A Contemporary Algebraic Attributes of $\mathfrak{m}$-Polar Q-Hesitant Fuzzy Sets in $\mathfrak{B}\mathbf{C}\mathbb{K}/\mathfrak{B}\mathbf{C}\mathbb{I}$ Algebras and Applications of Career Determination

Abstract

To systematically address the intricate multiple criteria decision-making (MCDM) challenges to practical situations where uncertain and hesitant information plays a critical role in guiding optimal choices. In this article, we introduce the concept of $\mathfrak{m}$-polar Q-hesitant fuzzy ($\mathcal{MPQHF}$) $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras, combining $\mathfrak{m}$-PFS theory with Q-hesitant fuzzy set theory in the framework of $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. This innovative approach enhances the attitudes of uncertainty, vagueness, and hesitance of data in decision-making processes. We investigate the features and actions of this proposed hybrid approach to fuzzy sets and hesitant fuzzy sets, focusing on $\mathcal{MPQHF}$ subalgebras, and explore the characteristics of several kinds of ideals under $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. It also showed that it can better represent complex levels of uncertainty than regular sets. The proposed method’s theoretical framework offers a better way to show uncertain data in areas like engineering, computer science, and computational mathematics. By linking theoretical advancements of $\mathcal{MPQHF}$ sets with practical applications, we highlight the benefits and challenges of this approach. Demonstrating the practical uses of the $\mathcal{MPQHF}$ sets aims to encourage broader adoption. Symmetry plays a vital role in algebraic structure and is used in various fields like decision-making, encryption, pattern recognition problems, and automata theory. Furthermore, this work enhances the understanding of algebraic structures and offers a robust tool for career exploration and development through improved decision-making methodologies.

1. Introduction

The algebraic structure relies significantly on symmetry, which is applied across various domains. Tung [1] used an algebraic approach to structuring patterns as extended symmetries. Rupe [2] generalized the accurate predictive regularity of symmetry groups and provided an algebraic theory of patterns based on a fundamental premise of future equivalence. In addition, the latter and its semigroup algebra extended the translation symmetry to partial and hidden symmetries. The algebra with arbitrary internal symmetry and local translations was set up by Corral [3] for a general gauge theory of gravity. Decision-making (DM) typically faces challenges associated with uncertain circumstances, which are particularly demanding due to the complex modeling and management of such uncertainties. Fenz and Guo [4] introduced FS set-based group DM and gave the frame related to group DM. Multiple criteria decision-making (MCDM) challenges exist in a variety of practical domains [5,6,7], and techniques are employed to collect decision data and build the framework to get at an acceptable solution to the challenges at hand. For specific purposes, these models make a significant contribution to precise DM, time savings, and cost reduction. We have used a number of theories and mechanisms to overcome these problems. Uncertainty, like probability, differs from probability in several aspects. Nevertheless, to hold such an assumption, when two or more roots of imprecision occur together, it results in vagueness and imprecision in the modeling process. To address the ambiguity of evaluative data, fuzzy sets introduced by Zadeh [8] marked a pivotal development in addressing the intricacies of uncertainty and vagueness within datasets and DM processes defined as $S=\{(o,{\nu }_P\left(o\right)),o\in S\},$ where U is the universal set and ${\nu }_S:S\to [0,1].$ The intuitionist FS, presented by Atanassov [9] as an extension of the FS, provides two-dimensional information of data by increasing the non-membership value known as IFS, defined as $S=\{(o,{\nu }_S\left(o\right),{\gamma }_S\left(o\right)),o\in S\}$.

Shabir et al. [10] explained the algebraic structure of q-rung ortho-pair fuzzy relations and showed how they could be used in real life in DM. Also presented a comparative analysis with current theories. Aguiló [11] showed several binary tools that can be used on well-known families of fuzzy implications. These tools also create lattice substructures in certain sets when the implication generators are limited. Zheng [12] presented a novel category of membership functions that integrates subjective perception with objective information via the Gaussian belonging function grounded on probability density. GPDMF established five operators and offered various examples to elucidate theoretical findings. In 1994, Zhang [13] introduced the bipolar FS, which is an extension of the FS with a degree of belonging range of [−1, 1]. For a bipolar FS, an element with a degree of belonging 0 is considered not relevant to the corresponding property. An entity with a degree of belonging (0, 1] is considered partially satisfied with the property, while an entity with a degree of belonging [−1, 0] is considered partially satisfied with the implicit counter-property. Multipolar information is critical in many real-world challenges, including those in technology and neurobiology. Chen et al. introduced a new concept called $\mathfrak{m}$-PFS theory [14], where the grade of the membership function is extended from the interval $[0,1]$ to $\mathfrak{m}$-polar intervals of $[0,1]$ or ${[0,1]}^m$. Additionally, it illustrates the diverse ideas established by bipolar fuzzy sets and their implementation in $\mathfrak{m}$-PFSs. It also gave examples of how $\mathfrak{m}$-PFSs can be used in the real world. The basic idea behind this strategy is that multipolar knowledge emerges from the possibility of encountering information and data related to real-world issues from multiple n agents ($n\ge 2$). This new approach inspired scholars to propose many innovative notions for $\mathfrak{m}$-PFSs and hybrid models, such as mathematics and artificial intelligence.

Molodtsov introduced an innovative extension of FS theory, which he called the soft set theory [15]. The soft set theory has been useful in many areas because it is naturally flexible and adaptable. It also provided a strong framework for quantifying and analyzing vague data, which led to more accurate and trustworthy assessments. This further enhances the characteristics of FS sets as delineated and examined in the research of prior scholars [16]. Application DM Furthermore, the HFS set is examined in [17]. ÂIseki and co-authors introduced the concept of $\mathfrak{B}\mathrm{C}\mathbb{I}/\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra. $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra is founded on two distinct theories: set theory and propositional calculus [18,19,20]. Xi [21] initially introduced fuzzy $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra by combining the fundamental concepts of FS and $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra. Ahmad [22] developed a strong connection between the concept of FSs and $\mathfrak{B}\mathrm{C}\mathbb{I}$ algebra. As a result, other scholars [23,24,25] conducted extensive studies on $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebra and ideals in the context of FSs. Masarwah et al. [26] talked about generalized $\mathfrak{m}$-polar fuzzy positive implicative ideals in the context of $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebras and provided important contributions to the field of algebra by doing this.

Hesitant FSs are an empirical extension of FSs introduced by Torra [27]. These sets were created to address the challenge of determining an element’s membership in a set, which can be complicated due to uncertainty among several valid values. For example, when two experts debate integrating x in A, one intends to allocate a value of 0.3 while the other recommends 0.4. Therefore, the degree of uncertainty about possible values is slightly reduced. IFS are not suitable for modeling this particular scenario. However, HFSs can be represented using multisets, which provide more accurate estimation in the DM process. Alsager and Alshehri introduced the concept of a single-valued neutrosophic $\mathcal{HF}$ rough set by coupling a single-valued neutrosophic HFS with a rough set. In information systems, the use of single-valued neutrosophic HFSs and rough sets is a highly effective approach for addressing uncertainty, granularity, and knowledge incompleteness. Moreover Song et al. [28] established HF set in $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{K}$ and investigated a multi-Q-dual $\mathcal{HF}$ soft rough model for DM, offering valuable approaches [29,30,31]. The investigation related to the Q-$\mathcal{HF}$ ideal in $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra expanded the theoretical framework [32]. Moreover, the concept of anti-fuzzy ideal in $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebras and soft $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras has been introduced, and fundamental properties are discussed [33]. The practical application of fuzzy soft set theory in $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras has also been explored, highlighting its versatility in various domains [34]. Furthermore, in the context of $\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras, research on fuzzy commutative ideals and the application of HFSs to filters in MTL algerbra have a wide range of the scope of algebraic concepts [35].

