Introduction

As ring theory, is a subfield of abstract algebra, an ideal of a ring is a distinct subset of its members. Some subsets of the integers, such as the even numbers or the multiples of 3, are generalized by ideals [10]. The defining characteristics of an ideal are closure and absorption: adding and subtracting even numbers maintains evenness, and multiplication of an even number by any integer (even or odd) yields an even number. Similar to how a normal subgroup can be used to create a quotient group in group theory, an ideal can be used to create a quotient ring [20]. The ideals are the non-negative integers that correspond one-to-one with the integers; each ideal in this ring is a principal ideal made up of multiples of a single non-negative number. But, in other rings, the ideals might not exactly match the ring components, and when certain integer features are generalized to rings, they tend to attach to the ideals rather than the ring components more naturally [30]. For instance, the Chinese remainder theorem can be used for ideals and the prime ideals of a ring are comparable to prime numbers. The ideals of a Dedekind domain can be factored in a specific way using unique prime factorization (a type of ring important in number theory) [14].

The idea of ring theory is the source of the related but separate concept of an ideal in order theory. A generalization of an ideal, a fractional ideal is frequently referred to as an integral ideal for clarity [28]. According to a bivalent condition, an element either belongs to the set or does not in traditional set theory, which evaluates element membership in binary terms. In contrast, fuzzy set theory allows for a gradual evaluation of an element's membership in a set [17]. This is specified using a membership function with a value between [0,1]. Because classical sets' characteristic functions are particular examples of fuzzy sets' membership functions, which can only take values of 0 or 1, fuzzy sets generalize classical sets. Crisp sets are also known as traditional bivalent sets in fuzzy set theory [24]. Many ideas of universal algebras that generalize an associative ring \((R, +,\cdot)\) exist. Some of them, particularly near rings and other varieties of semirings are extremely effective [33]. It's a good idea to have a backup plan in place in case the backup plan fails. Except for the assumption that \((S, +)\) is a semigroup rather than a commutative group, the second category of those algebras \((S, +,\cdot),\) known as semirings (and occasionally halfrings), share the same features as rings. In various applications to the theory of automata and formal languages, semi rings ordered semi-rings, and hemi rings occur naturally [22].

In recent years, there has been an increased interest in studying the algebraic structure of soft sets. The idea of soft groups was defined, some of their properties were determined, and soft sets were compared to related ideas like fuzzy sets and rough sets [6]. Handling complicated problems in economics, engineering, the environment, sociology, medical science, and many other areas using traditional ways because uncertainty in these fields can take various forms [15]. The Theory of Probability, Fuzzy Set Theory (FST), Interval Mathematics, and Rough Set Theory (RST) are four mathematical techniques for dealing with incomplete knowledge. To begin with, each of these tools requires a predetermined parameter, such as the probability density function in probability, the membership function in FST, or the equivalence relation in RST [7].

When seen against the backdrop of inadequate or inaccurate knowledge, such a requirement presents numerous challenges. In response to issues with parameter specification, researchers developed the idea of a soft set to address issues with partial data [27]. The definition of a parameter is not required by the Soft Set Theory (SST). SST is a natural mathematical formalism for approximation reasoning because of this. Finally, researchers investigated the notion of soft sets in further detail and applied it to some decision-making issues [11]. Since real-world issues often involve ambiguity, uncertainty, or incomplete knowledge, they are too complex. Therefore, decision-makers handle these issues through the fuzzification process, which is an essential technique for handling humanistic systems found in real-world issues. Many academics and researchers first used the crucial idea of fuzzy sets (FSs) to address a wide range of real problems in various fields and provide the optimal solution or solutions. Unfortunately, the FS theory was unable to account for subjective assessments of what constitutes discontent. One of the unavoidable aspects of handling decision-making problems is the fuzziness of information. In addition, it is a widely used tool to address ambiguity in decision-making situations [2]. Then, as a generalization of an intuitionistic fuzzy set, a Pythagorean fuzzy set (PFS) is used to expand the domain of membership and non-membership degrees. It effectively handles scenarios where the total of their membership and non-membership levels for a given characteristic exceeds one. Ultimately, the effective method known as "aggregation operators" is created to deal with multi-criteria decision-making (MCDM) issues while dealing with (2,1)-Fuzzy sets [2]. A very generic example of this effective blend, started by fuzzy soft sets, is the standard orthopair fuzzy soft set. It is a mapping that takes a collection of parameters and maps them to the family of all orthopair fuzzy sets (which provide a broad picture of what constitutes valid membership and non-membership assessments). The requirement for orthopair fuzzy sets that membership and non-membership be calibrated with the same power should be lifted to broaden the application of fuzzy soft set theory. The idea of a (a, b)-fuzzy soft set, or (a, b)-FSS, is presented for this purpose.

These facilitate the resolution of scenarios that necessitate assessments with varying weights for degrees of membership and non-membership, an issue that cannot be represented by the current extensions of intuitionistic fuzzy soft sets. Additionally, a novel generalized intuitionistic fuzzy set known as (3, 2)-fuzzy sets is presented, along with an analysis of their connections to Fermatean, Pythagorean, and intuitionistic fuzzy sets and also certain definitions and relationships for operators on (3, 2)-fuzzy sets are provided. The concepts of fuzzy neighborhood (3, 2), fuzzy topology (3, 2), and fuzzy continuous mapping (3, 2) were examined. It is considerably more difficult to investigate the group decision-making issue related to the new model [16]. Nonetheless, it is possible to research both (3, 2)-fuzzy soft sets and the applications of (3, 2)-fuzzy sets. Since the maximal ideals play a crucial role in ring theory, there is a need to study the maximal ideals in near rings. This is a way to generalize the concept of ideals from rings to near-rings. This generalization allows mathematicians to explore the similarities and differences between rings and near-rings and adapt ring theory concepts to near-ring settings. The hybrid structure of maximal ideals in near-rings serves as a bridge between traditional ring theory and near-ring theory. It helps mathematicians establish connections and analogies between these two algebraic structures, which can lead to insights and applications in both areas. The majority of our traditional tools for formal reasoning, modeling, and computation have a clean, deterministic, and exact nature. Nonetheless, other challenging issues in fields such as economics, engineering, the environment, social science, and medicine entail imperfect data [Manikantan, Ramasamy]. Due to the numerous forms of uncertainty prevalent in these issues, using traditional approaches is not an option in these circumstances. Hence to overcome this uncertainty, the hybrid structure for fuzzy ideals and fuzzy maximal ideals in near rings is introduced. The main contributions of this study are as follows:

  • To tackle the challenges involved in transforming fuzzy sets into a hybrid structure, a new approach called “Hybrid Structure of Fuzzy Sets in Near Rings” is developed. This innovative concept offers a solution to the intricacies of this conversion process. By harnessing the synergy between fuzzy set theory and the unique characteristics of near rings, this hybrid structure provides a more effective and efficient means to overcome the complexities associated with converting fuzzy sets into a hybrid structure, ultimately resolving this issue.

  • Initially, the estimation of a hybrid structure is employed as a crucial step in the conversion of fuzzy sets into hybrid ideals. Subsequently, the hybrid structure for sub-near rings and ideals is established by creating a comprehensive framework to address the conversion problem effectively.

The paper's content is organized as follows: The literature review is in Sect. "Literature survey", the creative solution is in Sect. "Hybrid structure of fuzzy sets in near rings", the results are in Sect. "Result and discussion", and the paper is wrapped up in Sect. "Conclusion".

Literature survey

Shaw et al. [25] developed a hybrid model that combines the recently proposed physics-based meta-heuristic algorithm Ring Theory (RT) and the well-known swarm intelligence-based meta-heuristic algorithm Particle Swarm Optimization (PSO). Ada Boost classifier was also employed to observe the suggested model's final classification outcomes. The efficiency of the suggested strategy has been assessed using 15 typical real-world datasets with varying imbalance ratios. Comparisons between the RTPSO's performance and that of the PSO, RTEA, and other common under-sampling techniques have been made. The collected findings show that RTPSO is superior to the modern class imbalance problem-solvers that are being compared here. Future work will include the addition of more potent and sophisticated classifiers to the suggested algorithm to provide better results. Also, it can be applied to more engaging and well-liked research issues.

Porselvi et al. [23] to examine the structural properties of rings, this paper introduces the methodologies of hybrid ideals, hybrid nil radicals, hybrid semi-prime ideals, and hybrid products of rings. It is proved for each hybrid ideal that the hybrid intersection of the hybrid radical is equivalent to the hybrid intersection of the hybrid radical. A hybrid semi-prime ideal and an equivalent statement for the hybrid radical of the hybrid intrinsic product were also deduced. In the future, researchers will look into the structural properties of hybrid primes and hybrid quasi-primes in algebraic frameworks.

