Exploring Hybrid H-bi-Ideals in Hemirings: Characterizations and Applications in Decision Making

: The concept of the hybrid structure, as an extension of both soft sets and fuzzy sets, has gained signiﬁcant attention in various mathematical and decision-making domains. In this paper, we delve into the realm of hemirings and investigate the properties of hybrid h-bi-ideals, including prime, strongly prime, semiprime, irreducible, and strongly irreducible ones. By employing these hybrid h-bi-ideals, we provide insightful characterizations of h-hemiregular and h-intra-hemiregular hemirings, offering a deeper understanding of their algebraic structures. Beyond theoretical implications, we demonstrate the practical value of hybrid structures and decision-making theory in handling real-world problems under imprecise environments. Using the proposed decision-making algorithm based on hybrid structures, we have successfully addressed a signiﬁcant real-world problem, showcasing the efﬁcacy of this approach in providing robust solutions.


Introduction
L. A. Zadeh in 1965 [1] initiated the concept of fuzzy sets, the best framework for addressing uncertainties and imprecise information. Fuzzy set is defined by its membership function, whose values are defined on the closed interval [0,1]. This approach extends the generalized theory of uncertainty described in [2] to a broader context. Numerous works [3][4][5] are based on the idea of fuzzy set theory, its extensions, and applications.
Modelling uncertain data is a challenge for researchers in a variety of disciplines, including economics, engineering, environmental science, sociology, and medical science. Classical methods might not always be adequate to deal with uncertainties appearing in these domains. Although mathematical methods like rough sets [6], fuzzy sets [1], and other mathematical tools [7] are frequently employed to express uncertainty, Molodtsov [8] highlighted that each has its own challenges. Molodtsov offered a novel method to model ambiguity and uncertainty as a result [8]. Many researchers contributed to extending soft sets with fuzzy set theory [9][10][11]. Fuzzy soft sets are an extension introduced by Maji et al. [12] and provide a more flexible approach to handling uncertainty. Since then there has been a rapid growth of interest in soft sets and their various applications to algebraic systems [13][14][15][16][17][18][19][20], data analysis [21], and decision making under uncertainty [22][23][24][25][26].
Hemirings are thought of as a generalization of rings and refer to an additively commutative semiring with zero members. Ideals of hemirings play a significant part

Preliminaries
We give a concise overview of the fundamental ideas and concepts employed in hemirings.
A nonempty set N with " " and " " as binary operations on N is said to be a semiring if (N, ) and (N, ) are semigroups and the following laws A member 0 of a semiring is said to be zero if and only if all of its members satisfy the conditions 0 ϕ P = ϕ P 0 = ϕ P and 0 ϕ P = ϕ P 0 = 0. Hemirings are semirings (N, , ) that contain zero members and are commutative with regard to addition " ".
The sum and product of k and R where ∅ = k ⊆ N and ∅ = R ⊆ N are provided in a hemiring (N, , ) by k R = {ϕ P ϕ Q : ϕ P ∈ k and ϕ Q ∈ R} kR = {ϕ P ϕ Q : ϕ P ∈ k and ϕ Q ∈ R}.
When a subset Q of a hemiring N is closed under addition and multiplication while QNQ ⊆ Q, the subset is said to be a bi-ideal.
Let ∅ = Q ⊆ N, the set Q = {φ P ∈ N : φ P ϕ L φ R = ϕ M φ R for some ϕ L , ϕ M ∈ Q, φ R ∈ N} is referred to as h-closure of Q.
For a bi-ideal Q of a hemiring N, if φ P , φ R ∈ N, ϕ L , ϕ M ∈ Q and φ P ϕ L φ R = ϕ M φ R implies φ P ∈ Q then Q is said to be an h-bi-ideal (H-BI) of N.
An ideal Q in N satisfying Q = Q 2 is referred to as an h-idempotent ideal of a hemiring N. Proposition 1 ([33]). If k and R are the H-BI of a hemiring N, following that kR is an H-BI of N. Definition 1 ([33]). If kR ⊆ Q (k 2 ⊆ Q) implies k ⊆ Q or R ⊆ Q (k ⊆ Q ) for all H-BI k and R of N, following that H-BI Q of a hemiring N is known as a prime (semiprime) H-BI of N. Definition 2 ([33]). If kR ∩ Rk ⊆ Q indicating k ⊆ Q or R ⊆ Q for all H-BI k and R of a hemiring N then the H-BI Q of N is said to be strongly prime.
ϕ M j φ 2 P ϕ N j φ R , then a hemiring N is said to be h-intra-hemiregular (H-IHemiR) for each φ P ∈ N.
By a fuzzy subset L of a non-empty set N, we mean a mapping L : N −→ [0, 1], from a non-empty set N within [0, 1] unit interval. If L and F are fuzzy subsets of N then the fuzzy subsets L F andL F are defined as: The term "soft set" (£, ) over C is a mapping of £ into the set of all subsets of C i.e., L : −→ ℘(C), where C is the initial universe set, is a collection of attributes that the entities in C hold, and ℘(C) is the power set of C.