The following are the motivations of the novel approach and objectives:

The methods presented in previous literature fail to generate any meaningful information when the provided data consists of distinct numerical fuzzy values within their crisp domain. Decision-makers have developed various hybrid approaches to address different forms of uncertainties, data variability, and multiple-criteria DM challenges arising from data with numerous numerical fuzzy values:

1.

Torra proposed the concept of HFSs, which is an empirical extension of FSs [27]. Chen et al. [14] introduced an extension of bipolar FS known as $\mathfrak{m}$-PFS, where the grade of the membership function is extended from the interval $[0,1]$ to $\mathfrak{m}$-polar intervals of $[0,1]$ or ${[0,1]}^m$. Xi [21] initially introduced fuzzy $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra by combining the fundamental concepts of FS and $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra, and  Ahmad [22] developed a strong connection between the concept of FSs and $\mathfrak{B}\mathrm{C}\mathbb{I}$ algebra.

2.

The proposed hybrid method addresses limitations in existing theories:

• The ability to handle multi-dimensional uncertainty and hesitancy is limited.
• Complex real-world problems, such as career determination, lack adequate algebraic frameworks for combining m-polarity and Q-hesitancy in decision-making.
• Currently, there are insufficient tools available for modeling and analyzing uncertainty in structured algebraic systems such as $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$-algebras. This hybrid approach aims to bridge these gaps by providing a more comprehensive and flexible mathematical framework.

3.

To generate a more advanced hybrid approach known as $\mathfrak{m}$-Polar Q-HFSs, we coupled $\mathfrak{m}$-PFS and Q-HFS, then applied it to the $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebra and demonstrated the application of career selection to check its effectiveness and accuracy.

4.

Using this hybrid approach, $\mathfrak{m}$-Polar Q-HFSs, we discuss several fundamental results under the influence of the proposed model, $\mathfrak{m}$-Polar Q-HFSs, in $\mathfrak{B}\mathrm{C}\mathbb{K}$/$\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras.

5.

The proposed approach demonstrates its novelty by yielding more authentic and comparable results when evaluating alternatives with multi-polar membership set information in the application of career selection. In other words, this technique can effectively tackle problems and systems with multiple membership values. The suggested methodology demonstrates more compatibility compared with numerous modifications of the FS and several other DM theories. Figure 1 shows the proposed model of the hybrid approach, and

Table 1. Comparison of $\mathcal{MPQHF}$ with the existing approaches.

Sets	Researchers	Uper & Lower Approximation	Degree Membership	Degree of Non-Membership	Other Constraints
FSs	Zadeh [8]	×	✓	×	×
IFSs	Atanassov [9]	×	✓	✓	×
IVFS	Zadeh [36]	×	✓	×	Interval grading
BFS	Zhang [37]	×	×	×	Positive grading ${\mu }^{+}\in [0,1]$ and negative ${\mu }^{-}\in [-1,0]$
m-PFS	Juni et al. [14]	×	✓	×	Positive grading ${\mu }^{+}\in {[0,1]}^m$ and negative ${\mu }^{-}\in {[-1,0]}^m$
HFS	Jun [38]	×	✓	×	Elements have set of membership degrees
$\mathcal{MPQHF}$	Proposed Approach	✓	✓	✓	Hybrid model of m-PFS and HFS

shows the comparison with the existing approaches.

The organization of this research paper is as follows. In Section 2, we review some fundamental concepts and preliminary information about the proposed method. In Section 3, we discuss the $\mathcal{MPQHF}$ sets and examine their several fundamental properties. Moreover, we investigate specific variations, including $\mathcal{MPQHF}$ subalgebras, ideals, and numerical examples. In Section 4, practical applications are based on $\mathcal{MPQHF}$ sets in DM problems, and we demonstrate how these concepts find relevance in solving real-world decision-support challenges. As we conclude, our Conclusion section not only summarizes our findings but also underscores the broader significance of the $\mathcal{MPQHF}$ sets in the realm of mathematics and beyond.

2. Preliminaries

In this section, we discuss the fundamental definitions and related concepts of $\mathfrak{m}$-PFS, HFS, and Q-HFS that serve as building blocks for our research. These definitions provide the necessary foundation on which our work is based.

Definition 1.

An algebra $(\mathfrak{B};\ast ,0)$ of type $(2,0)$ is known as a $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra if it fulfills the specified criteria:

1. 

For all $v,\kappa ,\eta \in \mathfrak{B}:\left(v\kappa \right)\ast (v\ast \eta )\ast (\eta \ast \kappa )=0$.

2. 

For all $v,\kappa \in \mathfrak{B}:(v\ast (v\ast \kappa )\ast \kappa =0$.

3. 

For all $v\in \mathfrak{B}:v\ast v=0$.

4. 

For all $v,\kappa \in \mathfrak{B}:(v\ast \kappa =0)\wedge (\kappa \ast v=0).\to v=\kappa$.

5. 

For all $v\in \mathfrak{B}:0\ast v=0$.

• If $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra $\mathfrak{B}$ meets the following additional conditions:

1. 

For all $v\in \mathfrak{B}:v\ast 0=v$.

2. 

For all $v,\kappa ,\eta \in \mathfrak{B}:(v\le \kappa )\to (v\ast \eta \le \kappa \ast \eta )\wedge (\eta \ast \kappa \le \eta \ast v)$

3. 

For all $v,\kappa ,\eta \in \mathfrak{B}:(v\ast \kappa )\ast \eta =(v\ast \eta )\ast \kappa$

4. 

For all $v,\kappa ,\eta \in \mathfrak{B}:(v\ast \eta )\ast (\kappa \ast \eta )\le v\ast \kappa$

• Additionally, any $\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra $\mathfrak{B}$ fulfills the following condition:
• For all $v,\kappa ,\eta \in \mathfrak{B}:0\ast (0\ast ((v\ast \eta )\ast (\kappa \ast \eta )\left)\right)=(0\ast \kappa )\ast (0\ast v)$

Definition 2

([19]). A subalgebra of a $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra, denoted as $\mathfrak{S}$, is defined as a non-empty subset that conforms to the following equation:

$$v\ast \kappa \in \mathfrak{S}\mathit{for}\mathit{all}v,\kappa \in \mathfrak{S}$$

Definition 3

([19]). An ideal in a $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra $\mathfrak{B}$ is defined as a non-empty subset $\mathfrak{I}$ that meets the following condition:

• ID1:     $0\in \mathfrak{I}$
• ID2:   for all $v,\kappa \in \mathfrak{B}v\ast \kappa \in \mathfrak{I},\kappa \in \mathfrak{I}\to v\in \mathfrak{I}$

Definition 4

([38]). Assuming $\mathfrak{B}$ is a $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra, a HFS $\varrho =\{(v,{\varsigma }_{\varrho }\left(v\right))\mid v\in \mathfrak{B}\}$ defined on $\mathfrak{B}$ is labeled a $\mathcal{HF}$-subalgebra of $\mathfrak{B}$ if it adheres to the following criteria:

$$\forall v,\kappa \in \mathfrak{B},{\varsigma }_{\varrho }(v\ast \kappa )\supseteq {\varsigma }_{\varrho }\left(v\right)\cap {\varsigma }_{\varrho }\left(\kappa \right)$$

Definition 5

([38]). Suppose that $\mathfrak{B}$ is a $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra, a HFS $\varrho =\{(v,{\varsigma }_{\varrho }\left(v\right))\mid v\in \mathfrak{B}\}$ defined on $\mathfrak{B}$ is referred to as a $\mathcal{HF}$ ideal of $\mathfrak{B}$ if it fulfills the criteria:

$$\forall v,\kappa \in \mathfrak{B},({\varsigma }_{\varrho }(v\ast \kappa )\cap {\varsigma }_{\varrho }\left(\kappa \right))\subseteq {\varsigma }_{\varrho }\left(v\right)\subseteq {\varsigma }_{\varrho }\left(0\right)$$

Definition 6

([39]). Let a non-empty finite discourse set $\mathfrak{B}$ and a non-empty set Q. A Q-HFS, denoted as ${\varrho }_Q$, can be described as follows:

$${\varrho }_Q=\{((v,\mathfrak{q}),{\varsigma }_Q(v,\mathfrak{q}))\mid v\in \mathfrak{B},\mathfrak{q}\in Q\}\mathrm{where}{\varrho }_Q:\mathfrak{B}\times Q\to [0,1].$$

3. $\mathfrak{m}$-Polar $\mathbf{Q}$-Hesitant Fuzzy Sets

In this part, we develop the $\mathcal{MPQHF}$ sets and expand on traditional fuzzy and HFSs, offering a multidimensional way to deal with uncertainty and unwillingness. We study the core principles, properties, and modifications of subalgebras and ideals. Also, it aims to provide an overview of $\mathcal{MPQHF}$ sets and their applications in DM and mathematics. Furthermore, it examines $\mathcal{MPQHF}$ subalgebras, which are important in $\mathcal{MPQHF}$ set theory, to understand their fundamental properties and significance within this mathematical context.