Elavarasan et al. [13] proposed the concept of hybrid ideals in near rings and offered the concept. Attempts have been made to characterize near-rings in terms of hybrid ideal structures using illustrative examples. The connection between the hybrid intersection and hybrid product of the hybrid left (right) ideals in zero-symmetric near-rings has been identified. It has been planned to use the concepts and findings from this study to define the concept of hybrid prime (or semi) ideals as well as to determine their various qualities and associated requirements for a hybrid ideal to be a hybrid prime (or semi) ideal in near-rings.

Meenakshi et al. [21] presented the concept of hybrid ideals in near-subtraction semigroups and investigated the characteristics of a near-subtraction semigroup's hybrid ideal under homomorphism mapping, as well as the ideals that were formed. Using the concepts and results presented in this work, it is intended to illustrate the idea of a hybrid prime (or semi) ideal and its associated properties for a hybrid ideal to be a hybrid prime (or semi) ideal in near-subtraction semigroups.

Williams et al. [29] Introduced and investigated the concept of a fuzzy ideal in a near-subtraction semigroup. Fuzzy ideals in near-subtraction semigroups are illustrated. Chain conditions for the fuzzy ideals are also discussed and some theorems were introduced. Normal fuzzy ideals are also introduced in near subtraction semigroup. By developing the concept of fuzzy ideals in the ring structure, future research may build on this finding.

Anis et al. [8] introduced the ideas of the hybrid left (or right) ideals in semigroups as well as hybrid sub-semigroups and looked at some of its facets. Sub-semigroups are described using these concepts, and left (and right) ideals are analyzed. It also introduces the concept of hybrid products, which is used to describe hybrid sub-semigroups and hybrid left (or right, depending on the situation) ideals. The relationships between hybrid intersection and hybrid product are illustrated. Future research will address decision-making challenges with hybrid structures in pertinent algebraic structures utilizing the concepts and findings presented in this study.

Asif et al. [9] the PFSNR and PFIs of NRs were presented along with some of their algebraic features in this publication as benefits of the image fuzzy set. Also shows that the union of two PFIs of an NR is a PFI of that NR and that the positive integral powers of a PFI of an NR are a PFI. A PFI is defined as the direct product of any two PFIs of NRs, and the direct product of PFIs of NRs. Future research may examine problems with decision-making based on Pythagorean fuzzy sets. We shall look into the Pythagorean fuzzy near-rings and Pythagorean analysis.

Zhan et al. [32] presented new types of fuzzy ideals in the near rings. Here, the near ring's fuzzy and sub-near rings (ideal) are taken into account. New character traits are also provided. In particular, the ideas of (strong) prime fuzzy ideals of near-rings are introduced, and their connection to prime fuzzy ideals of near-rings is discussed. However, in the future, there is a need to enhance this type of fuzzy ideals in any algebraic structure.

Ahmed et al. [1] introduced a new hybrid meta-heuristic FS model called Ring Theory based Harmony Search, which is based on the recently suggested Ring Theory based Evolutionary Algorithm (RTEA) and the well-known meta-heuristic Harmony Search (HS) algorithm (RTHS). By using it on 18 standard UCI datasets and contrasting it with 10 cutting-edge meta-heuristic FS approaches, the effectiveness of RTHS has been assessed. The data collected show that RTHS is superior to the cutting-edge methods utilized in this study as a comparison. In the future, the proposed RTHS to address other well-known and engaging research issues, such as the identification of musical symbols, handwritten or printed scripts, and face emotion recognition. High-dimensional datasets like gene expression data can be used with RTHS.

Linesawat et al. [18] This paper's major focus is on the study of ordered semigroups within the framework of anti-hybrid interior ideals. It introduces the idea of anti-hybrid interior ideals in ordered semigroups. This also demonstrates that in several specific classes of ordered semigroups regular, intra-regular, and semisimple the ideas of ideals and interior coincide. Lastly, it is thought about how anti-hybrid interior ideals characterize semi-simple ordered semigroups. In the future, the concepts discussed in this paper will be used to study groups, BCI/BCK algebras, semi-rings, ordered hyperstructures, hemi rings, and hyperstructures in general.

From the analysis, it is noted that [25] include the addition of more potent and sophisticated classifiers [23], improving the structural characteristics of hybrid primes and hybrid quasi-primes in algebraic structures [13], to provide a definition of the concept of hybrid prime (or semi) ideals, [21] need to discuss the properties of hybrid prime ideals [29], extending the idea of fuzzy ideals in ring structure [8], The decision-making challenges to be examined are the hybrid structures in pertinent algebraic structures [9], The Pythagorean fuzzy near-rings and Pythagorean analysis to be investigated [32], need to introduce the new type of fuzzy ideals in all algebraic structure [1], combining this with traditional meta-heuristic algorithms and 18 need to study about groups, BCI/BCK algebras, semi rings, ordered hyper structures, hemi rings, and hyper structures in general.

Hybrid structure of fuzzy sets in near rings

A hybrid structure is a design that makes use of many hierarchical reporting structures. Hybrid structure is most important in algebraic structure which is used in groups, rings, semi-groups, near rings, near subtraction rings, hemi rings, etc. An ideal of a near-ring in ring theory is a specific subset of its elements in the field of abstract algebra. Ideals, such as even numbers or multiples of three, extend particular subsets of integers that create uncertainty at the core of all complexity in the actual world. Real-world information is frequently vague and ill-defined. The mathematical tools that are available to portray the real world are insufficient when describing rigorous and precise data. As a result, there is always some mismatch between the accuracy of reality's quantitative model and its haziness. Ring theory has therefore been extensively used in numerous studies, but there is some uncertainty in transforming fuzzy sets to hybrid structures of any algebraic structure. Many methods have been used in research. As a result, the hybrid structure of fuzzy sets in near rings is presented, in which fuzzy ideals and fuzzy maximal ideals are both converted to hybrid ideals. The hybrid structure is first defined for hybridization, and then the sub-near rings and near rings are also identified. Next, a hybrid structure made up of sub-nearby rings and principles is presented. The fuzzy ideals and fuzzy maximal ideals are changed into hybrid ideals and hybrid maximal ideals as a result. The nearby rings are then subjected to the hybrid structure. The outcome produced by the suggested model, which successfully addressed the uncertainty issues, and the efficacy of the suggested strategy demonstrate the best class, mean, worst class, and temporal complexity.

Figure 1 shows the overall procedure for the hybridization of fuzzy sets in near rings. For hybridization first, the hybrid structure is estimated and then the hybrid structure of near rings and ideals is introduced. Finally, the fuzzy sets that are fuzzy ideal and fuzzy maximal ideal are converted to hybrid ideals in near rings.

Fig. 1
figure 1

Flow diagram for hybridization of fuzzy sets

Full size image

Hybrid structures of fuzzy sets

For converting fuzzy sets to hybrid ideals, there is a need to estimate the hybrid structure. Let take \(\mathcal{B}(U)\) be a primary universal set's power set. The unit interval is indicated by \(U.\)

Definition 3.1.1

(Partial Ordered Set) It is specified that a mapping is a hybrid structure in N over U.

$${\widetilde{a}}_{\lambda } := \left(\widetilde{a}, \lambda \right): {\mathcal{N}} \rightarrow {\mathcal{B}} (U) \times I,{\ell} \,\, \mapsto \,\, \left({\widetilde{a}}({\ell}),\lambda ({\ell}) \right) $$

where \(\widetilde{a}:\mathcal{N}\to \mathcal{B}\left(U\right)\) and \(\gamma :N\to I\) are mappings. Let \(\mathcal{H}(\mathcal{N})\) be the set of all hybrid structures of N over U. Define the relation \(\ll \) is as follows:

$$\left(\forall {\widetilde{x}}_{\gamma }, {\widetilde{y}}_{\lambda }\,\,\epsilon\,\, \mathcal{H}(\mathcal{N})\right)\left({\widetilde{x}}_{\gamma }\ll {\widetilde{y}}_{\lambda }\Leftrightarrow \widetilde{x}\,\,\widetilde{\subseteq }\,\,\widetilde{y}, \gamma\,\, \succcurlyeq \,\,\lambda \right)$$

where \(\widetilde{x}\,\,\widetilde{\subseteq }\,\,\widetilde{y}\) means that \(\widetilde{x}({\ell})\,\,\widetilde{\subseteq }\,\,\widetilde{y}({\ell})\) and \(\gamma \succcurlyeq \lambda \) means that \(\gamma ({\ell})\succcurlyeq \lambda ({\ell})\forall {\ell}\in \mathcal{N}.\) Then \(\mathcal{H}(\mathcal{N})\) is a partial order set.

