An efficient approach to study multi-polar fuzzy ideals of semirings

The multi polar fuzzy (m-PF) set has an extensive range of implementations in real world problems related to the multi-polar information, multi-index and multi-attributes data. This paper introduces innovative extensions to algebraic structures. We present the definitions and some important results of m-polar fuzzy subsemirings (m-PFSSs), m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings. The main contributions of the paper include the derivation and proof of key theorems that shed light on the algebraic interplay and computational aspects of m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings along with examples. Moreover, this paper deals with several important properties of m-PFIs and characterizes regular and intra-regular semirings by the properties of these ideals.

The theory of fuzzy set is used in many fields including medical diagnosis, artificial intelligence, computer networks and decision making problems 1 .A membership function with a range of [0, 1] is used to illustrate a fuzzy set 2 .Zadeh introduced the concept of fuzzy set in his paper 3 .Rosenfeld applied this theory on groups 4 .Throughout, the history of fuzzy set, there are many types of fuzzy set extensions [5][6][7][8] , for example interval-valued fuzzy sets, vague sets etc.To differentiate the irrelevant and contrary elements in a fuzzy set, Zhang 9 came up with the term BF set on the basis of this consideration.For illustration, profit and loss, effects and side effects of medicines, both are two-sided aspects of a situation.Lee 10 introduced the idea of bipolar fuzzy ideals.The BF set is in fact an extension of a fuzzy set with a membership degree range [−1, 1] .Many researchers have done various works on BF sets 11 .
The BFS and the 2-polar fuzzy set have a natural one-to-one correspondence.Here, the degree of memberships will be positive for each element 12 .The BFS and 2-polar fuzzy set are cryptomorphic concepts, and one of them can be obtained concisely.For more applications, see [13][14][15] .Let K = {1, 2, 3, . . .m} is the set of contexts.Then, for each k ∈ K , the satisfaction degree of an element will be signified by an m-PF set with respect to kth context 16 .For example, the fuzzy set for the property "brilliant" has various interpretations among students of any class.
These sets can also be utilized as a model for clustering or classification and to define multi relation.An m -PF sets have lot of usage in decision making, co-operative games, diagnosis datum, etc.We will give an example to demonstrate it.
Let X = {v, w, x, y, z} be the set of 5 students.We shall grade them according to six qualities in the form of 6-PF subset given in Table 1.

Organization of the paper
This paper is organized as follows: In Section "Preliminaries", we give the fundamental definitions related to semiring and fuzzy set on semirings.Section "Characterization of semirings by m-polar fuzzy sets" is the main section in which m-PFSS, m-PFI, m-PFBI and m-PFQI of semirings are discussed in detail.In Section "Characterization of regular and intra-regular semirings by m-polar fuzzy ideals", we characterized regular and intraregular semirings by m-PFIs.In Section "Comparative study" we discuss the comparative study and in last we make conclusion as well as future plans.The list of acronyms is given in Table 2.

Preliminaries
This section includes simple but necessary definitions and preliminary results based on semirings that are important in their own right.These are prerequisite for later sections.A non-empty subset T of a semring (W, +, •) is stated as subsemiring of (W, +, •) if T itself is a semiring under the same operations.Throughout this research work, W will denote semiring unless otherwise specified.In this research work, subsets mean non- empty subsets.A subset T of W is stated as left ideal (LI) (resp.right ideal (RI)) of W if T is closed under + and WT ⊆ T(TW ⊆ T) .If T is both LI and RI of W then T is two-sided ideal or simply an ideal of W 33 .A subset is a mapping for all κ ∈ {1, 2, 3, . . ., m} .An m-PF power set of W , that is the set of all m-PF subsets of W is denoted by m(W) .We define relation on m(W) as follows: For any m-PF subsets Example 1 Consider a semiring W = {u, v, w} under the operations as given in Tables 3 and 4.
We d e f i n e 3 -P F s u b s e t s By definition, we have   (1) An m-PF subset ζ of W is called an m-PF two-sided ideal or an m-PFI of W if it is both m-PFLI and m-PFRI of W . Example 2 is of ideals of 3-PF subset.
Example 2 Consider a semiring W = {r, s, t} under the operations as given in Tables 5 and 6.
Lemma 1 Let T and U be subsets of W. Then the followings hold 31 . (1) Lemma 2 Let T be a subset of W. Then the following assertions are true.
(1) T is subsemiring of (2) Suppose that T is an LI of W .We have to show that C T (l + m) ≥ C T (l) ∧ C T (m) and C T (lm) ≥ C T (m) for all l, m ∈ W. We consider the following two cases. Case Similarly, we can prove for RI and two sided ideal.□ Then the following assertions are true.