Definition 7.

Assuming $\mathfrak{B}$ is a $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra, the $\mathcal{MPQHF}$ set can be described as follows:

$${\varrho }_i:=\{(v,\mathfrak{m}),{\varsigma }_{\varrho }^i(v,\mathfrak{m})\mid v\in \mathfrak{B},\mathfrak{m}\in Q\}$$

on $\mathfrak{B}$ is said to be an $\mathcal{MPQHF}$-subalgebra if it fulfills the condition: $\forall v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q$

$${\varsigma }_{{\varrho }_i}(v\ast \kappa ·\mathfrak{m})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{m})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{m})\forall i=1,2,...,m.$$

Example 1.

Suppose that $\mathfrak{B}=\{{\mathfrak{t}}_1,{\mathfrak{t}}_2,{\mathfrak{t}}_3\}$ is a $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra with a binary action "*", as shown in

Table 2. Binary operation table for $\mathfrak{B}$.

*	${\mathfrak{t}}_1$	${\mathfrak{t}}_2$	${\mathfrak{t}}_3$
${\mathfrak{t}}_1$	${\mathfrak{t}}_1$	${\mathfrak{t}}_1$	${\mathfrak{t}}_1$
${\mathfrak{t}}_2$	${\mathfrak{t}}_2$	${\mathfrak{t}}_1$	${\mathfrak{t}}_2$
${\mathfrak{t}}_3$	${\mathfrak{t}}_3$	${\mathfrak{t}}_3$	${\mathfrak{t}}_1$

below:

Define the set $Q=\{{\mathfrak{t}}_1,{\mathfrak{t}}_2,{\mathfrak{t}}_3,{\mathfrak{t}}_4\}$ and a 2-polar FS on $\mathfrak{B}$ as follows (

Table 3. 2-Polar FS on $\mathfrak{B}$.

$({\mathfrak{t}}_1,{\mathfrak{t}}_1)$	$\left\{\right\{0.9,0.8\},(0.9\left)\right\}$
$({\mathfrak{t}}_1,{\mathfrak{t}}_2)$	$\left\{\right(0.7,0.8),(0.8,0.8\left)\right\}$
$({\mathfrak{t}}_1,{\mathfrak{t}}_3)$	$\left\{\right(0.8,0.9,0.7),\{0.9\left\}\right\}$
$({\mathfrak{t}}_1,{\mathfrak{t}}_4)$	$\left\{\right(0.7,0.9),\{0.8\left\}\right\}$
$({\mathfrak{t}}_2,{\mathfrak{t}}_1)$	$\left\{\right(0.5,0.6),(0.3,0.2\left)\right\}$
$({\mathfrak{t}}_2,{\mathfrak{t}}_2)$	$\left\{\right(0.1,0.4),(0.5,0.3\left)\right\}$
$({\mathfrak{t}}_2,{\mathfrak{t}}_3)$	$\left\{\right(0.2,0.1),(0.5,0.1\left)\right\}$
$({\mathfrak{t}}_2,{\mathfrak{t}}_4)$	$\left\{\right(0.4,0.3,0.1),(0.5,0.1\left)\right\}$
$({\mathfrak{t}}_3,{\mathfrak{t}}_1)$	$\left\{\right\{0.7\},(0.8,0.7\left)\right\}$
$({\mathfrak{t}}_3,{\mathfrak{t}}_2)$	$\left\{\right(0.6,0.5),(0.7,0.6\left)\right\}$
$({\mathfrak{t}}_3,{\mathfrak{t}}_3)$	$\left\{\right(0.6,0.4,0.3),(0.8,0.7\left)\right\}$
$({\mathfrak{t}}_3,{\mathfrak{t}}_4)$	$\left\{\right(0.7,0.5),(0.7,0.6\left)\right\}$

):

Thus, ${\varsigma }_{{\varrho }_i}$ is an $\mathfrak{m}$-polar Q-$\mathcal{HF}$ subalgebra.

Proposition 1.

Every $\mathcal{MPQHF}$ subalgebra of $\mathfrak{B}$ meets the following: $\forall v\in \mathfrak{B},\mathfrak{q}\in Q$

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\subseteq {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\forall i=1,2,...,m$$

Proof. 

$\forall v\in \mathfrak{B},\mathfrak{q}\in Q$, we obtain the following:

$${\varsigma }_{{\varrho }_i}(0,\mathfrak{q})={\varsigma }_{{\varrho }_i}(v\ast v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})={\varsigma }_{{\varrho }_i}(v,\mathfrak{q})$$

This completes the proof.    □

Proposition 2.

If every $\mathcal{MPQHF}$ subalgebra of $\mathfrak{B}$ fulfills this inequality: $\forall v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q$

$$\begin{array}{ccc}\hfill {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})& \subseteq & {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\hfill \\ \hfill \mathit{then}, \mathit{we} \mathit{have} {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})& =& {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\forall i=1,2,...,m.\hfill \end{array}$$

Proof. 

By using the definition of the $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra, we obtain $\forall v\in \mathfrak{B},v\ast 0=v$. Then, $\forall v\in \mathfrak{B},\mathfrak{q}\in Q$, we obtain the following:

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})={\varsigma }_{{\varrho }_i}(v\ast 0,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})$$

   □

Proposition 1 implies that ${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})={\varsigma }_{{\varrho }_i}(0,\mathfrak{q})$ Now, investigate the $\mathcal{MPQHF}$ ideals, which have significant importance in $\mathcal{MPQHF}$ sets. We will study their properties and practical uses to see how they fit into the mathematical framework we are discussing.

Definition 8.

Let ${\varrho }_i=(v,\mathfrak{q}),\left({\varrho }_{\varsigma }^i(v,\mathfrak{q})\mid v\in ⨁,\mathfrak{q}\in Q\right\}.$ be a Q-HFS in $\mathfrak{B}$. Then, ${\varsigma }_{{\varrho }_i}$ is known as $\mathcal{MPQHF}$ ideal of $\mathfrak{B}$ if it meets the following axiom:

1. 

${\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})$.

2. 

For all $v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q,{\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})$ where $i=1,2\dots m$.

Example 2.

Let $\mathfrak{B}=\{s,l,m,n\}$ be a set with a binary action "*", which is described in the following

Table 4. Binary operation table for $\mathfrak{B}$.

*	s	l	m	n
s	m	m	s	s
l	l	m	l	l
m	m	m	m	m
n	n	n	n	m

:

Then, $\mathfrak{B}$ is a $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra. Define a set $Q=\{1,2,3\}$ and a 3-polar FS ${\varsigma }_{{\varrho }_i}$ on $\mathfrak{B}$ as follows

Table 5. Three -Polar FS ${\varsigma }_{{\varrho }_i}$ on $\mathfrak{B}$.