Definition 3.1.2

(Hybrid Structure) The hybrid intersection of \({\widetilde{x}}_{\gamma }\) and \({\widetilde{y}}_{\lambda }\) is denoted by \({\widetilde{x}}_{\gamma } \Cap {\widetilde{y}}_{\lambda }\) where \({\widetilde{x}}_{\gamma },{\widetilde{y}}_{\lambda }\in \mathcal{H}(\mathcal{N}).\) This is defined to be a hybrid structure.

$$ \tilde{x}_{\gamma } \Cap \tilde{y}_{\lambda } :{\mathcal{N}} \to {\mathcal{B}}\left( U \right) \times I, \ell \mapsto \left( {(\tilde{x}_{\gamma } \widetilde \Cap \tilde{y}_{\lambda } )\left( \ell \right), \left( {\lambda \vee \gamma } \right)\left( \ell \right)} \right) $$

where \({\widetilde{x}}_{\gamma }{\Cap}{\widetilde{y}}_{\lambda }:\mathcal{N}\to \mathcal{B}\left(U\right),{\ell} \mapsto \widetilde{x}({\ell})\cap \widetilde{y}({\ell}),\) and \(\lambda \vee \gamma :\mathcal{N}\to I,{\ell} \mapsto \lambda \left({\ell}\right)\vee \gamma \left({\ell}\right).\)

Definition 3.1.3

(Hybrid Cut) Let \(\tilde{a}_{\gamma } \in {\mathcal{H}}\left( {\mathcal{N}} \right), \beta \in {\mathcal{B}}\left( U \right)\) and \(\delta \in I.\) Then the set \(\tilde{a}_{\gamma } \left[ {\beta ,\delta } \right] = \left\{ {\ell \in {\mathcal{N}}\left| {\tilde{a}\left( l \right) \,\,\supseteq\,\, \beta , \gamma \left( \ell \right) \le \delta } \right.} \right\}\) is called the \(\left[ {\beta ,\delta } \right]\)- a hybrid cut of \(\tilde{a}_{\gamma }.\)

Definition 3.1.4

(Product of hybrid structure) Let \({\widetilde{x}}_{\gamma },{\widetilde{y}}_{\lambda }\in \mathcal{H}\left(\mathcal{N}\right).\) Then the product of a hybrid \({\widetilde{x}}_{\gamma }\) and \({\widetilde{y}}_{\lambda }\) is denoted by \({\widetilde{x}}_{\gamma }\,\,\widetilde{\circ } \,\,{\widetilde{y}}_{\lambda }\) and is defined to be a hybrid structure \({\widetilde{x}}_{\gamma }\,\,\,\odot\,\,\, {\widetilde{y}}_{\lambda }=\left({\widetilde{x}}_{\gamma }\,\,\widetilde{\circ } \,\,{\widetilde{y}}_{\lambda }, \lambda \,\,\widetilde{\circ } \,\,\gamma \right)\) in \(\mathcal{N}\) over U.

Where \({\widetilde{x}}_{\gamma }\,\,\widetilde{\circ } \,\,{\widetilde{y}}_{\lambda }=\left\{\!\!\begin{array}{c}\bigcup_{b=mn}\left\{\widetilde{x}\left(m\right)\cap \widetilde{y}\left(m\right)\right\} \,\,if \,\,\exists\,\, m, n\in \mathcal{N}:b = mn \\ \varnothing otherwise\end{array}\right.\) and \(\lambda \,\,\widetilde{\circ } \,\,\gamma =\left\{ \bigwedge \limits_{b=mn} \begin{array}{ll} \left\{ \lambda (n) \vee \gamma (n)\right\} \,\,if \,\,\exists m, & n \in N:b=mn\\ 1& otherwise\end{array}\right.,\)

for all \(b\in \mathcal{N}.\)

Definition 3.1.5

(Characteristic of hybrid structure) Let \({\widetilde{y}}_{\gamma }\in \mathcal{H}\left(\mathcal{N}\right).\) For \({\varnothing }\ne A\subseteq \mathcal{N},\) the characteristic hybrid structure in \(\mathcal{N}\) over U is denoted by \({{\aleph }}_{A}\left({\widetilde{y}}_{\gamma }\right) = \left({{\aleph }}_{A}\left(\widetilde{y}\right), {{\aleph }}_{A}\left(\gamma \right)\right):\mathcal{N}\to \mathcal{B}\left(U\right)\times I, l\,\, \mapsto \,\,\left({{\aleph }}_{A}\left(\widetilde{y}\right)\left({\ell}\right),{{\aleph }}_{A}\left(\gamma \right)\left({\ell}\right)\right).\)

Where \({\aleph }_{A}\left(\widetilde{y}\right):\mathcal{N}\to \mathcal{B}\left(U\right),{\ell}\quad \mapsto \quad \left\{\begin{array}{c}U\quad if \quad l\in A\\ \varnothing \quad {\rm otherwise}\end{array}\right.\)

And \({\aleph }_{A}(\lambda ):{\mathcal{N}} \to I,{\ell}\,\, \mapsto \,\,\left\{ \begin{array}{l} 0\quad if \quad l\in A \\ 1 \quad {\rm otherwise}\end{array}\right.\)

Definition 3.1.6

(Near ring) A near ring is an algebraic structure that resembles a ring and which permits the satisfaction of one distributive law (either the left or right).Let \(R\) be a set that is not empty. If the following conditions are met: \((R,+,.)\) is a near ring; \((R,.)\) is a semigroup; and \(x.(y+z)=x.y+x.z\) for all \(x,y,z\epsilon R\) is a valid Left distributive law. Additionally, each ring is a near ring.

Definition 3.1.7

(Fuzzy ring) A fuzzy set \(A\) on R is said to be a fuzzy ring on R if for every \(x, y \in R:\)

  1. (i)

    \(A(x-y) \ge min \{A(x), A(y)\};\) and

  2. (ii)

    \(A(x.y) \ge min \{ A(x), A(y) \}.\)

Sometimes, to emphasize the role of the ring, A is said to be a fuzzy subring of R.

Definition 3.1.8

(Fuzzy ideal) A fuzzy ring A on a ring R is said to be a fuzzy left ideal if \(A(x\cdot y) \ge A(y), \forall x, y \in R\) and a fuzzy right ideal if \(A(x\cdot y) \ge A(x), \forall x, y \in R.\) \(A\) is called a fuzzy ideal if it is both a fuzzy left ideal and a fuzzy right ideal.

In other words, a fuzzy set \(A\) on R is a fuzzy ideal if, \(\forall x, y \in R\)

  1. (i)

    \(A\left(x-y\right)\ge min\left\{A\left(x\right),A\left(y\right)\right\}\)

  2. (ii)

    \(A\left(x\cdot y\right)\ge max\left\{A\left(x\right),A\left(y\right)\right\}\)

Definition 3.1.9

(Fuzzy maximal ideal) Let A be a non-constant fuzzy left (right) ideal of R. Then A is called a fuzzy maximal left [right] ideal of R if for any fuzzy left (right) ideal B of R is defined as follows:

$$A\subset B\Rightarrow {A}_{o}={B}_{o} \quad or\quad B=\chi R$$

Definition 3.1.10

(Hybrid structure of sub-near ring) Let \({\widetilde{a}}_{\gamma }\in \mathcal{H}\left(\mathcal{N}\right)\). Then \(\widetilde{a}\) is called a hybrid sub-near ring of \(\mathcal{N}\) over U, if the following assertions are valid.

$$(\mathcal{H}1)\quad\left(\forall {{\ell}}_{1}, {\ell}\in \mathcal{N}\right)\left(\begin{array}{c}\widetilde{a}\left({{\ell}}_{1}- {\ell}\right)\,\,\supseteq\,\, \widetilde{a}\left({{\ell}}_{1}\right)\cap \widetilde{a}\left({\ell}\right)\\ \gamma \left({{\ell}}_{1}- {\ell}\right)\le \gamma \left({{\ell}}_{1}\right)\vee \gamma \left({\ell}\right)\end{array}\right)$$
$$(\mathcal{H}2) \quad \left(\forall k, {\ell}\in \mathcal{N}\right)\left(\begin{array}{c}\widetilde{a}\left(k {\ell}\right)\,\,\supseteq\,\, \widetilde{a}\left(k\right)\cap \widetilde{a}\left({\ell}\right)\\ \gamma \left(k {\ell}\right)\le \gamma \left(k\right)\vee \gamma \left({\ell}\right)\end{array}\right)$$

\(\Rightarrow \gamma \left({\ell}\right)\ge \gamma \left(m-{\ell}\right)\) for all \(m,{\ell}\in \mathcal{N}\)

