1. 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.
- (2)
To derive the application of possibility fuzzy soft ordered semigroup in decision sciences.
- (3)
To introduce a concept of possibility fuzzy soft left (resp. right) ideals and possibility fuzzy soft interior ideals and establish a relationship between these types of ideals. This relationship is well explained via suitable examples.
- (4)
To investigate various classes of ordered semigroups including regular and intra-regular in terms of possibility fuzzy soft ideals and possibility fuzzy soft interior ideals.
- (5)
To present a new concept of possibility fuzzy soft simple ordered semigroups and characterize this class of ordered semigroups in terms of possibility fuzzy soft ideals and possibility fuzzy soft interior ideals.
- (6)
To introduce a novel concept of possibility fuzzy soft ordered semigroup in terms of semiprime ideals.
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.
2. Preliminaries
In this section, we recapitulate some basic definitions and lemmas that will further be used to derive some new and fundamental concepts.
For an ordered semigroup
Z, represents a structure
that satisfies the following conditions:
defines a semigroup,
is an ordered set and for all,
implies
or
[
38]. A non-empty subset
X of
is called a subsemigroup of
Z if
. 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
there exists
such that
or in other words
we have
and
,
. Let
there exists
then
Z is called left (resp. right) regular if
(resp.
) or in other words
we have
(resp.
) and (
,
(resp.
). If an ordered semigroup
Z is both left and right regular then it is known as completely regular.
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.
Definition 1. [12] Let, for a universal set U, there exist a set of parameters represented as N. If, thenis called a soft set, where f is a mapping from M to power set of U, i.e.,.
Definition 2. [34] Letbe a soft set over Z, thenis called a soft ordered semigroup if(is a subsemigroup of Z. Definition 3. [34] Ifandare two soft sets over U then their union is represented bywith the following conditions: - (1)
- (2)
Definition 4. [34] Ifandare two soft sets over U then their intersection is represented aswith the following conditions: - (1)
- (2)
Definition 5. [42] Leta pair of sets, where U and N represent a universal set and a set of parameters, respectively. If,whereconsists of all fuzzy subsets of U, thenis a fuzzy soft set. Definition 6. [43] A soft setover Z is a soft left (resp. right) ideal over Z if and only ifis a fuzzy soft left (resp. right) ideal over Z. Definition 7. [43] A soft setover Z is a soft left (resp. right) ideal over Z if and only ifis a fuzzy soft left (resp. right) ideal over Z. Definition 8. [43] A soft setover Z is a semiprime soft left (resp. right) ideal over Z if and only ifis a semiprime ideal over Z for all. Additionally, for eachand,.
Definition 9. [32] Letbe a universal set andbe the set of parameters, then the possibility fuzzy soft setis defined by a mapping(Is the fuzzy subset of N) for all, where. Here,anddefine the degree of the membership value and the possibility of the degree of the membership value, respectively. In generalized form the possibility fuzzy soft is represented by: The possibility fuzzy soft set is among the generalized techniques used to resolve decision-making problems.
Definition 10. [32] The union of any two possibility fuzzy soft setsandover U is represented aswhereand is defined as.
must satisfy the following relation: - (1)
- (2)
Definition 11. [32] The intersection of any two possibility fuzzy soft setsandover U is represented aswhereand is defined as,must satisfy the following relation: - (1)
- (2)
Definition 12. Letandbe the possibility fuzzy soft sets over U, then the AND operation is denoted asand is defined as,, where,such that:
- (1)
- (2)
4. 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.
Definition 15. Letbe a PFSS over Z then a possibility fuzzy soft setover Z is called a possibility fuzzy soft l-ideal (resp. r-ideal) ofrepresented as(resp. ()), if it follows:
- (1)
- (2)
,is a fuzzy soft left ideal (resp. right ideal) ofimplies(resp. ())
Ifis l-ideal and r-ideal of, thenis a possibility fuzzy soft ideal ofcan be further denoted as.
Example 2. Letbe an ordered semigroup as defined in Table 3 and ordered relation as represented by Figure 3: Letbe a possibility fuzzy soft set defined under, whereand. Let L be a set of parameters such thatand then definewhere As in Equation (7),henceis a fuzzy soft right ideal of. Similarly,impliesis a fuzzy soft left ideal of. Thus,and, henceis a Possibility fuzzy soft ideal of.
Theorem 6. Letbe a PFSS over Z, then for any two possibility fuzzy soft setsandover Z, wherethen following relations can be proved:
- 1.
Ifandthen
- 2.
Ifandthen
Proof. Intersection of any two possibility fuzzy soft sets is defined as , where then implies and so either or also hence is a PFSS over Z using Theorem 3 Since implies as .
Similarly, we can prove the 2nd relation. □
Theorem 7. Letbe a PFSS over S, then for any two possibility fuzzy soft setsandover S, where, then following relations can be proved:
- 1.