$(\mathit{s},1)$	$\left\{\right\{0.6\},(0.5,0.6),(0.3,0.4\left)\right\}$
$(s,2)$	$\left\{\right(0.5,0.6),(0.4),(0.4,0.5\left)\right\}$
$(s,3)$	$\left\{\right(0.4,0.5,0.4),\{0.4,0.3\},(0.6,0.7\left)\right\}$
$(l,1)$	$\left\{\right(0.3,0.2),(0.4,0.3),(0.2,0.1,0.2\left)\right\}$
$(l,2)$	$\left\{\right(0.4,0.3),(0.3),(0.2,0.1\left)\right\}$
$(l,3)$	$\left\{\right(0.1,0.3),(0.2),(0.5\left)\right\}$
$(m,1)$	$\left\{\right\{0.9\},(0.8,0.7),\{0.8\left\}\right\}$
$(m,2)$	$\left\{\right(0.9,0.8),[0.9],\{0.7,0.9\left\}\right\}$
$(m,3)$	$\left\{\right(0.8),[0.8,0.9],\{0.7,0.8\left\}\right)\}$
$(n,1)$	$\left\{\right\{0.7\},(0.7,0.6),(0.5\left)\right\}$
$(n,2)$	$\left\{\right(0.7,0.6,0.6),(0.5,0.6),(0.6\left)\right\}$
$(n,3)$	$\left\{\right(0.5,0.7),(0.5),[0.7\left]\right\}$

:

It is customary to confirm that ${\varsigma }_{{\varrho }_i}$ is a 3-polar Q-$\mathcal{HF}$ ideal.

Proposition 3.

All of $\mathcal{MPQHF}$ ideal over $\mathfrak{B}$ satisfies the following condition:   $v,\kappa \le \eta \to {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})$   $\forall v,\kappa ,\eta \in$$\mathfrak{B},\mathfrak{q}\in Q$

Proof. 

Let $\forall v,\kappa ,\eta \in \mathfrak{B},\mathfrak{q}\in Q$ such that $v,\kappa \le \eta$, then $(v\ast \kappa )\ast \eta =0$:

$$\begin{array}{cc}\hfill {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})& \supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\hfill \\ \hfill =& {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\hfill \end{array}$$

$$={\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})$$

It follows that

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\\ \supseteq {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\end{array}$$

This ends the proof.   □

Theorem 1.

In $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra $\mathfrak{B}$, every $\mathcal{MPQHF}$ ideal is $\mathcal{MPQHF}$ subalgebra.

Proof. 

Assume that $\mathfrak{q}\in Q$ and ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ ideal over $\mathfrak{B}$,

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast v,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}((v\ast v)\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(0\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\end{array}$$

for $v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q$ and for all $i=1,2\dots m$. Then, ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ subalgebra over $\mathfrak{B}$. This complete the proof.    □

Proposition 4.

Every $\mathcal{MPQHF}$ ideal over $\mathfrak{B}\mathrm{C}\mathbb{I}$ algebra $\mathfrak{B}$ fulfills the below inequality: $\forall v\in \mathfrak{B},\mathfrak{q}\in Q$

$${\varsigma }_{{\varrho }_i}(0\ast (0\ast v),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})$$

Proof. 

Let ${\varsigma }_{{\underline{Q}}_i}$ be an $\mathcal{MPQHF}$ ideal; therefore, for $\mathfrak{q}\in Q$ and $v\in \mathfrak{B}$

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(0\ast (0\ast v),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((0\ast (0\ast v))\ast v,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\forall i=1,2,...,m\end{array}$$

   □

Proposition 5.

Each $\mathcal{MPQHF}$ ideal meets the following axioms: $\forall v,\kappa ,\eta \in \mathfrak{B},\mathfrak{q}\in Q$.

1. 

If $v\le \kappa \to {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})$.

2. 

${\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta \ast \kappa ,\mathfrak{q})$.

3. 

If ${\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})={\varsigma }_{{\varrho }_i}(0,\mathfrak{q})$. Then,

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})$$

Proof. 

Assume that $\mathfrak{q}\in Q$ and $v,\kappa ,\eta \in \mathfrak{B}$

(1) If $v\le \kappa$, then $v\ast \kappa =0$ since ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ ideal of $\mathfrak{B}$

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\end{array}$$

(2) Since $(v\ast \kappa )\ast (v\ast \eta )⩽\eta \ast \kappa$ from (1) we have ${\varsigma }_{{\underline{Q}}_i}((v\ast \kappa )\ast (v\ast \eta ),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(\eta \ast \kappa ,\mathfrak{q})$. Hence,

$$\begin{array}{cc}\hfill {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})& \supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast (v\ast \eta ),\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\hfill \\ & \supseteq {\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta \ast \kappa ,\mathfrak{q})\hfill \end{array}$$

(3) If ${\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})={\varsigma }_{{\varrho }_i}(0,\mathfrak{q})$, then

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\end{array}$$

Exploring $\mathcal{MPQHF}$ commutative ideals, we will see how they are important in $\mathcal{MPQHF}$ sets and DM. Understanding them helps us grasp their importance in our math framework.    □

Definition 9.

An $\mathcal{MPQHF}$ set

$${\varrho }_i=\left\{\left((v,\mathfrak{q}),{\varsigma }_{\varrho }^i(v,\mathfrak{q})\right)\mid v\in \mathfrak{B},\mathfrak{q}\in Q\right\}$$

in $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra is said to be an $\mathcal{MPQHF}$ commutative ideal of $\mathfrak{B}$ if it holds these conditions:

1. 

For all $v\in \mathfrak{B}$

$${\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})$$

2. 

For all $v,\kappa ,\eta \in \mathfrak{B}$ and $i=1,2,...m$.

$$\left.{\varsigma }_{{\varrho }_i}(v\ast (\kappa \ast (\kappa \ast v)),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ),\mathfrak{q}\right)\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})$$

Example 3.

Let us consider the $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra $\mathfrak{B}=\{{\mathfrak{t}}_1,{\mathfrak{t}}_2,{\mathfrak{t}}_3\}$ and the set $Q=\{{\mathfrak{t}}_1,{\mathfrak{t}}_2,{\mathfrak{t}}_3,{\mathfrak{t}}_4\}$ as defined in the previous example. We will now define an $\mathfrak{m}$-polar Q-$\mathcal{HF}$ commutative ideal based on this algebra.

First, let us consider the $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra $\mathfrak{B}=\{{\mathfrak{t}}_1,{\mathfrak{t}}_2,{\mathfrak{t}}_3\}$ with the binary operation "*", which is given in the following

Table 6. Binary operation table for $\mathfrak{B}$.

*	${\mathfrak{t}}_1$	${\mathfrak{t}}_2$	${\mathfrak{t}}_3$
${\mathfrak{t}}_1$	${\mathfrak{t}}_1$	${\mathfrak{t}}_1$	${\mathfrak{t}}_1$
${\mathfrak{t}}_2$	${\mathfrak{t}}_2$	${\mathfrak{t}}_1$	${\mathfrak{t}}_2$
${\mathfrak{t}}_3$	${\mathfrak{t}}_3$	${\mathfrak{t}}_3$	${\mathfrak{t}}_1$

:

The set $Q=\{{\mathfrak{t}}_1,{\mathfrak{t}}_2,{\mathfrak{t}}_3,{\mathfrak{t}}_4\}$, and  2-polar FS on $\mathfrak{B}$, is described as follows:

For each pair of elements in $\mathfrak{B}$, we define membership values. Here’s how we define membership values for the FS $M_1$ 

Table 7. Membership values for FS $M_1$.