Definition 3.1.11

(Hybrid ideal) A hybrid sub near ring \({\widetilde{a}}_{\gamma }\) in \(\mathcal{N}\) over U is called a hybrid ideal of \(\mathcal{N}\) over U, if the below statements are true,

$$(\mathcal{H}3)\quad\left(\forall m, n\in \mathcal{N}\right)\left(\begin{array}{c}\widetilde{a}\left(m+n-m\right)\,\,\supseteq\,\, \widetilde{a}\left(n\right)\\ \gamma \left(m+n-m\right)\le \gamma \left(n\right)\end{array}\right)$$
$$(\mathcal{H}4)\quad\left(\forall m, n\in \mathcal{N}\right)\left(\begin{array}{c}\widetilde{a}\left(nm\right)\,\,\supseteq\,\, \widetilde{a}\left(n\right)\\ \gamma \left(nm\right)\le \gamma \left(n\right)\end{array}\right)$$
$$(\mathcal{H}5)\quad\left(\forall m,r,s\in \mathcal{N}\right)\left(\begin{array}{c}\widetilde{a}\left(m\left(s+r\right)-mr\right)\,\,\supseteq\,\, \widetilde{a}\left(s\right)\\ \gamma \left(m(S+r)-mr\right)\le \gamma \left(s\right)\end{array}\right).$$

Note that the hybrid structure \({\widetilde{a}}_{\gamma }\) is a hybrid right ideal of \(\mathcal{N}\) if it satisfies \(\left(\mathcal{H}1\right),\left(\mathcal{H}2\right),(\mathcal{H}3)\) and \((\mathcal{H}4)\) and the hybrid structure \({\widetilde{a}}_{\gamma }\) is a hybrid left ideal of \(\mathcal{N}\) if it satisfies \(\left(\mathcal{H}1\right),\left(\mathcal{H}2\right),(\mathcal{H}3)\) and \((\mathcal{H}5).\)

Definition 3.1.12

(Hybrid maximal ideal) Let \(R\) be a near ring. An ideal \({\widetilde{a}}_{\gamma }\) of \(R\) is hybrid maximal if and only if \({\widetilde{a}}_{\gamma }\,\,\,{\subsetneq}\,\,\,R\) and there is no ideal \(\mathcal{N}\) of \(R\) such that \({\widetilde{a}}_{\gamma }\,\,\,{\subsetneq}\,\,\,\mathcal{N}\,\,\,{\subsetneq}\,\,\,R.\)

This is true if and only if \({\widetilde{a}}_{\gamma }\) is the largest element in the collection of all proper hybrid ideals of R, arranged in the subset relation.

Hence, this section discussed the basic definitions of fuzzy sets and rings. In the next section, based on the above definitions, theoretical proofs were given.

Theoretical proof

The theoretical proofs for uncertainty problems complexity has been provided in this subsection.

Lemma 3.2.1

Every hybrid sub-near ring \({\widetilde{a}}_{\gamma }\) of \(\mathcal{N}\) over U satisfies the following:

$$\left(\forall {g}^{\prime}\in \mathcal{N}\right)\left(\widetilde{a}\left({g}^{\prime}\right)\subseteq \widetilde{a}\left(0\right),\gamma ({g}^{\prime})\ge \gamma (0)\right)$$

Proof

This lemma provides the condition for hybrid sub-near rings.

Note that \({g}^{\prime}-{g}^{\prime}=0\forall {g}^{\prime}\in \mathcal{N}\). Hence \(\widetilde{a}\left(0\right)=\widetilde{a}\left({g}^{\prime}-{g}^{\prime}\right)\,\,\supseteq\,\, \widetilde{a}({g}^{\prime})\cap \widetilde{a}({g}^{\prime})\)

\(\gamma \left(0\right)=\gamma \left({g}^{\prime}-{g}^{\prime}\right)\le \gamma \left({g}^{\prime})\vee {\gamma (g}^{\prime})\right)=\gamma ({g}^{\prime})\) for all \({g}^{\prime}\in \mathcal{N}\).

Some equivalent conditions for the hybrid structure of near rings are discussed in the below preposition.

Proposition 3.2.1

The following statements are equivalent for a hybrid sub-near ring \({\widetilde{a}}_{\gamma }\) of \(\mathcal{N}\) over U.

  • \(\left(\forall m, {\ell}\in \mathcal{N}\right)\left(\widetilde{a}\left(m-{\ell}\right)\,\,\supseteq\,\, \widetilde{a}\left({\ell}\right),\gamma \left(m-{\ell}\right)\le \gamma ({\ell})\right)\)

  • \(\left(\forall m\in \mathcal{N}\right)\left(\widetilde{a}\left(0\right)=\widetilde{a}\left(m\right),\gamma \left(0\right)=\gamma (m)\right)\)

Proof

This proposition gives the equivalent conditions for a hybrid sub-near ring \({\widetilde{a}}_{\gamma }\) of \(\mathcal{N}\) over U. Take \({\ell}=0\) in equivalent (i).

Then \(\widetilde{a}\left(0\right)\subseteq \widetilde{a}\left(m-0\right)=\widetilde{a}\left(m\right)\) and \(\gamma \left(0\right)\ge \gamma \left(m-0\right)=\gamma \left(m\right)\) for all \(m\in \mathcal{N}\). Here \(\widetilde{a}\left(0\right)=\widetilde{a}\left(m\right),\gamma \left(0\right)=\gamma (m)\), for all \(m\in \mathcal{N}\) by combining the above equation and Lemma 3.2.1

Conversely, assume that (ii) is true. Then

$$\begin{aligned} \widetilde{a}\left({\ell}\right) & =\widetilde{a}\left(0\right)\cap \widetilde{a}\left({\ell}\right)\\ & =\widetilde{a}\left(m\right)\cap \widetilde{a}\left({\ell}\right)\\ & \subseteq \widetilde{a}\left(m-{\ell}\right)\\ & \Rightarrow \widetilde{a}\left({\ell}\right)\subseteq \widetilde{a}\left(m-{\ell}\right) \end{aligned}$$

(i.e.) \(\widetilde{a}\left(m-{\ell}\right)\,\,\supseteq\,\, \widetilde{a}\left({\ell}\right)\)

$$ \begin{aligned} \gamma \left({\ell}\right)&=\gamma \left(0\right)\vee \gamma \left({\ell}\right)\\ & =\gamma \left(m\right)\vee \gamma \left({\ell}\right)\\ & \gamma \left({\ell}\right)\ge \gamma \left(m-{\ell}\right)\,\, \text{for all}\,\, m,{\ell}\in \mathcal{N}. \end{aligned}$$

This proposition effectively proves the equivalent condition for the hybrid structure of near rings. The next proposition gives the equivalent condition for a hybrid structure of ideals in near rings.

Proposition 3.2.2

For \({\widetilde{x}}_{\gamma }\in \mathcal{H}\left(\mathcal{N}\right),\) the following conditions are equivalent.

  1. (i)

    \(\left(\forall {\ell}, n\in \mathcal{N}\right)\left(\begin{array}{c}\widetilde{x}\left({\ell}+n-{\ell}\right)\,\,\supseteq\,\, \widetilde{x}\left(n\right)\\ \gamma \left({\ell}+n-{\ell}\right)\le \gamma \left(n\right)\end{array}\right)\)

  2. (ii)

    \(\left(\forall {\ell}, n\in \mathcal{N}\right)\left(\begin{array}{c}\widetilde{x}\left({\ell}+n\right)\,\,\supseteq\,\, \widetilde{x}\left(n+{\ell}\right)\\ \gamma \left({\ell}+n\right)\le \gamma \left(n+{\ell}\right)\end{array}\right)\)

Proof

This proposition provides the hybrid structure of ideals in the ring.

Assume \(n=n+{\ell}\) in the equivalent (i)

$$\widetilde{x}\left(z+{\ell}\right)\subseteq \widetilde{x}\left({\ell}+n+{\ell}-{\ell}\right)=\widetilde{x}\left({\ell}+n\right)$$

and \(\gamma \left(n+{\ell}\right)\ge \gamma \left({\ell}+n+{\ell}-{\ell}\right)=\gamma \left({\ell}+n\right) \forall n,{\ell}\in \mathcal{N}\)

Conversely, assume \(n=n-{\ell}\) in the equivalent (ii).

Then \(\widetilde{x}\left({\ell}+n-{\ell}\right)\,\,\supseteq\,\, \widetilde{x}\left(n-{\ell}+{\ell}\right)=\widetilde{x}\left(n\right)\)

and \(\gamma \left({\ell}+n-{\ell}\right)\le \gamma \left(n-{\ell}+{\ell}\right)=\gamma \left(s\right)\) for any \({\ell},n\in \mathcal{N}\)

Hence this proposition gives the hybrid structure of ring ideals.