Lemma 4
The following assertions are true in W. ( Other cases can be proved on the same lines.Hence, we have proved that the intersection of any two m-PFSSs (resp.m-PFIs) is also an m-PFSS (resp.m-PFI).Also, if an m-PF subset of W is m-PFSS (resp.m-PFIs) then its level set is subsemiring (resp.ideal) and vice versa.□

m-Polar fuzzy generalized bi-ideals in semirings
Now, we define the m-PFGBI of semirings.
Definition 5 An m-PF subset ζ of W is stated as m-PFGBI of W. If for all l, m, n ∈ W it satisfies the following condition: Hence Here, it is shown that if an m-PF subset of W is m-PFGBI then its level set is GBI and vice versa.□

m-Polar fuzzy bi-ideals in semirings
Here, we define m-PFBIs in semirings.
Definition 6 An m-PF subset ζ of W is stated as m-PFBI of W, if for all l, m, n ∈ W it satisfies the following condi- tions: Proof Similar to Lemmas 2 and 5. □ Proof The proofs of (1) and ( 2) are follows from the proof of Lemma 3, and proof of (3) follows from Lemma 6.□ Proof Follows from Proposition 2. □ Remark 1 Every m-PFBI of W is an m-PFGBI of W

m-Polar fuzzy quasi-ideal in semirings
Now, we define m-PFQI of semirings.
Lemma 9 Let T be a subset of W. Then T is QI of W if and only if C T is an m-PFQI of W.
Proof Let T be a QI of W , we have to show that (i) For the parts (i) see Lemma 2.
ii) For l ∈ W, consider the following two cases.Case 1: Conversely, suppose C T is an m-PFQI of W and we have to prove that T is a QI of W. Let l, m ∈ T. Then C T (l) = (1, 1, ..., 1) and C T (m) = (1, 1, ..., 1).
By def init ion ).This implies that l + m ∈ T. Again, let n ∈ TW ∩ WT.Then n = el and n = fm where l, m ∈ W and e, f ∈ T.
Since C T is m-PFQI of W . So, and n ∈ Wζ t .Also n = al and n = mb for some l, m ∈ W and a, b Conversely, on contrary suppose ζ is not m-PFQI of W. Suppose for l ∈ W, (( Hence, it is shown that if an m-PF subset of W is m-PFQI then its level set is QI and vice versa.□ Lemma 10 Every m-PF one-sided ideal of W is an m-PFQI of W.

Proof
The Proof Follows from Lemma 3. □ In Example 3, it is shown that the converse of Lemma 10 may not be true.
Example 3 Consider the semiring W = {0, a, 1} under the operations as given in Tables 7 and 8.
We define a 3 Then simple calculations show that ζ t is a QI of W. Therefore by using Proposition 3 Table 7. Table of addition.

Characterization of regular and intra-regular semirings by m-polar fuzzy ideals
In this section, regular and intra-regular semirings are characterized by m-PFIs.Some theorems are proved regarding regular and intra-regular semirings in terms of m-PFIs, m-PFQIs and m-PFBIs.
Definition 8 W is said to be regular if for all a ∈ W, there exist an element l ∈ W such that a = ala. 11.
Theorem 1 For semiring W, the following assertions are equivalent 30 : (1) W is regular; (2) I ∩ J = IJ for every RI I and LI J of W.
Theorem 2 Every m-PFQI of W is an m-PFBI of W.  www.nature.com/scientificreports/ (1) W is regular; ) be an m-PFGBI of W and a ∈ W. Since W is regular, so there exists an element l of W such that a = ala.So we have.□ Definition 9 W is stated as an intra-regular semiring if for each l ∈ W we get l ∈ Wl 2 W that is l can be written as l = n i=1 a i l 2 b i for some a i , b i ∈ W 10 .