$({\mathfrak{t}}_1,{\mathfrak{t}}_1)$	$\{0.9,0.8\}$
$({\mathfrak{t}}_1,{\mathfrak{t}}_2)$	$(0.7,0.8)\ast (0.8,0.8)$
$({\mathfrak{t}}_1,{\mathfrak{t}}_3)$	$(0.8,0.9,0.7)$
$({\mathfrak{t}}_1,{\mathfrak{t}}_4)$	$(0.7,0.9)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_1)$	$(0.5,0.6)\ast (0.3,0.2)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_2)$	$(0.1,0.4)\ast (0.5,0.3)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_3)$	$(0.2,0.1)\ast (0.5,0.1)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_4)$	$(0.4,0.3,0.1)\ast (0.5,0.1)$
$({\mathfrak{t}}_3,{\mathfrak{t}}_1)$	$\left\{0.7\right\}$
$({\mathfrak{t}}_3,{\mathfrak{t}}_2)$	$(0.6,0.5)\ast (0.7,0.6)$
$({\mathfrak{t}}_3,{\mathfrak{t}}_3)$	$(0.6,0.4,0.3)\ast (0.8,0.7)$
$({\mathfrak{t}}_3,{\mathfrak{t}}_4)$	$(0.7,0.5)\ast (0.7,0.6)$

:

The membership values for the FS $M_2$ are defined as follows in

Table 8. Membership values for FS $M_2$.

$({\mathfrak{t}}_1,{\mathfrak{t}}_1)$	$\{0.8,0.9\}$
$({\mathfrak{t}}_1,{\mathfrak{t}}_2)$	$(0.7,0.8)\ast (0.9,0.7)$
$({\mathfrak{t}}_1,{\mathfrak{t}}_3)$	$(0.9,0.8,0.7)$
$({\mathfrak{t}}_1,{\mathfrak{t}}_4)$	$(0.7,0.9)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_1)$	$(0.6,0.5)\ast (0.8,0.6)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_2)$	$(0.1,0.5)\ast (0.9,0.7)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_3)$	$(0.3,0.2)\ast (0.9,0.8)$
$({\mathfrak{t}}_2,{\mathfrak{t}}_4)$	$(0.4,0.1,0.5)\ast (0.9,0.7)$
$({\mathfrak{t}}_3,{\mathfrak{t}}_1)$	$\left\{0.7\right\}$
$({\mathfrak{t}}_3,{\mathfrak{t}}_2)$	$(0.5,0.6)\ast (0.6,0.8)$
$({\mathfrak{t}}_3,{\mathfrak{t}}_3)$	$(0.6,0.7,0.4)\ast (0.7,0.9)$
$({\mathfrak{t}}_3,{\mathfrak{t}}_4)$	$(0.6,0.4)\ast (0.8,0.7)$

:

To construct the $\mathfrak{m}$-polar Q-$\mathcal{HF}$ commutative ideal, take the intersection of $M_1$ and $M_2$ for every pair of elements in $\mathfrak{B}$. This ideal represents the commutative qualities and unwilling nature of the operations in the supplied $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra. It enables a more nuanced understanding of algebraic behavior by taking into account various amounts of delay in operations.

Theorem 2.

An $\mathcal{MPQHF}$ ideal of a $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra is an $\mathcal{MPQHF}$ commutative ideal if it meets the properties given below:

$${\varsigma }_{{\varrho }_i}\left(v\ast (\kappa \ast (\kappa \ast v),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\right.$$

$v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q$.

Proof. 

Let $v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q$,

$(⟹)$ Suppose that ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ commutative-ideal of $\mathfrak{B}$. Taking $\eta =0$, then

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v\ast (\kappa \ast (\kappa \ast v)),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast 0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(v\ast \kappa ,q)\end{array}$$

$(⟸)$ conversely, Assume that ${\varsigma }_{{\varrho }_i}$ satisfies

$${\varsigma }_{{\varrho }_i}(v\ast (\kappa \ast (\kappa \ast v)),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})$$

then,

$${\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})$$

then, we obtain the following:

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v\ast (\kappa \ast (\kappa \ast v)),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap \\ {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\end{array}$$

$\forall v,\kappa ,\eta \in \mathfrak{B},\mathfrak{q}\in Q$. Hence, ${\varsigma }_{{\underline{Q}}_i}$ is an $\mathcal{MPQHF}$ commutative ideal.    □

Theorem 3.

All $\mathcal{MPQHF}$ commutative ideals are an $\mathcal{MPQHF}$ ideal of $\mathfrak{B}$.

Proof. 

Let $v,\kappa ,\eta \in \mathfrak{B},\mathfrak{q}\in Q$. Let ${\varsigma }_{{\varrho }_i}$ be an $\mathcal{MPQHF}$ commutative ideal of $\mathfrak{B}$.

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v,\mathfrak{q})={\varsigma }_{{\varrho }_i}(v\ast (0\ast (0\ast v)),\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast 0),\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}(\eta \ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\end{array}$$

for all $v,\eta \in \mathfrak{B},\mathfrak{q}\in Q$. Thus, ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ ideal.

Here, we will look at close $\mathcal{MPQHF}$ ideals, a special concept in our study. We will introduce them, explain what they are, and show how they are used, especially in decision-support systems. Understanding these ideals helps us see their role in our research.   □

Definition 10.

An $\mathcal{MPQHF}$ ideal

$${\varrho }_i:=\left\{\left((\mathfrak{t},\mathfrak{q}),{\varsigma }_{\varrho }^i(\mathfrak{t},\mathfrak{q})\right)\mid \mathfrak{t}\in \mathfrak{B},\mathfrak{q}\in Q\right\}$$

of $\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra is known to be closed if

$${\varsigma }_{{\varrho }_i}(\mathfrak{t},\mathfrak{q})\subseteq {\varsigma }_{{\varrho }_i}(0\ast \mathfrak{t},\mathfrak{q})$$

$\forall \mathfrak{t}\in \mathfrak{B},\mathfrak{q}\in Q$ and $i=1,2\dots m$.

Example 4.

The $\mathcal{MPQHF}$ ideal, which is established in (3.2), is a closed $\mathcal{MPQHF}$ ideal.

Theorem 4.

An $\mathcal{MPQHF}$ ideal ${\varsigma }_{\varrho }$ is known to be closed if it meets $\forall v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q$

$${\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})$$

Proof. 

$(⟹)$ Suppose that ${\varsigma }_{{\varrho }_i}$ is a closed $\mathcal{MPQHF}$ ideal over a $\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra $\mathfrak{B}$. Since $(v\ast \kappa )\ast v\le$$0\ast \kappa$$\forall v,\kappa \in \mathfrak{B}$

$$\begin{array}{cc}\hfill {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})& \supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\cap \mathrm{varsigma}{\varrho }_i(0\ast \kappa ,\mathfrak{q})\hfill \\ & \supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})\hfill \end{array}$$

$\forall v,\kappa \in \mathfrak{B},\mathfrak{q}\in Q$, and $i=1,2,\dots m$.

$(⟸)$ Conversely, let ${\varsigma }_{{\varrho }_i}$ be an $\mathcal{MPQHF}$ ideal in $\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra $\mathfrak{B}$, since

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\subseteq {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})$$

$\forall v\in \mathfrak{B},\mathfrak{q}\in Q$ we have:

$$\begin{array}{cc}\hfill {\varsigma }_{{\varrho }_i}(0\ast v,\mathfrak{q})& \supseteq {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\hfill \\ & ={\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\hfill \end{array}$$

for all $v\in \mathfrak{B}$. Therefore, ${\varsigma }_{{\varrho }_i}$ is a closed $\mathcal{MPQHF}$ ideal covering a $\mathfrak{B}\mathrm{C}\mathbb{I}$ algerbra $\mathfrak{B}$ for every $i=1,2,\dots ,m$.    □

In the following, we will explore $\mathcal{MPQHF}$ implicative ideals, defining and explaining them to understand their role in DM. These ideals are crucial in our research, showing their importance in our overall work.

Definition 11.

An $\mathcal{MPQHF}$ set

$${\varrho }_i:=\left\{\left((v,\mathfrak{q}),{\varsigma }_{\varrho }^i(v,\mathfrak{q})\right)\mid v\in \mathfrak{B},\mathfrak{q}\in Q\right\}$$

in $\mathfrak{B}$ is named as $\mathcal{MPQHF}$ implicative ideal if it fulfills the condition given below:

1. 