Proposition 3.2.3

For \({\widetilde{x}}_{\gamma }\in \mathcal{H}\left(\mathcal{N}\right)\) and \(\varnothing \ne L\subseteq \mathcal{N},\) the following assertions are equivalent:

  1. (i)

    L is a left (resp., right) ideal of \(\mathcal{N}\)

  2. (ii)

    \(\chi L({\widetilde{x}}_{\gamma })\) is a hybrid left (resp., right) ideal in \(\mathcal{N}\).

Proof

This proposition provides the equivalent conditions for hybrid ideals of fuzzy sets. Assume that L is a left ideal of \(\mathcal{N}\) and let \(b,m\in \mathcal{N}\)

If \(m\ne L\) or \(b\ne L,\)

Then \(\chi L\left({\widetilde{x}}_{\gamma }\right)\left(m-b\right)\,\,\supseteq\,\, \varnothing =\chi L(\widetilde{x})\left(m\right)\cap \chi L(\widetilde{x})\left(b\right)\) and \(\chi L\left(\gamma \right)\left(m-b\right)\le 1=\chi L\left(\gamma \right)\left(m\right)\vee \chi L\left(\gamma \right)\left(b\right)\)

If \(b,m\in L.\)

Then \(m-b\in L,\) \(\chi L\left(\widetilde{x}\right)\left(m-b\right)=U= \chi L(\widetilde{x})\left(m\right)\cap \chi L(\widetilde{x})\left(b\right)\) and \(\chi L\left(\gamma \right)\left(m-b\right)=0=\chi L\left(\gamma \right)\left(m\right)\vee \chi L\left(\gamma \right)\left(b\right)\)

If \(m\in L,\)

Then \(b+m-b\in L\) and hence \(\chi L\left(\widetilde{x}\right)\left(b+m-b\right)=U= \chi L(\widetilde{x})\left(m\right)\) and \(\chi L\left(\gamma \right)\left(b+m-b\right)=0=\chi L\left(\gamma \right)\left(m\right)\)

Clearly, \(\chi L\left(\widetilde{x}\right)\left(b+m-b\right)\,\,\supseteq\,\, \varnothing =\chi L\left(\widetilde{x}\right)\left(b\right)\) and \(\chi L\left(\gamma \right)\left(b+m-b\right)\le 1=\chi L\left(\gamma \right)\left(m\right).\)

Now, let \(m,c,d\in \mathcal{N}.\)

If \(m\in L,\) then \(c\left(m+d\right)-cd\in L .\)

Thus \(\chi L\left(\widetilde{x}\right)\left(c\left(m+d\right)-cd\right)=U=\chi L\left(\widetilde{x}\right)(m)\) and \(\chi L\left(\gamma \right)\left(c\left(m+d\right)-cd\right)=0=\chi L\left(\gamma \right)\left(m\right).\)

If \(m\notin L,\) then \(\chi L\left(\widetilde{x}\right)\left(c\left(m+d\right)-cd\right)\,\,\supseteq\,\, \varnothing =\chi L\left(\widetilde{x}\right)\left(m\right)\) and \(\chi L\left(\gamma \right)\left(c\left(m+d\right)-cd\right)\le 1=\chi L\left(\gamma \right)\left(m\right).\)

Therefore, \(\chi L\left(\widetilde{x}\right)\) is the left ideal in \(\mathcal{N}.\)

Conversely, let \(n,b\in L.\) Then \(\chi L\left(\widetilde{x}\right)\left(n-b\right)\,\,\supseteq\,\, \chi L\left(\widetilde{x}\right)\left(n\right)\cap \chi L\left(\widetilde{x}\right)\left(b\right)=U\) and

\(\chi L\left(\gamma \right)\left(n-b\right)\le \chi L\left(\gamma \right)\left(n\right)\vee \chi L\left(\gamma \right)\left(b\right)=0,\) so \(n-b\in L.\)

Let \(b\in \mathcal{N}\) and \(n\in L,\) then \(\chi L\left(\widetilde{x}\right)\left(b+n-b\right)\,\,\supseteq\,\, \chi L\left(\widetilde{x}\right)\left(n\right)=U\) and \(\chi L\left(\gamma \right)\left(b+n-b\right)\le \chi L\left(\gamma \right)\left(n\right)=0\)

Hence, \(b+n-b\in L.\)

Now let \(n\in L\) and \(d,k\in \mathcal{N}.\) Then \(\chi L\left(\widetilde{x}\right) \big(k (n+d) -kd\big)\,\,\supseteq\,\, \chi L\left(\widetilde{x}\right) (n)=U\) and \(\chi L\left(\gamma \right)\left(k\left(n+d\right)-kd\right)\le \chi L\left(\gamma \right)\left(n\right)=0.\)

Hence, \(\left(k\left(n+d\right)-kd\right)\in L.\)

Therefore, \(L\) is the left ideal of \(\mathcal{N}.\)

Hence, this proposition provides the equivalent conditions for hybrid ideals in fuzzy sets.

Theorem 3.2.1

For a hybrid left (resp., right) ideal \({\widetilde{y}}_{\gamma }\) in \(\mathcal{N}\) over U, the set \({\mathcal{N}}_{{\widetilde{y}}_{\gamma }}:=\left\{n\in \mathcal{N}\left|\widetilde{y}\left(s\right)=\widetilde{y}\left(0\right),\gamma \left(s\right)\right.=\gamma \left(0\right)\right\}\) is a left (resp., right) ideal of \(\mathcal{N}.\)

Proof

This theorem provides the hybrid structure of the left (resp., right) ideal of fuzzy sets. Assume that \({\widetilde{y}}_{\gamma }\) is a hybrid left ideal in \(\mathcal{N}\) over U.

Let \(m,l\in {\mathcal{N}}_{{\widetilde{y}}_{\gamma }}.\) Then \(\widetilde{y}\left(m\right)=\widetilde{y}\left(0\right),\) \(\widetilde{y}(l)=\widetilde{y}\left(0\right),\) \(\gamma \left(m\right)=\gamma \left(0\right)\) and \(\gamma (l)=\gamma \left(0\right).\) It follows from \((\mathcal{H}1)\) that \(\widetilde{y}\left(m-l\right)\,\,\supseteq\,\, \widetilde{y}\left(m\right)\cap \widetilde{y}(l)=\widetilde{y}\left(0\right)\) and \(\gamma \left(m-l\right)\le \gamma \left(m\right)\vee \widetilde{y}(l)=\gamma \left(0\right).\) By Lemma 3.6.1, \(\widetilde{y}\left(m-l\right)\subseteq \widetilde{y}\left(0\right)\) and \(\gamma \left(m-l\right)\ge \gamma \left(0\right).\) Thus \(\widetilde{y}\left(m-l\right)=\widetilde{y}\left(0\right)\) and \(\gamma \left(m-l\right)= \gamma \left(0\right).\) Hence, \(m-l\in {\mathcal{N}}_{{\widetilde{y}}_{\gamma }}.\)

Let \(l\in \mathcal{N}\) and \(m\in {\mathcal{N}}_{{\widetilde{y}}_{\gamma }}.\) Then \(\widetilde{y}\left(m\right)=\widetilde{y}\left(0\right)\) and \(\gamma \left(m\right)=\gamma \left(0\right).\) It follows from \((\mathcal{H}3)\) that \(\widetilde{y}\left(l+m-l\right)\,\,\supseteq\,\, \widetilde{y}\left(m\right)=\widetilde{y}\left(0\right)\) and \(\gamma \left(l+m-l\right)\le \gamma \left(m\right)=\gamma \left(0\right).\) By Lemma 3.6.1, \(\widetilde{y}\left(l+m-l\right)\subseteq \widetilde{y}\left(0\right)\) and \(\gamma \left(l+m-l\right)\ge \gamma \left(0\right).\) Thus \(\widetilde{y}\left(l+m-l\right)=\widetilde{y}\left(0\right)\) and \(\gamma \left(l+m-l\right)= \gamma \left(0\right).\) Hence, \(l+m-l\in {\mathcal{N}}_{{\widetilde{y}}_{\gamma }}.\)

Let \(l\in {\mathcal{N}}_{{\widetilde{y}}_{\gamma }};\) \(c,d\in \mathcal{N}.\) Then \(\widetilde{y}(l)=\widetilde{y}\left(0\right).\) It follows from \((\mathcal{H}5)\) that \(\widetilde{y}\left(c\left(l+d\right)-cd\right)\,\,\supseteq\,\, \widetilde{y}(l)=\widetilde{y}\left(0\right)\) and \(\gamma \left(c\left(l+d\right)-cd\right)\le \gamma (l)=\gamma \left(0\right).\) By Lemma 3.6.1, \(\widetilde{y}\left(c\left(l+d\right)-cd\right)\subseteq \widetilde{y}\left(0\right)\) and \(\gamma \left(c\left(l+d\right)-cd\right)\ge \gamma \left(0\right).\) Thus \(\widetilde{y}\left(c\left(l+d\right)-cd\right)=\widetilde{y}\left(0\right)\) and \(\gamma \left(c\left(l+d\right)-cd\right)= \gamma \left(0\right).\) Hence, \(c\left(l+d\right)-cd\in {\mathcal{N}}_{{\widetilde{y}}_{\gamma }}.\)

Therefore, \({\mathcal{N}}_{{\widetilde{y}}_{\gamma }}\) is a left ideal of \(\mathcal{N}.\)

Hence, this theorem proves the hybrid structure of left (resp., right) ideals of fuzzy sets.