Definition 3 ([34]).
A hybrid structure (ḦyS) in a set of parameters Q over an initial universe set C is defined as: where, ξ S 1z : Q −→ ℘(C) and η F 1z : Q −→ I are mappings, ℘(C) represents the set of all subsets of C and I = [0, 1].

Definition 4.
Let us represent the set of allḦyS in Q over C by H(Q). Here, we define an order ⊆ in H(Q): The hybrid intersection of two hybrid structures Z1 and Z2 in Q over C is defined as: Definition 5. The hybrid union of two hybrid structures Z1 and Z2 in Q over C is defined as: The hybrid framework described by is called identity hybrid mapping in Q over C.
where, the symbols for supremum and infimum are and ∏, respectively. Definition 7. Let Z1 and Z2 be twoḦyS in a hemiring N over C.The hybrid h-product Z1 Z2 is defined as 1z H ξ S 2z and η F 1z H η F 2z are given as:

Prime Hybrid H-bi-Ideals
In this section, the concept of prime, strongly prime, semiprime, irreducible and strongly irreducibleḦyH-BI are provided with examples. The characterization of H-HemiR and H-IHemiR hemirings by theseḦyH-BI is also discussed.
Example: Suppose that there are five houses in the initial universe set C given by C = {H 1 , H 2 , H 3 , H 4 , H 5 }. Let a set of parameters N = {ϕ A ,ϕ B ,ϕ C ,ϕ D } be a set of status of houses in which ϕ A stands for the parameter "beautiful", ϕ B stands for the parameter "cheap", ϕ C stands for the parameter "in good location", ϕ D stands for the parameter "in green surrounding". We define the binary operation and on N by the Cayley table  in Table 1.

Definition 12.
A semiprime hybrid h-bi-ideal (briefly SPḦyH-BI) is defined to be aḦyH-BI Z1 in N over C satisfying Z2 Z2 ⊆Z1 means Z2 ⊆Z1 for every singleḦyH-BI Z2 of N upon C (see the related example in Appendix A i.e., Example A2).
Further, we demonstrate that the hybrid h-product of any twoḦyH-BI of N over C is similar toḦyH-BI in the following proposition.
Proof. Let Z1 and Z2 be anyḦyH-BI of N over C and ϕ A , ϕ B ∈ N. Then Also, Consequently, Z1 Z2 is likewise aḦyH-BI of N over C.
The intersection of any collection ofḦyH-BI of N over C is shown to be aḦyH-BI in the ensuing Lemma 1.
The result given below tells us that the intersection of a family of prime hybrid h-biideals in a hemring N over C is semiprime.

Lemma 2.
If Z1 is a member of the family of PḦyH-BI of N over C and subsequently ∩ ι∈ζ (Z1) ι is a SPḦyH-BI of N over C.
Proof. Assume that Z1 is an element of a collection {(Z1) ι : ι ∈ ζ} of PḦyH-BI of N over C, Lemma 1 states that ∩ ι∈ζ (Z1) ι is aḦyH-BI of N over C. Consequently, we acquire that We demonstrate that every strongly irreducible semiprimeḦyH-BI of N upon C is a StPḦyH-BI in the paragraph that follows.

Hemirings in Which Each Hybrid H-bi-Ideal Is Strongly Prime
The hemirings in which eachḦyH-BI is semiprime are studied in this part of the paper. Furthermore, we talk about the hemirings in which eachḦyH-BI is strongly prime.
The following theorem investigates the H-HemiR and H-IHemiR hemirings for which eachḦyH-BI is semiprime.

Theorem 2. The following statements have similar results in N:
(1) N is H-HemiR and H-IHemiR hemiring at the same time.
In the following Theorem, it is shown that everyḦyH-BI of a totally ordered, H-HemiR and H-IHemiR hemiring N is strongly prime. Proof. Let Z1, Z2 and Z3 be anyḦyH-BI of N over C arranged in the way (Z1 Z2) ∩(Z2 Z1) ⊆Z3. Consider the case where N is H-HemiR and H-IHemiR and its elements are in total order. According to Theorem 2, ⊆Z3 or Z2 ⊆Z3 concludes as a result of assumption. Hence Z3 is StPḦyH-BI of N over C.
(1) Set ofḦyH-BI of N over C satisfies total order (TO).

Proposed Hybrid Structure-Based Algorithm in Decision Making
This section introduces a novel algorithm utilizing hybrid structures to handle uncertain information in real-world decision-making scenarios. By showcasing its practical application, we aim to demonstrate the algorithm's effectiveness and versatility, offering valuable insights for enhancing decision-making processes amid complex data uncertainties.