${\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})$

2. 

${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v))\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})$

for all $i=1,2\dots ,m$ and $v,\kappa ,\eta \in \mathfrak{B}$.

Example 5.

Let $\mathfrak{B}=\{{\alpha }_1,{\beta }_1,{\gamma }_1\}$ be a $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra under binary action " *", as illustrated in the clayey

Table 9. Binary operation table for $\mathfrak{B}$.

*	${\alpha }_1$	${\beta }_1$	${\gamma }_1$
${\alpha }_1$	${\alpha }_1$	${\alpha }_1$	${\alpha }_1$
${\beta }_1$	${\beta }_1$	${\alpha }_1$	${\beta }_1$
${\gamma }_1$	${\gamma }_1$	${\gamma }_1$	${\alpha }_1$

below.

Define the set $Q=\left\{{\psi }_1\right\}$ and a 3-polar FS on $\mathfrak{B}$ as follows (

Table 10. Three-polar FS on $\mathfrak{B}$.

$({\mathit{\alpha }}_1,{\mathit{\psi }}_1)$	$\left\{\right\{0.9,0.8\},(0.7,0.6,0.9),\{0.8\left\}\right\}\}$
$({\beta }_1,{\psi }_1)$	$\left\{\right(0.7,0.8)·(0.5,0.4),(0.6\left)\right\}$
$({\gamma }_1,{\psi }_1)$	$\left\{\right(0.6,0.6,0.5),\{0.3,0.1\},(0.1,0.2\left)\right\}$

):

Thus, ${\varsigma }_{{\varrho }_i}$ is a 3-polar Q-$\mathcal{HF}$ implicative ideal.

Proposition 6.

In $\mathfrak{B}\mathrm{C}\mathbb{K}$ algebra $\mathfrak{B}$, every $\mathcal{MPQHF}$ implicative ideal is an $\mathcal{MPQHF}$ ideal.

Proof. 

Let ${\varsigma }_{{\varrho }_i}$ be an $\mathcal{MPQHF}$ implicative ideal over $\mathfrak{B}$. Let $v,\kappa ,\eta \in \mathfrak{B}$, then:

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v))\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})$$

Replace $\kappa =v$, and using $v\ast v=0$, we obtain

$$\begin{array}{c}\hfill {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast (v\ast v))\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\\ \hfill ={\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\end{array}$$

for all $v,\eta \in \mathfrak{B}.{\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ ideal.    □

Theorem 5.

Assuming $\mathfrak{B}$ is an implicative $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra, all $\mathcal{MPQHF}$ ideals over $\mathfrak{B}$ are also $\mathcal{MPQHF}$ implicative ideals.

Proof. 

Assuming $\mathfrak{B}$ is an implicative $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra, it implies accordingly $v=v\ast (\kappa \ast v)\forall v$ and $\kappa$. Additionally, let ${\varsigma }_{{\varrho }_i}$ be an $\mathcal{MPQHF}$ ideal. Then, we can derive:

$$\begin{array}{cc}& {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\hfill \\ \hfill =& {\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v))\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\hfill \end{array}$$

for all $v,\kappa ,\eta \in \mathfrak{B}$. Thus, it is an $\mathcal{MPQHF}$ implicative ideal of $\mathfrak{B}$. Then, ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ implicative ideal of $\mathfrak{B}$.   □

Theorem 6.

Let ${\varsigma }_{{\varrho }_i}$ be an $\mathcal{MPQHF}$ ideal of a $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra $\mathfrak{B}$. Then ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ implicative ideal of $\mathfrak{B}$ if it meets the following criteria:

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast (\kappa \ast v),\mathfrak{q})$$

$\forall v,\kappa \in \mathfrak{B},\mathfrak{q}\in Qandi=1,2,\dots ,m$.

Proof. 

Assume that ${\varsigma }_{{\underline{Q}}_i}$ is an $\mathcal{MPQHF}$ implicative ideal of $\mathfrak{B}$. Take $\eta =0$ in

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v))\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v))\ast 0,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v)),\mathfrak{q})\end{array}$$

Conversely, assume that ${\varsigma }_{{\varrho }_i}$ fulfill the characteristics. As ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ ideal of $\mathfrak{B}$, we have

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v)),\mathfrak{q})\\ \supseteq {\varsigma }_{{\varrho }_i}((v\ast (\kappa \ast v))\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\eta ,\mathfrak{q})\end{array}$$

Then, ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ implicative ideal of $\mathfrak{B}$ and the proof is completed. Starting in the following section, we will explore $\mathcal{MPQHF}$ PI ideals. These ideals are important in our research, especially for DM, and explain these ideals to understand their characteristics.   □

Definition 12.

An $\mathcal{MPQHF}$ set

$${\varrho }_i:=\left\{\left((v,\mathfrak{q}),{\varsigma }_{\varrho }^i(v,\mathfrak{q})\right)\mid v\in \mathfrak{B},\mathfrak{q}\in Q\right\}$$

is $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra $\mathfrak{B}$ is named as $\mathcal{MPQHF}$ PI ideal of $\mathfrak{B}$

1. 

${\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})$

2. 

${\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \eta ,\mathfrak{q})$

$\forall v,\kappa ,\eta \in \mathfrak{B},\mathfrak{q}\in Q$ and $i=1,2,\dots ,m$.

Example 6.

Suppose that $\mathfrak{B}=\{\mathfrak{a},\mathfrak{b},\mathfrak{c},\mathfrak{d},\mathfrak{e}\}$ is an $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra with a binary action " *", describe in the clayey

Table 11. Binary operation table for $\mathfrak{B}$.

*	$\mathfrak{a}$	$\mathfrak{b}$	c	$\mathfrak{d}$	$\mathfrak{e}$
$\mathfrak{a}$	$\mathfrak{a}$	$\mathfrak{a}$	$\mathfrak{a}$	$\mathfrak{a}$	$\mathfrak{a}$
$\mathfrak{b}$	$\mathfrak{b}$	$\mathfrak{a}$	$\mathfrak{a}$	a	$\mathfrak{a}$
c	c	c	$\mathfrak{a}$	$\mathfrak{a}$	c
$\mathfrak{d}$	$\mathfrak{d}$	$\mathfrak{d}$	$\mathfrak{d}$	$\mathfrak{a}$	$\mathfrak{d}$
$\mathfrak{e}$	$\mathfrak{e}$	$\mathfrak{e}$	$\mathfrak{e}$	$\mathfrak{e}$	$\mathfrak{a}$

.

Defined the set $Q=\left\{{\mathfrak{a}}^{\prime },b^{\prime }\right\}$ and a 3-polar FS on $\mathfrak{B}$ as follows

Table 12. Two-polar FS on $\mathfrak{B}$.

$\left(\mathfrak{a},{\mathfrak{a}}^{\prime }\right)$	$\left\{\right\{0.9,0.8\},(0.9,0.9\left)\right\}$
$\left(\mathfrak{a},{\mathfrak{b}}^{\prime }\right)$	$\left\{\right(0.8,0.8,0.9),\{0.8\left\}\right\}$
$\left(\mathfrak{b},{\mathfrak{a}}^{\prime }\right)$	$\left\{\right(0.5,0.5),\{0.7,0.6,0.5\left\}\right\}$
$\left(\mathfrak{b},{\mathfrak{b}}^{\prime }\right)$	$\left\{\right(0.7),(0.7,0.6\left)\right\}$
$\left(c,{\mathfrak{a}}^{\prime }\right)$	$\left\{\right(0.3,0.2,0.3),(0.2\left)\right\}$
$\left(c,{\mathfrak{b}}^{\prime }\right)$	$\left\{\right(0.1,0.3,0.2,0.1),(0.2,0.2\left)\right\}$
$\left(d,{\mathfrak{a}}^{\prime }\right)$	$\left\{\right(0.2),(0.3\left)\right\}$
$\left(d,{\mathfrak{b}}^{\prime }\right)$	$\left\{\right(0.3,0.1),(0.2,0.1\left)\right\}$
$\left(e,{\mathfrak{a}}^{\prime }\right)$	$\left\{\right\{0.4\},(0.5,0.4\left)\right\}$
$\left(e,{\mathfrak{b}}^{\prime }\right)$	$\left\{\right(0.6,0.5),(0.4\left)\right\}$

:

So, ${\varsigma }_{{\varrho }_i}$ is a 2-polar Q-$\mathcal{HF}$ PI ideal.