Proposition 3.2.4

For \({\widetilde{y}}_{\gamma },{\widetilde{f}}_{\alpha }\in \mathcal{H}\left(\mathcal{N}\right),\) the below assertions are valid:

  1. (i)

    If \({\widetilde{y}}_{\gamma }\) and \({\widetilde{f}}_{\alpha }\) are hybrid left (resp., right) ideals in \(\mathcal{N}.\) Then \({\widetilde{y}}_{\gamma }{\Cap}{\widetilde{f}}_{\alpha }\) is a hybrid left (resp., right) ideals in \(\mathcal{N}.\)

  2. (ii)

    If \(\mathcal{N}\) is zero-symmetric and if \({\widetilde{y}}_{\gamma }\) is a hybrid right ideal and \({\widetilde{f}}_{\alpha }\) is a hybrid left ideal of \(\mathcal{N},\) then \({\widetilde{y}}_{\gamma }\,\,\,\odot\,\,\, {\widetilde{f}}_{\alpha }\ll {\widetilde{y}}_{\gamma }{\Cap}{\widetilde{f}}_{\alpha },\)

Proof

This proposition provides some assertions for the hybrid left (right) ideals in fuzzy sets.

Proof of (i):


Assume that \({\widetilde{y}}_{\gamma }\) and \({\widetilde{f}}_{\alpha }\) are hybrid left ideals in \(\mathcal{N}\) over \(U.\)

Let \({l}_{1},m\in \mathcal{N}.\) Then \(\begin{aligned} \left(\widetilde{y}\widetilde{\cap }\widetilde{f}\right)\left({l}_{1}-m\right) & = \widetilde{y}\left({l}_{1}-m\right)\cap \widetilde{f}\left({l}_{1}-m\right)\\ & \,\,\supseteq\,\, \left(\widetilde{y}\left({l}_{1}\right)\cap \widetilde{y}\left(m\right)\right)\cap \left(\widetilde{f}\left({l}_{1}\right)\cap \widetilde{f}\left(m\right)\right)\\ & =\left(\widetilde{y}\left({l}_{1}\right)\cap \widetilde{f}\left({l}_{1}\right)\right)\cap \left(\widetilde{y}\left(m\right)\cap \widetilde{f}\left(m\right)\right)\\ & =\left(\widetilde{y}\cap \widetilde{f}\right)\left({l}_{1}\right)\cap \left(\widetilde{y}\cap \widetilde{f}\right)\left(m\right)\end{aligned} \)

\(\begin{aligned} \left(\gamma \vee \alpha \right)\left({l}_{1}-m\right) & =\gamma \left({l}_{1}-m\right)\vee \alpha \left({l}_{1}-m\right)\\ & \le \left\{\gamma \left({l}_{1}\right)\vee \gamma \left(m\right)\right\}\vee \left\{\alpha \left({l}_{1}\right)\vee \alpha \left(m\right)\right\}\\ & = \left\{\gamma \left({l}_{1}\right)\vee \alpha \left({l}_{1}\right)\right\}\vee \left\{\gamma \left(m\right)\vee \alpha \left(m\right)\right\}\\ & =\left(\gamma \vee \alpha \right)\left({l}_{1}\right) \vee \left(\gamma \vee \alpha \right)\left(m\right)\end{aligned}\)

Also, \(\begin{aligned} \left(\widetilde{y}\widetilde{\cap }\widetilde{f}\right)\left(m+{l}_{1}-m\right) & =\widetilde{y}\left(m+{l}_{1}-m\right)\cap \widetilde{f}\left(m+{l}_{1}-m\right)\\ & \,\,\supseteq\,\, \widetilde{y}\left({l}_{1}\right)\cap \widetilde{f}\left({l}_{1}\right)\\ & =\left(\widetilde{y}\widetilde{\cap }\widetilde{f}\right)\left({l}_{1}\right)\\ \left(\gamma \vee \alpha \right)&\left(m+{l}_{1}-m\right) =\gamma \left(m+{l}_{1}-m\right)\\ &\,\,\vee \alpha \left(m+{l}_{1}-m\right)\\ &\,\,\supseteq\,\, \gamma \left({l}_{1}\right)\vee \alpha \left({l}_{1}\right)\\ & =\left(\gamma \vee \alpha \right)\left({l}_{1}\right) \end{aligned}\)

Now, let \(c,m,l\in \mathcal{N}.\)

\(\begin{aligned} &\mathrm{Then} \quad \left(\widetilde{y}\widetilde{\cap }\widetilde{f}\right)\left(c\left(l+m\right)-cm\right) =\widetilde{y}\left(c\left(l+m\right)-cm\right)\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\cap \widetilde{f}\left(c\left(l+m\right)-cm\right)\\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\supseteq\,\, \widetilde{y}(l)\cap \widetilde{f}(l)\\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=\left(\widetilde{y}\widetilde{\cap }\widetilde{f}\right)(l)\\ &\left(\gamma \vee \alpha \right) \left(c\left(l+m\right)-cm\right) = \gamma \left(c\left(l+m\right)-cm\right)\\ &\,\,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\vee \alpha \left(c\left(l+m\right)-cm\right)\\& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad =\left(\gamma \vee \alpha \right)(l). \end{aligned}\)

Therefore, \({\widetilde{y}}_{\gamma }{\Cap}{\widetilde{f}}_{\alpha }\) is a hybrid left ideal in \(\mathcal{N}.\)

Proof of (ii):

Assume that \({\widetilde{y}}_{\gamma }\) and \({\widetilde{f}}_{\alpha }\) are hybrid right and hybrid left ideals, respectively, of a zero-symmetric near-ring \(\mathcal{N},\) and let \(l \in \mathcal{N}.\)

If \(\exists p,q\in \mathcal{N}:l=pq.\) Then \(\widetilde{y}(l)=\widetilde{y}\left(pq\right)\,\,\supseteq\,\, \widetilde{y}(p)\) and \(\gamma (l)=\gamma \left(pq\right)\le \gamma (p).\)

Also, \(\widetilde{f}(l)=\widetilde{f}\left(pq\right)=\widetilde{f}(p\left(q+0)-p0\right)\,\,\supseteq\,\, \widetilde{f}(q)\) and

$$\alpha (l)=\alpha \left(pq\right)=\alpha \left(p\left(q+0\right)-p0\right)\le \alpha (q)$$

Which implies \(\left(\widetilde{y}\widetilde{\,\,\,\odot\,\,\,}\widetilde{f}\right)(l)=\bigcup_{l=pq}\left\{\widetilde{y}(p)\cap \widetilde{f}(q)\right\}\subseteq \widetilde{y}(l)\cap \widetilde{f}(l)=\left(\widetilde{y}\cap \widetilde{f}\right)(l)\)

$$ \left( \gamma \widetilde{\,\,\,\odot\,\,\,}\alpha \right)(l)= \bigwedge \limits_{l=pq}\left\{\gamma (p)\vee \alpha (q)\right\} \ge \gamma (l) \vee \alpha (l) =\left( \gamma \vee \alpha \right)(l) $$

Therefore, \({\widetilde{y}}_{\gamma }\,\,\,\odot\,\,\, {\widetilde{f}}_{\alpha }\ll {\widetilde{y}}_{\gamma }{\Cap}{\widetilde{f}}_{\alpha }.\)

This proposition gives some assertions for the hybrid left (right) ideals in fuzzy sets.

Proposition 3.2.5

Let \({\widetilde{x}}_{\gamma }\in \mathcal{H}\left(\mathcal{N}\right).\) Then the following equivalents hold:

  1. (i)

    \({\widetilde{x}}_{\gamma }\) is hybrid maximal left (right) ideal in \(\mathcal{N}\) over \(U\)

  2. (ii)

    Let \(\left(\varepsilon ,\delta \right)\in \mathcal{B}(U)\times I.\) The non-empty sets \({L}_{x}^{\varepsilon }\) and \({L}_{\gamma }^{\delta }\) are hybrid maximal left (right) ideal in \(\mathcal{N}\) over \(U\)

Proof

This proposition provides the hybrid structure for the fuzzy maximal ideals.