Tabular Representation of Hybrid Structure
Here, we describe the general tabular form before proposing a hybrid structurebased algorithm.
Let Z1 = ξ S 1z , η F 1z be anyḦyS in a hemiring N over an initial universal set C. Then, Z1 is presented as: where ξ S 1z : N −→ ℘(C) and η F 1z : N −→ I are mappings, ℘(C) represents the set of all subsets of C and I = [0, 1].

Proposed Algorithm
In this section, we propose a hybrid structure-based algorithm (Algorithm 1). The proposed algorithm combines both soft sets and fuzzy sets to effectively handle uncertainties and imprecision in a variety of domains, resulting in a more stable and flexible modelling framework. The proposed algorithm consists of the following steps:

Algorithm 1 Hybrid structure-based algorithm
Step 1. Input the hybrid structure in tabular form (defined in Section 5.1).
Step 2. Represent the tabular form of the soft set ξ S 1z : N −→ ℘(C) as given in [9] and fuzzy set η F 1z : N −→ ℘(C) in separate tables.
Step 3. Construct the priority Step 4. Construct the comparison table (CT) according to [42]. This can be achieved by finding the entries as differences of each row sum in priority table with those of all other rows.
Step 5. Find the row sum of each row in the comparison table to obtain the score.
Step 6. Finally, the highest score is chosen.
The flowchart in Figure 1 summarizes the step-by-step process of the proposed algorithm and logical flow towards achieving its objectives.

Example
To empirically evaluate the effectiveness of our proposed algorithm, we give an example of the selection of a school for a cochlear-implanted child from a real-life scenario. The decision of selecting a school for a child with a cochlear implant is a critical and common decision that parents of such children often face.
A cochlear implant is an electronic device that can provide partial hearing to individuals with severe to profound hearing loss. Advances in early identification, implant technology, and early intensive therapy have enabled the implanted child to study in mainstream schools. Visual distraction, background noise, or any other environmental sounds may interfere with the understanding speech for a child with an implant. So, the decision of the selection of a school for an implanted child is based on the individual needs of the child, their capacity to learn in a spoken language environment, the environment of the school, and cooperation of the teaching staff.
Suppose Mr and Mrs Ali are in search of a school for their implanted child. To complete this task, the parents visit some schools and collect the required information about different schools in their area. They choose the five schools, C = {S 1 , S 2 , S 3 , S 4 , S 5 } namely, "Beacon House" (S 1 ) " The city school" (S 2 ) "Pak-American" (S 3 ), "Educators"(S 4 ), and "Superior Montessori" (S 5 ) who are willing to give admission to the child. The parents configure five attributes N = {ϕ A , ϕ B , ϕ C , ϕ D , ϕ E }, where "access" (ϕ A ),"environment" (ϕ B ), "learning" (ϕ C ), "staff cooperation" (ϕ D ), and "no of students in class" (ϕ E ) as a set of parameters which they think are crucial for making the best option and ensuring their child's adequate education. A hierarchical structure is shown in Figure 2, presenting the five selection criteria (i.e., access, environment, learning, staff cooperation, and the number of pupils in the class). Based on these criteria, the family wants to select the best school. The following hybrid structure illustrates the information of the schools based on the chosen criteria.
We define the binary operation and on N by the Cayley table in Table 6. Table 6. Cayley tableof the binary operations and .
Then (N, , ) is a hemiring. For the implementation of our proposed Algorithm 1, the following steps are used.
Step 1: Based on the hybrid structure, all the possible values are estimated for the attributes, given in Table 7.
Step 2. Construct seperate tables for the soft set ξ S 1z and fuzzy set η F 1z .
Step 3. By multiplying the corresponding values of ξ S 1z in Table 8 and η F 1z in Table 9, Table 10 computes the priority table. Table 8. Tabular representation of ξ S 1z : N −→ ℘(C).   Step 4. In Table 11, each attribute is obtained as the difference of row sum with the all the other rows.
Step 5. Calculate the sum of each row in the comparison table to obtain the score of schools as shown in Table 12.
Step 6. From the Table 12, we can see that alternative S 1 is the best selection. Scores of alternatives for the selection of best school for implanted child are shown in Figure 3. In Figure 3, the bar chart serves as a visual representation of the alternative values derived from the evaluation criteria outlined in Table 12. The chart allows for a clear and concise comparison of the different schools (S 1 , S 2 , S 3 , S 4 , and S 5 ) based on their respective scores.
At the top of the bar chart, we can observe that Beacon House School (S 1 ) stands out with the highest score of 3.7 among all the schools. This score reflects its excellent performance in the evaluation criteria, indicating that it outperformed the other schools in the assessment.
The City School (S 2 ) follows closely behind Beacon House School, securing the secondhighest score. This suggests that The City School is also a strong competitor and performed admirably in the comparison.
However, the alternatives at the bottom of the bar chart, namely The Pak-American School (S 3 ), The Educators School (S 4 ), and The Superior Montessori School (S 5 ), obtained relatively lower scores. These scores indicate that these schools had comparatively lesser acceptability or performance based on the assessment criteria.
The bar chart, as an essential part of the decision-making process, provides a visual tool to discern the varying levels of performance among the alternatives. It simplifies the comparison, allowing decision-makers to identify the top-performing school (Beacon House School) and observe the relative positions of the other schools in terms of their scores.
By incorporating a bar chart into the decision-making process, the evaluation becomes more intuitive and accessible. Decision-makers can make well-informed choices by considering the graphical representation of the schools' performance, facilitating a clearer understanding of their respective strengths and weaknesses based on the evaluation criteria.
The visual comparison aids in selecting the most appropriate school that aligns with the decision-maker's preferences and requirements.
As compared to Asmat et al. [40,41] our proposed method has assessed the hybrid structure in hemirings and investigated several properties of hybrid h-bi-ideals. The proposed prime hybrid h-bi-ideals are an extension of hybrid h-bi-ideals and we have investigated various aspects of hemirings. Also, this work has conducted a characterization of certain classes of hemirings based on these prime hybrid h-bi-ideals. Furthermore, we have utilized the hybrid structure in the decision-making process with the help of examples from real-world situations to explore new directions in algebraic development and tackle practical problems with improved uncertainty-handling abilities by utilizing hybrid structures in hemirings. Furthermore, our proposed method distinguishes itself from the existing literature by adopting a hybrid structure-based model rather than focusing solely on algebraic structures like BCK/BCI algebras and semigroups, as seen in previous studies [35][36][37][38][39].