Proposition 7.

In $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra $\mathfrak{B}$ each $\mathcal{MPQHF}$ PI ideal is $\mathfrak{m}$-polar Q $\mathcal{HF}$ ideal.

Proof. 

Suppose that ${\varsigma }_{{\varrho }_i}$ be $\mathcal{MPQHF}$ PI ideal of $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra $\mathfrak{B}$. $\forall v,\kappa ,\eta \in$$\mathfrak{B},\mathfrak{q}\in Q$, such that

$${\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \eta ,\mathfrak{q})$$

put $\eta =0$

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})$$

Therefore, ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ ideal. That end the proof.   □

Theorem 7.

Let ${\varsigma }_{{\varrho }_i}$$\mathcal{MPQHF}$ ideal on $\mathfrak{B}$. Then, ${\varsigma }_{{\varrho }_i}$$\mathcal{MPQHF}$ PI ideal if

$${\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \kappa ,\mathfrak{q})$$

$\forall v,\kappa ,\eta \in \mathfrak{B}$ and $i=1,2,\dots ,m$.

Proof. 

$(⟹)$ Assume that the $\mathcal{MPQHF}$ ideal ${\varsigma }_{{\varrho }_i}$ of $\mathfrak{B}$ is an $\mathcal{MPQHF}$ filter PI ideal of $\mathfrak{B}$, So

$${\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \eta ,\mathfrak{q})$$

if we put $\eta =\kappa$, we have

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \kappa ,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \kappa ,\mathfrak{q})\end{array}$$

$\forall v,\kappa \in \mathfrak{B}$ and $i=1,2,\dots ,m$.

$(⟸)$ Conversely, assume that ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ ideal over $\mathfrak{B}$ and satisfies the inequality

$${\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \kappa ,\mathfrak{q})$$

Since ${\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v,\mathfrak{q})$ Now we can prove that

$${\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \eta ,\mathfrak{q})$$

for all $v,\kappa ,\eta \in \mathfrak{B}$. In contrast, there exist $v^{\prime },{\kappa }^{\prime }\in \mathfrak{B}$ Such that

$$\begin{array}{c}{\varsigma }_{{\varrho }_i}\left(v^{\prime }\ast {\kappa }^{\prime },\mathfrak{q}\right)\subseteq {\varsigma }_{{\varrho }_i}\left(\left(v^{\prime }\ast {\kappa }^{\prime }\right)\ast {\kappa }^{\prime },\mathfrak{q}\right)\cap {\varsigma }_{{\varrho }_i}\left({\kappa }^{\prime }\ast {\kappa }^{\prime },\mathfrak{q}\right)\\ ={\varsigma }_{{\varrho }_i}\left(\left(v^{\prime }\ast {\kappa }^{\prime }\right)\ast {\kappa }^{\prime },\mathfrak{q}\right)\cap {\varsigma }_{{\varrho }_i}(0,\mathfrak{q})\\ ={\varsigma }_{{\varrho }_i}\left(\left(v^{\prime }\ast {\kappa }^{\prime }\right)\ast {\kappa }^{\prime },\mathfrak{q}\right)\end{array}$$

Which is a contradiction; therefore,

$${\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \eta ,\mathfrak{q})$$

$\forall v,\kappa ,\eta \in \mathfrak{B}$ and $\mathfrak{q}\in Q$. Thus ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ PI ideal of $\mathfrak{B}$ for all $i=1,2,\dots .m$.   □

Proposition 8.

If $\mathfrak{B}$ is a PI $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra, then all $\mathcal{MPQHF}$ ideal of $\mathfrak{B}$ is an $\mathcal{MPQHF}$ PI ideal of $\mathfrak{B}$.

Proof. 

Suppose that ${\varsigma }_{{\varrho }_i}$ is an $\mathcal{MPQHF}$ PI ideal of $\mathfrak{B}$, for all $v,\kappa \in \mathfrak{B}$ and $\mathfrak{q}\in Q$, and then

$${\varsigma }_{{\varrho }_i}(v,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}(v\ast \kappa ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa ,\mathfrak{q})$$

By replacing v by $v\ast \eta$ and $\kappa$ by $\kappa \ast \eta$, we get

$${\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \eta )\ast (\kappa \ast \eta ),\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \eta ,\mathfrak{q})$$

Therefore, $\mathfrak{B}$ is a PI $\mathfrak{B}\mathrm{C}\mathbb{K}$ algerbra, $(v\ast \eta )\ast (\kappa \ast \eta )=(v\ast \kappa )\ast \eta$ for all $v,\kappa ,\eta \in \mathfrak{B}$. Hence,

$${\varsigma }_{{\varrho }_i}(v\ast \eta ,\mathfrak{q})\supseteq {\varsigma }_{{\varrho }_i}((v\ast \kappa )\ast \eta ,\mathfrak{q})\cap {\varsigma }_{{\varrho }_i}(\kappa \ast \eta ,\mathfrak{q})$$

Thus, ${\varsigma }_{{\varrho }_i}$$\mathcal{MPQHF}$ PI ideal of $\mathfrak{B}$ for all $i=1,2,\dots \dots ,m$.   □

4. Application of $\mathfrak{m}$-Polar $\mathit{Q}$-Hesitant Fuzzy Sets in Career Determination Problems

In this section, we shift our focus from the theoretical aspects of $\mathcal{MPQHF}$ sets to their practical applications in career determination problems. The theoretical foundations laid out in the previous sections serve as the building blocks for understanding how $\mathcal{MPQHF}$ sets can be effectively utilized in real-world scenarios. Career determination is a critical domain where these concepts find their relevance, providing innovative solutions to complex problems.

We are going to address how $\mathcal{MPQHF}$ is employed for enhancing career decision-support systems and problem-solving methodologies, illustrating its usefulness in resolving practical challenges. The comparison table is  with an equal number of rows and columns; both rows and columns are distinguished with the names of the companies, $a_1,a_2,\dots \dots ,a_n$, and the elements are $t_{ij},i,j=1,2,\dots ,n$, given by $t_{ij}=$ the number of the belonging degree of $a_i$ exceeds or equals the belonging degree of $a_j$. The row sum of a company $a_i$ is indicated by $r_i$

$$\begin{array}{ccc}\hfill r_i& =& \sum _{j=1}^n{t}_{ij}\hfill \end{array}$$

The column sum of a company $a_j$ is denoted by $c_j$:

$$\begin{array}{ccc}\hfill c_j& =& \sum _{i=1}^n{t}_{ij}\hfill \end{array}$$

The score of a company $a_j$ is $S_j$ and this may be offered as:

$$S_j=r_j-c_j$$

We provide a procedure based on sets of $\mathcal{MPQHF}$. Algorithm 1 defined a step-by-step strategy to employ MQHFs in career selection circumstances. It provided a basic and methodical approach to tackling uncertainty and apprehension in career decision services. This algorithm provides practical guidelines for implementing MQHFs and is a valuable tool for scholars and professionals working in a range of disciplines where career decisions must be made under uncertain conditions. The following Figure 2 shows the model for the career determination algorithm.