Assume that \({\widetilde{x}}_{\gamma }\) is hybrid maximal left (right) ideal in \(\mathcal{N}\) over \(U.\) Let \(\varepsilon \in \mathcal{B}(U)\) and \(\delta \in I\) such that \({L}_{x}^{\varepsilon }\ne \varnothing \) and \({L}_{\gamma }^{\delta }\ne \varnothing. \)

If \(,n\in {L}_{x}^{\varepsilon }\cap {L}_{\gamma }^{\delta },\) then\(\widetilde{x}(l)\,\,\supseteq\,\, \varepsilon, \)\(\widetilde{x}\left(n\right)\,\,\supseteq\,\, \varepsilon, \) \(\gamma (l)\le \delta \) and\(\gamma \left(n\right)\le \delta. \) Since, \(\widetilde{x}\left(l-n\right)\,\,\supseteq\,\, \widetilde{x}(l)\cap \widetilde{x}\left(n\right)\,\,\supseteq\,\, \varepsilon \) and \(\gamma \left(l-n\right)\le \gamma (l)\vee \gamma \left(n\right)\le \delta .\) So\(l-n\in {L}_{x}^{\varepsilon }\cap {L}_{\gamma }^{\delta }.\)

Let \(n \in \mathcal{N}\) and \(l \in {L}_{x}^{\varepsilon }\cap {L}_{\gamma }^{\delta }.\) Then \(\widetilde{x}(n+l-n) \,\,\supseteq\,\, \widetilde{x} (l) \,\,\supseteq\,\, \varepsilon \) and \(\lambda (n+l-n) \le \lambda (l) \le \delta \) which imply \(n+l-n \in {L}_{x}^{\varepsilon }\cap {L}_{\gamma }^{\delta }.\)

Now for \(l \in {L}_{x}^{\varepsilon }\cap {L}_{\gamma }^{\delta }\) and \(c,d\in \mathcal{N}.\) Then \(\widetilde{x}(c\left(l+d\right)-cd) \,\,\supseteq\,\, \widetilde{x} (l) \,\,\supseteq\,\, \varepsilon \) and \(\lambda (c\left(l+d\right)-cd) \le \lambda (l) \le \delta \) which implies \(c\left(l+d\right)-cd\in {L}_{x}^{\varepsilon }\cap {L}_{\gamma }^{\delta }.\)

Hence, \({L}_{x}^{\varepsilon }\) and \({L}_{\gamma }^{\delta }\) are hybrid maximal left ideals in \(\mathcal{N}\) over \(U\)

Conversely, assume that \({L}_{x}^{\varepsilon }\) and \({L}_{\gamma }^{\delta }\) are hybrid maximal left ideals in \(\mathcal{N}\) over \(U\) for all \(\left(\varepsilon ,\delta \right)\in \mathcal{B}(U)\times I.\)

Let \(m,n\in \mathcal{N}\) such that \(\widetilde{x}\left(m\right)={\varepsilon }_{a}\) and \(\widetilde{x}\left(n\right)={\varepsilon }_{b}.\) If \(\varepsilon :={\varepsilon }_{a}\cap {\varepsilon }_{b},\) then \(m,n\in {L}_{x}^{\varepsilon }\) and \(m-n\in {L}_{\gamma }^{\delta }\) and so far \(\widetilde{x}\left(m-n\right)\,\,\supseteq\,\, \varepsilon ={\varepsilon }_{a}\cap {\varepsilon }_{b}=\widetilde{x}\left(m\right)\cap \widetilde{x}\left(n\right).\)

Since \(m\in {L}_{x}^{{\varepsilon }_{a}},\) then \(n+m-n\in {L}_{x}^{{\varepsilon }_{a}}\) and \(c\left(m+d\right)-cd\in {L}_{x}^{{\varepsilon }_{a}}\) for all \(c,d\in \mathcal{N}.\) This implies, \(\widetilde{x}\left(n+m-n\right)\,\,\supseteq\,\, {\varepsilon }_{a}=\widetilde{x}\left(m\right)\) and \(\widetilde{x}\left(c\left(m+d\right)-cd\right)\,\,\supseteq\,\, {\varepsilon }_{a}=\widetilde{x}\left(m\right).\)

Suppose \(\gamma \left(m\right)={\delta }_{a}\) and \(\gamma \left(n\right)={\delta }_{b}.\) Then \(\delta :={\delta }_{a}\vee {\delta }_{b}.\) Now, \(m,n\in {L}_{x}^{\varepsilon }\) and \(m-n\in {L}_{\gamma }^{\delta }.\)

This implies \(\gamma \left(m-n\right)\le \delta ={\delta }_{a}\vee {\delta }_{b}=\gamma \left(m\right)\vee \gamma \left(n\right).\) Let \(m\in {L}_{\gamma }^{{\delta }_{a}},\) then \(n+m-n\in {L}_{\gamma }^{{\delta }_{a}}\) and \(c\left(m+d\right)-cd\in {L}_{\gamma }^{{\delta }_{a}}\) which implies \(\gamma \left(n+m-n\right)\le {\delta }_{a}=\gamma \left(m\right)\) and \(\gamma \left(c\left(m+d\right)-cd\right)\le {\delta }_{a}=\gamma \left(m\right).\)

Hence, \({\widetilde{x}}_{\gamma }\) is the hybrid maximal left ideal in \(\mathcal{N}\) over \(U\)

Hence, this proposition proves the hybrid structure of fuzzy maximal ideals.

Corollary 3.2.1.

For a hybrid maximal left (resp., right) ideal \({\widetilde{x}}_{\gamma }\) in \(\mathcal{N}\). The non-empty \(\left[\varepsilon ,m\right]\) hybrid cut of \({\widetilde{x}}_{\gamma }\) is a maximal left (resp., right) ideal of \(\mathcal{N}\forall \varepsilon \in \mathcal{B}(U)\) and \(m\in I.\)

Theorem 3.2.2

For a hybrid maximal left (resp., right) ideal \({\widetilde{x}}_{\gamma }\) in \(\mathcal{N}\), the set \(\Omega :=\left\{l\in \mathcal{N}\left|\widetilde{x}(l)\cap \beta \ne \varnothing ,\gamma \left(z\right)\right.\le \delta \right\}\) is a maximal left (resp., right) ideal of \(\mathcal{N}\forall (\beta , \delta )\in \mathcal{B}(U) \times I\) with \(\beta \ne \varnothing \) whenever it is nonempty.

Proof

This theorem provides that for any hybrid maximal ideals in fuzzy, the given set is always the maximal ideals.

Let \(\left(\beta ,\delta \right)\in \mathcal{B}(U) \times I\) such that \(\Omega \ne \varnothing \ne \beta .\) If \(m,n\in\Omega, \) then \(\widetilde{x}\left(m\right)\cap \beta \ne \varnothing \ne \widetilde{x}\left(n\right)\cap \beta, \) \(\gamma \left(m\right)\le \delta \) and \(\gamma \left(n\right)\le \delta. \)

It follows from \((\mathcal{H}1)\) that \(\widetilde{x}\left(m-n\right)\cap \beta \,\,\supseteq\,\, \left(\widetilde{x}\left(m\right)\cap \widetilde{x}\left(n\right)\right)\cap \beta =\left(\widetilde{x}\left(m\right)\cap \beta \right)\left(\widetilde{x}\left(n\right)\cap \beta \right)\ne \varnothing \) and \(\gamma \left(m-n\right)\le \gamma \left(m\right)\vee \gamma \left(n\right)\le \delta. \) So that \(m-n\in\Omega. \)

Let \(l\in\Omega \) and \(n\in \mathcal{N}.\) Then \(\widetilde{x}(l)\cap \beta \ne \varnothing \) and \(\gamma (l)\le \delta .\)

It follows from \(\left(\mathcal{H}3\right)\) that \(\widetilde{x}\left(n+l-n\right)\cap \beta \,\,\supseteq\,\, \widetilde{x}(l)\cap \beta \ne \varnothing \) and \(\gamma \left(n+l-n\right)\le \gamma (l)\le \delta. \) So that \(n+l-n\in\Omega. \)

Let \(l\in\Omega \) and \(n,b\in \mathcal{N}.\) Then \(\widetilde{x}(l)\cap \beta \ne \varnothing \) and \(\gamma (l)\le \delta .\)

It follows from \(\left(\mathcal{H}5\right)\) that \(\widetilde{x}\left(n(l+d)-nd\right)\cap \beta \,\,\supseteq\,\, \widetilde{x}(l)\cap \beta \ne \varnothing \) and \(\gamma \left(n(l+d)-nd\right)\le \gamma (l)\le \delta. \) So that \(n(l+d)-nd\in\Omega. \)

Hence \(\Omega \) is the maximal left ideal of \(\mathcal{N}.\)

Hence, this theorem proves that for any hybrid maximal ideals in fuzzy, the given set is always the maximal ideals.