Theorem 7
The following assertions are equivalent for W.
Proof (1) ⇒ (2) Let W be an intra-regular semiring and ζ , ξ be any m-PFLI and m-PFRI of W respec- tively.For l ∈ W there exist a i , b i ∈ W such that l = n i=1 a i l 2 b i for some a i , b i ∈ W. Hence we have (1) W is both regular and intra-regular; (2) T = TT for each BI T of W.

Theorem 9
The following assertions are equivalent for W.
(1) W is both regular and intra-regular; (

Comparative study
In this section we provide the comparative study between our newly generated results and the other results addressed by Bashir et al. 31,32 .While comparing we would like to emphasize on couple of points.An m-PF set in semirings is presented to overcome the restrictions involved in single-valued and two-valued fuzzification find the ideals of semigroup in terms of m-polar fuzzy ideals.However, we introduce the concept of m-polar fuzzy ideals in the structure of semirings.No doubt our provided results are efficient and more generalized because it tackles more complicated problems.Our methodology offers a broad variety of applications.

Conclusion
An m-PF (multi-polar fuzzy) set theory is a powerful mathematical tool for decision-making and handling uncertainty in real-world scenarios where data stems from m-factors (m > 2).Bashir et al. introduced m-polar fuzzy ideals of semigroup and multi-polar fuzzy ideals of ternary semigroup 31,32 .By extending the algebraic structure, we introduced m-polar fuzzy ideals of semiring with two binary operations "addition" and "multiplication".Within this framework, we have established significant findings relating to m-PF subsemirings, ideals, generalized bi-ideals, bi-ideals, and quasi-ideals within the context of semirings.Additionally, we have explored the characterizations of regular and intra-regular semirings in terms of m-PFIs.
In future, we will extend our research to m-polar Intuitionistic fuzzy set, m-polar Pythagorean fuzzy set, m-polar Picture fuzzy set and m-polar Spherical fuzzy set of many algebraic structures.An m-PF set has only membership degree for any situation but in real life there are many problems which are handled using nonmembership degree.For this reason, we will use more advanced techniques in future.Further, we will explore the m-PF ideals in hyperstructures, and investigate the concept of roughness as it applies to m-PF ideals of semirings.These foundations not only strengthen the mathematical framework but also cover the way for practical applications in various fields.
for all l ∈ W and κ ∈ {1, 2, 3, ..., m} 1 .Let ζ be fuzzy subset of W and t ∈ (0, 1].The set ζ t = {l ∈ W|ζ(l) ≥ t} is stated as level subset of ζ.Let ξ and ζ be fuzzy subsets of W . Define the fuzzy subset ξ • ζ of W by for all l ∈ W. Let ξ and ζ be fuzzy subsets of W . Define the fuzzy subset ξ + ζ of W by The next example shows the multiplication and addition of m-PF subsets.

Theorem 8
) ⇒ (1) Let T, U be LI and RI of W, respectively.Then C T and C U are m-PFLI and m-PFRI of W, respectively.Now, by the hypothesis,C T∩U = C T ∧ C U ≤ C T • C U = C TU .Thus T ∩ U ⊆ TU .So by Theorem 2, W is intra- regular.□The following assertions are equivalent for W 35 .
Table of qualities with their membership values.
Figure 1.Graphical representation of a 6-PF subset.

Table 2 .
List of acronyms.

Table 3 .
Table of addition.

Table 5 .
Table of addition.
Theorem 6 W is intra-regular if and only if I ∩ J ⊆ IJ for all RI I and LI J of W 35 .