Ifandthen.
- 2.
Ifandthen
Proof. The union of any two possibility fuzzy soft sets is defined
where
:
Since here so either or . If then where is a left ideal of . Thus, is also a left ideal of . Thus, . If then where is a left ideal of . Thus, is also a left ideal of . Thus, . Hence, we have .
Similarly, we can prove . □
Definition 16. Letbe a possibility fuzzy soft ordered semigroup of Z. Thenis said to be possibility fuzzy soft interior ideal of Z if:
- (1)
is fuzzy subsemigroup of Z
- (2)
- (2)
Example 3. Letis an ordered semigroup of Z as defined in Table 4 and ordered relation. Letbe a possibility fuzzy soft set of, wherethen we can define a mappingas:
Asare fuzzy subsemigroup of Z: Thus,is possibility fuzzy soft interior ideal of Z.
Definition 17. A possibility fuzzy soft setis called possibility fuzzy soft interior ideal over Z if and only ifis a possibility fuzzy soft interior ideal over Z.
Lemma 1. Every possibility fuzzy soft ideal of Z is also a possibility fuzzy soft interior ideal of Z.
Proof. The proof for Lemma 1 is straightforward and can easily be understand by Example 4. □
Example 4. Letis an ordered semigroup of Z as represented in Table 5 and ordered relation. Letbe a possibility fuzzy soft set defined under, whereand. Let N be a set of parameters and then define Ashenceis a fuzzy soft right ideal of Z. Similarly,impliesis a fuzzy soft left ideal of Z. Hence,is possibility fuzzy soft ideal of Z. Here. Thus,is also possibility fuzzy soft interior ideal of Z.
Hence, it is obvious that every possibility fuzzy soft ideal of Z is also a possibility fuzzy soft interior ideal of Z. But the converse is not true for all as elaborated in the following example.
Example 5. Letis an ordered semigroup of Z as represented in Example 4. Letbe a possibility fuzzy soft set defined under, whereand. Let N be a set of parameters and then define:
Thus, from Definition 16is possibility fuzzy soft interior ideal of Z. But the inequalitydoesn’t satisfy here for all the value of Z impliesis not a fuzzy soft left ideal of Z. Hence,is not possibility fuzzy soft ideal of Z.
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:
Thus, from the above two relations it can be proved that also for all and . Hence, is a possibility fuzzy soft ideal of Z. □
Corollary 1. For regular ordered semigroups, both the notion of possibility fuzzy soft ideal and possibility fuzzy soft interior ideals over Z must satisfy.
Theorem 8. Let for an ordered semigroup Z,and (be two possibility fuzzy soft left (resp. right) ideals of Z then for all,is also a possibility fuzzy soft left (resp. right) ideal of Z.
Proof. Let and be two possibility fuzzy soft left (resp. right) ideals of Z. By using Theorem 4, there exist and since and are both possibility fuzzy soft left (resp. right) ideal of Z their intersection is also a possibility fuzzy soft left (resp. right) ideal of Z. Thus, is a possibility fuzzy soft left (resp. right) ideal of Z. Accordingly, is a possibility fuzzy soft left (resp. right) ideal of Z. □
Theorem 9. Let for an ordered semigroup Z,and (be two possibility fuzzy soft left (resp. right) ideals of Z then for all,is also a possibility fuzzy soft left(resp. right) ideal of Z.
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.
Letbe 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. Let the set of parameters N includes the qualities list that CEO is looking for.
Hereis a set of parameters, whereis qualification;is behavior; and n3 is achievements.
Then we can defineas:
Hererepresents the membership values and possible membership values attain by the internees and given by the CEO.
Hence,, alsois a fuzzy subsemigroup of Z. Thus,is a possibility fuzzy soft ordered semigroup of Z. Additionally, noticing, we clearly concluded that the third internee attains maximum scores for all the parameters with respect to ordered semigroup relation defined on the base of their regularity. Alsoimpliesis a possibility fuzzy soft right ideal of Z. However,does not satisfy here for all the value of Z. Thus, this impliesis not a fuzzy soft left ideal of Z. Similarly,,does not satisfy here for all values of Z Thus, this impliesis not a possibility fuzzy soft interior ideal of Z. Possibility fuzzy soft ordered semigroups help in providing more precise and accurate results that help in solving decision sciences problems.
5. 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, .
Proposition 2. Let Z be an ordered semigroup andbe a possibility fuzzy soft right ideal over Z, thenis a soft right ideal of Z.
Proof. Let
be a possibility fuzzy soft right ideal over
Z and if
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
:
Hence, such that , so is a soft right ideal of Z.
Similarly, the proposition can easily be proved for left ideals following the above proof. □
Theorem 10. Let Z be an ordered semigroup andbe a possibility fuzzy soft ideal over Z, thenis a soft ideal of Z, for all.