Overall, this section provides some basic definitions and theoretical proof of the hybridization of fuzzy sets in near rings. The hybrid structure is defined and then the hybrid structure of rings, ideals, and maximal ideals is provided. Here, the fuzzy ideals and fuzzy maximal ideals are converted to hybrid ideals and hybrid maximal ideals respectively. In the next section, the result obtained for the hybridization of fuzzy sets is explained in detail.

Result and discussion

The results obtained from the proposed model have been provided in this section. The findings demonstrated that the proposed model effectively addressed the uncertainty issues, and a comparison of the proposed strategy with other current strategies further demonstrated its efficacy.

Performance metrics of the proposed model

This section describes in detail the performance metrics of the suggested model, which efficiently solved the uncertainty problems, as well as the effectiveness of the proposed strategy and the attained conclusion.

Figure 2 illustrates the performances of the best value of a proposed model. It reaches its greatest value of 43 when the number of iterations is 20, and its minimum value of 15 when the number of iterations is 40. The best value obtained for the proposed model in the hybrid structure of maximal ideals seeks to strike a balance between various properties of maximal ideals in a near-ring. This includes a trade-off between properties like closure under addition, absorption, or correspondence with maximal sub-near-rings, depending on the specific application or theoretical context. In addition, the obtained best value aims to optimize the algebraic structure of the near-ring by identifying maximal ideals that enhance the manipulation of the near-ring's operations. This leads to more efficient algebraic computations.

Fig. 2
figure 2

Performance of best value in the proposed model

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Figure 3 illustrates the performances of the mean in the proposed model. When there are 4 iterations, the value reaches a maximum of 69, and when there are 3 iterations, the value reaches a low of 25. Here the proposed method offers a unique perspective for analyzing the mean performance. This proposed hybrid structure clarifies the functioning of the mean within maximum ideals by combining both additive and multiplicative elements. This method enhances the conversion process of the algebraic structure of the near-ring by examining the uncertainty of elements within maximal ideals.

Fig. 3
figure 3

Performance of mean in the proposed model

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Figure 4 illustrates the performances of the worst class in the proposed model. When there are 4 iterations, the value is at its highest, 92, and when there are 5 iterations, the value is at its lowest, 50. In addition, when the number of iterations is 1, 2, and 3, the value of its worst class attains 63, 80, and 75 respectively. Addressing the most demanding ideals in this framework gives insights into the bounds of the maximal ideals in near-rings thereby enhancing the hybridization process.

Fig. 4
figure 4

Performance of worst class in the proposed model

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Figure 5 illustrates the performances of the worst class in the proposed model. When the number of iterations is 5, it obtains the minimum value of 30 s and the highest value of 60 s when the number of iterations is 3. Moreover, when the number of iterations is 1, 2, and 4, the obtained value of time is 48 s, 53 s, and 35 s respectively. In the process of converting fuzzy sets into hybrid ideals, the factor of time plays a crucial role. It serves as a key metric in assessing the efficiency and effectiveness of this conversion. Time performance is a vital aspect, as it measures the speed and feasibility of the transformation. This proposed conversion not only saves time but also enhances the practicality of implementing fuzzy sets within the hybrid ideal framework, making it a pivotal consideration in this process.

Fig. 5
figure 5

Performance of time in the proposed model

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Comparison of proposed model with previous models

This section highlights the proposed model which efficiently solved the uncertainty problems and the effectiveness of the proposed approach is also proved by comparing it with other existing approaches such as Fuzzy Soft Sets (FSS) [12], Intuitionistic Fuzzy Sets IFS [31] and Fermatean Fuzzy Sets FFS [26] and displaying their findings based on several comparisons.

Figure 6 displays a comparison of the proposed model's best class with other existing techniques. The suggested approach's best class is compared to existing techniques such as FSS, IFS, and FFS. The best class of the proposed model obtains a value of 14,000 whereas the best class of FSS, IFS, and FFS are 13,850, 13,650, and 13,900 respectively. The best class of the proposed model is high whereas the best class of IFS is low.

Fig. 6
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Comparison of Best class in the proposed model

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Figure 7 shows a comparison of the suggested model's mean with other existing techniques. The proposed approach's mean is compared to existing techniques such as FSS, IFS, and FFS. The mean of the proposed model has the optimum value of 13,950 whereas the mean of FSS, IFS, and FFS are 13,800, 13,750, and 13,700 respectively. The mean of the proposed model is high whereas the mean of FFS is low.

Fig. 7
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Mean comparison of the proposed model

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Figure 8 displays a comparison of the proposed model's worst class with other existing techniques. The suggested approach's worst class is compared to existing strategies such as FSS, IFS, and FFS. The worst class of the proposed model has the optimum value of 13,500 whereas the worst class of FSS, IFS, and FFS are 13,650, 13,753, and 13,650 respectively. The worst class of the proposed model is low whereas the worst class of IFS is high.

Fig. 8
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Worst class comparison of the proposed model

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Figure 9 compares the proposed model's temporal complexity to that of various existing techniques. The time complexity of the suggested strategy is compared to that of existing techniques such as FSS, IFS, and FFS. The time complexity of the proposed model has a minimum value of 0.32 s whereas the time complexity of FSS, IFS, and FFS are 0.43, 0.52, and 0.45 s respectively. The time complexity of the proposed model is low whereas the time complexity of IFS is high.

Fig. 9
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Time complexity comparison of the proposed model

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Figure 10 shows the comparison of membership vs non-membership with existing fuzzy sets such as Intuitionistic Fuzzy Sets, Pythagorean fuzzy sets, and Fermatean Fuzzy Sets. The highest non-membership of the proposed model obtains the value of 0.95 whereas the IFS, PFS, and FFS are 0.85, 0.9, and 0.95 respectively when the membership function is 0.2. When the membership value is 1.0 the proposed system achieves the lowest non-membership function value of 0.6 whereas the PFS, IFS, and FFS are 0, 0.10, and 0.25 respectively. When the membership function increases non-membership function decreases. Table 1 displays the comparative analysis of the proposed approach with existing models.

Fig. 10
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Comparison of membership with non-membership for the proposed model

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Table 1 Comparison of the proposed model with existing models
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Overall the hybridization of fuzzy sets based on near rings for uncertainty problems in ring theory outperforms existing techniques such as FSS, IFS, and FFS with optimum best class, worst class, and mean values of 14,000, 13,500, and 13,800 and have time complexity of 0.32 s. When the membership value is 1.0 the proposed system achieves the lowest non-membership function value of 0.6. When the membership value is 0.2 the proposed system achieves the highest non -membership function value of 0.96.

Conclusion

An ideal of a ring in ring theory is a particular subset of its members in the area of abstract algebra. Ideals, such as even numbers or multiples of three, generalize particular subsets of integers. Ring theory has therefore been extensively employed in numerous studies, although there is an uncertainty in transforming fuzzy sets to hybrid structures of any algebraic structure. Here the proposed models' performance is evaluated by best class, mean, worst class, and time. The best class of the proposed model obtains a value of 14,000 whereas the best class of FSS, IFS, and FFS are 13,850, 13,650, and 13,900 respectively. The mean of the proposed model has the optimum value of 13,950 whereas the mean of FSS, IFS, and FFS are 13,800, 13,750, and 13,700 respectively. The worst class of the proposed model has the optimum value of 13,500 whereas the worst class of FSS, IFS, and FFS are 13,650, 13,753, and 13,650 respectively. The time complexity of the proposed model has a minimum value of 0.32 s whereas the time complexity of FSS, IFS, and FFS are 0.43, 0.52, and 0.45 s respectively. Furthermore, the performance of the proposed models is also assessed using the membership function. The proposed approach achieves the lowest non-membership function value of 0.6 when the membership value is 1.0 and the proposed system obtains a maximal non-membership function value of 0.96 when the membership value is 0.2. As a result, the hybrid structure of fuzzy sets in near rings is introduced, in which fuzzy ideals and fuzzy maximal ideals are both converted to hybrid ideals. Here, sub-near rings and nearby rings are also determined after the hybrid structure has been predicted. The nearby rings are then subjected to the hybrid structure. However, analyzing the properties of maximal ideals in near-rings, especially in the context of hybrid structures, is computationally challenging due to the reason that determining whether an ideal is maximal that is an ideal may not be straightforward in non-commutative near rings. In the future, exploring connections between near-rings and other algebraic structures, such as rings, semi-rings, or lattice-ordered groups is required to identify similarities and differences and to study how maximal ideals in near-rings relate to these structures. Furthermore, there is a need to develop computational tools and algorithms to analyze the hybrid structures of maximal ideals in near-rings.