ∈ L and n ∈ L so either g (n) = g | (n) or g (n) = g | (n) also L ⊑ N hence (g , L) is a PFSS over Z using Theorem 3 Since (g | , L ) ⊲ (f ̅ , N) Similarly, we can prove the 2nd relation. □   Table 4 and ordered relation. .

Proposition 1. If Z is a regular ordered semigroup then for all possibility fuzzy soft interior ideal is a possibility fuzzy soft ideal of Z.
Proof. Let ( ̅ , ) be a possibility fuzzy soft interior ideal of Z and for all , ∈ , ∃ ∈ such that ≤ then by using the Lemma 1:  Proof. The proof directly follows Theorem 5. □ Example 6. Suppose a company is hiring an employer and the CEO will choose between three internees working there for past six months. Let = { , , } be a set of three internees and we have defined in Table 6 and ordered relation on the basis of their regularity in last six months.

Possibility Fuzzy Soft Simple Ordered Semigroup
Let ( , . , ≤) be an ordered semigroup and ( ̅ , ) be a possibility fuzzy soft set over Z. Then for all ∈ and ∈ , a mapping ̅ : → such that ( ( ), ) is a PFSS over Z and is defined as, Proof. Let ( ̅ , ) be a possibility fuzzy soft right ideal over Z and if z Z ∈ there exists ( ) ≠ ∅ ∈ then we have ̅ ( )( ) ≥ ̅ ( )( ) so ∈ ( ) and ( ) ≠ ∅. Let ∈ ( ) and for any ∈ , ≤ then ∈ ( ) ∀ ∈ . Since ( ̅ , ) is a possibility fuzzy soft right ideal over Z and ≤ then for every ∈ we have ̅ ( )( ) ≥ ̅ ( )( ) as ∈ ( ) implies, ̅ ( )( ) ≥ ̅ ( )( ) and so, ̅ ( )( ) ≥ ̅ ( )( ) Hence, ∈ ( ). Now let ∈ ( ) for every ∈ , ∈ then here we prove that ∈ ( ). As ( ̅ , ) is a possibility fuzzy soft right ideal over Z here ∈ Similarly, the proposition can easily be proved for left ideals following the above proof.  Table 7 and ordered relation.  Proof. Let an ordered semigroup Z is soft simple Assume that ( ̅ , ) be a possibility fuzzy soft interior ideal over Z and , ∈ . As Z is a simple and ∀ ∈ , = ( ], then 1 2 z az b ≤ for any , ∈ . As ( ̅ , ) is a possibility fuzzy soft interior ideal over Z, Thus, we can write, Proof. Let for an intra-regular ordered semigroup Z, suppose ( ̅ , ) is a possibility fuzzy soft ideal then for any ∈ there exist , ∈ such that ≤ . As ( ̅ , ) is a possibility fuzzy soft ideal Thus, we can write,  Table 8 and ordered relation respectively.  Hence, ( ̅ , ) is a semiprime possibility fuzzy soft ideal over Z.

Conclusions
This article was established to investigate a new subsystem named as possibility fuzzy soft ordered semigroups. An application to possibility fuzzy soft ordered semigroups in decision-making has been explained. As this theory relates two types of membership values that precisely characterize all the elements along with the suitable ordered semigroup relations, this leads to the correct and more appropriate choice by experts in decision-making. The presented work has introduced the novel concept of possibility fuzzy soft interior ideals and their relation with possibility fuzzy soft left (resp. right) ideals. Various algebraic relations related with the proposed theory have been investigated. Different applications of algebraic structures in applied sciences is a new direction of entrust for the researcher these days. This research can also be applied for solving such problems of applied sciences. This research will also lead: (1) To initiate new notions including possibility fuzzy soft bi-ideals, possibility fuzzy soft generalized bi-ideals, possibility fuzzy soft quasi ideals, and so on. Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.