Definition 18. Let Z be an ordered semigroup andbe a possibility fuzzy soft ordered semigroup over Z then Z is called possibility fuzzy left (resp. right) simple if and only if it is possibility fuzzy soft left (resp. right) idealover Z withfor alland.
An ordered semigroup Z is called possibility fuzzy soft simple if it is both possibility fuzzy soft left and right simple.
Example 7. Letis an ordered semigroup of Z as represented in Table 7 and ordered relation. Let a possibility fuzzy soft setis defined by a mapping. Where N is a set of parameters and then(n) (),where: Asare fuzzy subsemigroups of Z, thus,is a possibility fuzzy soft left (resp. right) ideal of Z. Additionally,for alland. Hence,is a possibility fuzzy soft simple ideal of Z.
Theorem 11. An ordered semigroup Z is called soft simple if it is possibility fuzzy soft simple.
Proof. Let be a possibility fuzzy soft ideal over Z and let then by Theorem 10 is a soft ideal of Z. As Z is a soft simple, it means that and . We have similarly, , hence . Thus, Z is a possibility fuzzy soft simple.
Conversely, let be a soft ideal over Z such that . Since is a soft ideal over Z, then it is obvious that is a soft ideal over Z. By Definition 6, every soft ideal is also a fuzzy soft ideal hence is a fuzzy ideal of Z also is fuzzy ideal of Z, implies is possibility fuzzy ideal of Z. Since Z is fuzzy simple Thus, must be a constant function that is for all , Thus, for any we can write and so thus, therefore, , which is a contradiction, so Z is a soft simple. □
Theorem 12. An ordered semigroup Z is soft simple if and only if for every possibility fuzzy soft interior idealover Z,for alland.
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 for any . As is a possibility fuzzy soft interior ideal over Z, Thus, we can write, as implies , Hence, . Following the same steps we can prove . Thus, .
Conversely, be a possibility fuzzy soft interior ideal over Z with , then by Proposition 1 is also a possibility fuzzy soft ideal over Z. Thus, by Theorem 11, Z is a possibility fuzzy soft simple with . Hence, Z is a soft simple. □
Theorem 13. An ordered semigroup Z is intra-regular if for every possibility fuzzy soft idealover Z,for alland.
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, as implies ( is a possibility fuzzy soft left ideal). Also , is a possibility fuzzy soft right ideal. Thus, } and, hence, . □
6. Semiprime Possibility Fuzzy Soft Ideals
Let Z be an ordered semigroup, a possibility fuzzy soft ideal is called a semiprime possibility fuzzy soft ideal over Z if and only if is a semiprime possibility fuzzy ideal of Z. That is for all and , .
Lemma 2. Let Z be an ordered semigroup then a soft setover Z is a semiprime soft ideal over Z if and only ifis a semiprime possibility fuzzy soft ideal over Z.
Proof. Let is a semiprime soft ideal over Z. By Definition 8 is also a semiprime soft ideal over Z then suppose such that also for possibility membership value concluding = 1, . Since is a semiprime ideal so then . Contrary, suppose , then . Hence, is a semiprime possibility fuzzy soft ideal over Z. Thus, is a semiprime possibility fuzzy soft ideal over Z.
Conversely, let is a semiprime possibility fuzzy soft ideal over Z. Let for all then = 1. As = 1 concluded that = 1, thus, . Hence, is a semiprime soft ideal over Z implies is also a semiprime soft ideal over Z and so does , following Definition 8. □
Theorem 14. An ordered semigroup Z is left (resp. right) regular if every possibility fuzzy soft left (resp. right) ideal over Z is semiprime.
Proof. Let Z be a left regular ordered semigroup and is a possibility fuzzy soft left ideal over Z. As Z is left regular, by definition such that Then we have . Additionally, is a possibility fuzzy soft left ideal, hence, is a semiprime possibility fuzzy soft left ideal over Z. Similarly, we can prove for right regular ordered semigroups as well. □
Theorem 15. An ordered semigroup Z is intra-regular if every possibility fuzzy soft left (resp. right) ideal over Z is semiprime.
Proof. The proof directly follows the Theorem 14. □
Example 8. Letbe an ordered semigroup defined in Table 8 and ordered relation respectively.Then possibility fuzzy soft setfor parametersis defined by mapping Asis non-empty alsoare all fuzzy subsemigroup of Z then as per the Definition 13,is possibility fuzzy soft ordered semigroup over Z. Asimpliesis a fuzzy soft left ideal of Z. Hence,is possibility fuzzy soft left ideal of Z.is also semiprime possibility fuzzy ideal of Z as per the Definition 8, along with the conditionfor alland.
Hence,is a semiprime possibility fuzzy soft ideal over Z.