Algorithm 1: Career determination algorithm using $\mathcal{MPQHF}$ set

Input the $\mathcal{MPQHF}$ set. Input the non-empty sets A, B, and C. Compute the score of $a_i$. Calculate the corresponding resultant $\mathcal{MPQHF}$ set and present it in tabular form. Create a comparison-table for the $\mathcal{MPQHF}$ set and calculate $r_i$ and $c_i$. The career choice is $S_k$ if it represents the maximum value among $S_i$. In case there are multiple values in $\mathrm{K}$, you can choose any of the career options $a_k$.

An illustrative example: In this example, we will show how the algorithm works and how it may be used in real-world data extraction situations. This actual application will allow us to assess the algorithm’s effectiveness and potential for providing useful insights in a variety of decision-support environments.

Example 7.

Enhanced Example: Career Determination Using $\mathcal{MPQHF}$ Sets for University Students.

In a university career counseling scenario, a group of students is facing a significant decision: selecting the most suitable career path. They approach this challenging task by considering three crucial factors: academic interest, job prospects, and personal aspirations. To effectively evaluate their career options, the students employ $\mathcal{MPQHF}$ sets to account for their individual preferences and uncertainties.

The universe set U represents the available career paths, denoted as $U=\{C_1,C_2,C_3,C_4,C_5\}$, and the key evaluation criteria are as follows:

• Academic Interest (A):

  -

  Academic interest levels are assessed using $\mathcal{MPQHF}$ sets. The following table shows the assessments for each career option (

  Table 13. Academic interest assessments for career options.

  	$a_1$	$a_2$	$a_3$	$a_4$
  $C_1$	$\{0.3,0.4\}\left(0.6\right)$	$\{0.3,0.1\}\left(0.5\right)$	$\left[0.5\right]\left(0.7\right)$	$\left[0.3\right]\{0.2,0.6\}$
  $C_2$	$(0.5,0.6,0.8)\left(0.1\right)$	$\left(0.8\right)[0.2,0.1]$	$\left\{0.6\right\}\left[0.7\right]$	$\left[0.3\right][0.2,0.1]$
  $C_3$	$\left[0.6\right](0.1,0.2,0.1)$	$(0.5,0.4)\left[0.5\right]$	$\left[0.6\right]\left(0.3\right)$	$\left[0.2\right]\left(0.1\right)$
  $C_4$	$(0.3,0.2]\left\{0.7\right\}$	$[0.8,0.7,0.9]\left\{0.9\right\}$	$(0.7,0.6)\left[0.5\right]$	$\left[0.2\right]\left[0.2\right]$
  $C_5$	$[0.7,0.3](0.2,0.6)$	$\left[0.3\right]\left\{0.2\right\}$	$\left[0.3\right]\left[0.2\right]$	$\left[0.3\right]\left(0.4\right)$

  ):

• Job Prospects (J):

  -

  Job prospect assessments, represented using $\mathcal{MPQHF}$ sets, are vital for career determination. The next table shows the evaluations for each career possibility (

  Table 14. Job prospect assessments for career options.

  	$j_1$	$j_2$	$j_3$
  $C_1$	$(0.3,0.1,0.3)\left\{0.2\right\}$	$\left\{0.6\right\}(0.7,0.6)$	$(0.3,0.1)(0.2,0.3)$
  $C_2$	$\left(0.3\right)\left\{0.2\right\}$	$\left\{0.7\right\}\left\{0.8\right\}$	$(0.9,0.8)\left\{0.9\right\}$
  $C_3$	$(0.1,0.2)\left\{0.2\right\}$	$(0.8,0.6)\left\{0.3\right\}$	$(0.2,0.3,0.2)\left\{0.6\right\}$
  $C_4$	$(0.8,0.6)\left\{0.5\right\}$	$(0.1,0.3)\left(0.2\right)$	$\left(0.5\right)\left(0.5\right)$
  $C_5$	$\left(0.6\right)\left(0.7\right)$	$\left(0.4\right)(0.3,0.2)$	$\left(0.1\right)\left(0.5\right)$

  ).

• Personal Aspirations (P):

  -

  Personal aspiration assessments, captured through $\mathcal{MPQHF}$ sets, are essential considerations for career decisions. The assessments for each career option are detailed in the following

  Table 15. Personal aspiration assessments for career options.

  	$p_1$	$p_2$
  $C_1$	$(0.4,0.3)\left(0.4\right)$	$(0.6,0.5,0.6)\left(0.7\right)$
  $C_2$	$(0.1,0.2)(0.3,0.2)$	$\left(0.9\right)\left(0.8\right)$
  $C_3$	$\{0.3,0.1\}\left(0.2\right)$	$(0.5,0.5)\left(0.3\right)$
  $C_4$	$(0.6,0.7,0.6)\left(0.8\right)$	$\left\{0.1\right\}\left(0.3\right)$
  $C_5$	$\left[0.7\right]\left(0.8\right)$	$\left(0.9\right)(0.8,0.7)$

  :

Performing Aggregation (AND) Operation:To identify the best job option depending on the combined factors of academic interest (A), job prospects (J), and personal aspirations (P), an aggregation operation is performed. The "AND" operation is employed to aggregate the scores of each career option across all three factors. The "AND" operation is defined as follows:

$$\begin{array}{cc}\hfill T\left(C_i\right)& =A\left(C_i\right)\mathrm{AND}J\left(C_i\right)\mathrm{AND}P\left(C_i\right)\hfill \\ \hfill T\left(C_i\right)& =min\left[A(C_i,a_j)·J(C_i,j_k)·P(C_i,p_l)\right]\hfill \end{array}$$

Now, let us calculate the aggregated scores for all career options using the "AND" operation.

• Aggregated Scores:
• For $C_1$:

  $$\begin{array}{cc}\hfill T\left(C_1\right)& =min\left[A(C_1,a_j)·J(C_1,j_k)·P(C_1,p_l)\right]\hfill \\ \hfill & =min\left[0.001728,0.02142,0.014112\right]=0.001728\mathit{(}\mathit{Minimum}\mathit{score}\mathit{)}\hfill \end{array}$$

  Perform similar calculations for $C_2$, $C_3$, $C_4$, and $C_5$ to find their respective aggregated scores and Figure 3 represents the graphical representation of aggregation scores.

• Aggregated Scores for All Career Options:
• For $C_2$:

  $$T\left(C_2\right)=min\left[0.0045,0.03,0.035\right]=0.0045$$
  For $C_3$:

  $$T\left(C_3\right)=min\left[0.006,0.0216,0.0275\right]=0.006$$
  For $C_4$:

  $$T\left(C_4\right)=min\left[0.012,0.03,0.042\right]=0.012$$
  For $C_5$:

  $$T\left(C_5\right)=min\left[0.0375,0.063,0.098\right]=0.0375$$

Decision: 

The most suitable career option based on the combined criteria of academic interest, job prospects, and personal aspirations is $C_5$, as it has the highest aggregated score. This method empowers the students to make well-informed career decisions that align with their individual preferences and uncertainties, ultimately leading to a more satisfying and fulfilling professional path. Figure 4 shows the complete steps of how our algorithm finds the highest aggregated score.

5. Conclusions

The concept of $\mathcal{MPQHF}$ sets is a valuable hybrid approach to discussing uncertainty and vagueness of information. In this article, we established $\mathcal{MPQHF}$-$\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras and discussed the fundamental characteristics of $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. In addition, we presented different types of $\mathcal{MPQHF}$ ideals, revealing their importance of $\mathfrak{B}\mathrm{C}\mathbb{K}/\mathfrak{B}\mathrm{C}\mathbb{I}$ algebras. Moreover, we intend to utilize this theory on other mathematical models and study basic operations between $\mathcal{MPQHF}$ sets. This will allow us to better understand and utilize these sets in a variety of situations. Furthermore, we constructed the complete algorithm steps under the influence of the proposed approach. Therefore, we applied this concept to the career selection option and illustrated some examples to find the accuracy and efficiency of our proposed model. In the future, we intend to integrate this technique with computational mathematics to improve accuracy and obtain robust results in the DM process.