Conclusions
In summary, the paper highlights the significance of hybrid structures in mathematical and decision-making domains. This study focuses on investigating the properties of hybrid h-bi-ideals within the context of hemirings. These hybrid h-bi-ideals include prime, strongly prime, semiprime, irreducible, and strongly irreducible. By employing the hybrid h-bi-ideals, the paper provides insightful characterizations of h-hemiregular and h-intra-hemiregular hemirings. This analysis contributes to a deeper understanding of the algebraic structures associated with these types of hemirings. To this end, we present a decision-making algorithm based on hybrid structures that has been successfully applied to solve a significant real-world problem. This showcases the effectiveness of the proposed approach in providing robust solutions in situations involving imprecise or uncertain data. The practical utility of the findings is demonstrated through the successful application of the proposed decision-making algorithm to solve real-world problems under imprecise conditions.
The proposed hybrid structure-based model empowers decision-makers in complex situations with uncertainty. It helps them understand uncertainties comprehensively and make effective choices in diverse scenarios. By using both crisp and fuzzy information, it improves decision outcomes in various domains. Also, the proposed algorithm considers both quantitative and qualitative information, enhancing the decision-making process and reducing risks.
On the contrary to this, defining membership functions for fuzzy and soft sets requires extensive domain knowledge, and interpreting dual representation demands specialized expertise, making implementation and maintenance of the approach more challenging. Additionally, gathering precise and accurate data, especially for subjective or qualitative information, can be difficult. The effectiveness of the proposed hybrid structures relies on sufficient and reliable data; in situations with scarce or unreliable data, their accuracy and effectiveness may be compromised.
However, in the future, the applicability of hybrid structures may be assessed in different algebraic structures including rings, semirings, and lattices, acquiring insightful knowledge into their adaptability and efficiency. To handle complex situational decisions with multiple objectives and criteria more successfully, one could combine these hybrid structures with multi-criteria decision-making methodologies. Furthermore, comparative studies between the existing models and hybrid structures could be considered to understand their respective strengths and limitations in various decision-making scenarios. Appendix A Example A1. Suppose that there are six houses in the initial universe set C given by C = {H 1 , H 2 , H 3 , H 4 , H 5 , H 6 }. Let a set of parameters N = {ϕ A ,ϕ B ,ϕ C ,ϕ D } be a set of status of houses in which ϕ A stands for the parameter "beautiful", ϕ B stands for the parameter "cheap", ϕ C stands for the parameter "in good location", ϕ D stands for the parameter "in green surrounding". We define the binary operation and on N by the Cayley table in Table A1.
Example A2. Suppose that there are six houses in the initial universe set C given by C = {H 1 , H 2 , H 3 , H 4 , H 5 , H 6 }. Let a set of parameters N = {ϕ A ,ϕ B ,ϕ C ,ϕ D } be a set of status of houses in which ϕ A stands for the parameter "beautiful", ϕ B stands for the parameter "cheap", ϕ C stands for the parameter "in good location", ϕ D stands for the parameter "in green surrounding". We define the binary operation and on N by the Cayley table in Table A5.