Soft Relations Applied to the Substructures of Quantale Module and Their Approximation

Qurashi, Saqib Mazher; Gilani, Khushboo Zahra; Shabir, Muhammad; Gulzar, Muhammad; Alam, Md. Ashraful

doi:10.1155/2022/9073503

Cited by 3 publications

(4 citation statements)

References 29 publications

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“…3. Literature already in existence studies various crisp and soft substructures under rough environment of quantale modules via soft relations ( [43], [44]). Moreover, such type of roughness via soft relations has also been discussed in different structures like semigroups and quantale ( [15], [17]).…”

Section: B Deficit and Research Gap In Existing Literaturementioning

confidence: 99%

“…5. Some fundamental and important theorems of quantale module homomorphism and soft quantale module homomorphism were discussed in ( [43], [44]). Consequently, a discussion of these important theorems in the context of quantale module homomorphism via fuzzy sets is imperative.…”

Section: B Deficit and Research Gap In Existing Literaturementioning

confidence: 99%

“…Although roughness with congruence relations and set-valued homomorphism [7] is available in the literature yet there is no such attempt to find roughness which is without the above techniques. More generalized form of roughness of substructures and soft substructures were performed in [43] and [44]. But these generalized models in quantale modules did not contain roughness of fuzzy substructures.…”

Section: Motivation and Goal Of The Proposed Workmentioning

confidence: 99%

“…Considering that soft relations were employed to more effectively that minimize some types of information was extended by Shabir et al [42]. Additionally, decision-making methods including soft relations were connected to algebraic structures and gave rise to fascinating new study areas ( [43], [44], [45], [46]).…”

Section: Introductionmentioning

confidence: 99%

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Rough Fuzzy Substructures of Quantale Module Under Soft Relations and Corresponding Decision-Making Methods

Qurashi,

Almutairi,

Kanwal

et al. 2023

IEEE Access

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This research paper has developed a way of roughness of fuzzy substructures by using soft relations for developing rough fuzzy substructures in Quantale module. Thus, an innovative concept of fuzzy substructures of Quantale module under rough environment by soft relations, is presented. The lower and upper approximations of fuzzy subsets of quantale module are defined by aftersets and foresets. This relationship leads to various characterizations of the rough fuzzy substructures of quantale modules. Besides of more comprehensive results, soft compatible and soft complete relations are required with foresets and aftersets. Soft relations are further being used to determine upper (lower) approximation of fuzzy subsets of quantale module using foreset and afterset. Moreover, several characterizations of rough fuzzy substructures of quantale module are investigated. Furthermore, the algebraic relations of upper (lower) approximations of fuzzy quantale submodule and fuzzy quantale submodule ideals are studied with the help of soft relations under weak quantale module homomorphism. To illustrate that the suggested approach is superior to the given methods, examples are provided. At last, we describe decision-making methods by using rough fuzzy substructures of Quantale module under soft relations to deal with uncertainties in the real-world problems. To demonstrate the validity, applicability, and efficacy of the suggested method, a detailed example of the decision-making process is provided.INDEX TERMS Quantale module and its substructures, rough sets, soft sets and fuzzy sets.

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Section: B Deficit and Research Gap In Existing Literaturementioning

confidence: 99%

Section: B Deficit and Research Gap In Existing Literaturementioning

confidence: 99%

Section: Motivation and Goal Of The Proposed Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rough Fuzzy Substructures of Quantale Module Under Soft Relations and Corresponding Decision-Making Methods

Qurashi,

Almutairi,

Kanwal

et al. 2023

IEEE Access

Self Cite

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Binary relations applied to the fuzzy substructures of quantales under rough environment

Mazher Qurashi,

Almutairi,

Xin

et al. 2024

Demonstratio Mathematica

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Binary relations (BIRs) have many applications in computer science, graph theory, and rough set theory. This study discusses the combination of BIRs, fuzzy substructures of quantale, and rough fuzzy sets. Approximation of fuzzy subsets of quantale with the help of BIRs is introduced. In quantale, compatible and complete relations in terms of aftersets and foresets with the help of BIRs are defined. Furthermore, we use compatible and complete relations to approximate fuzzy substructures of quantale, and these approximations are interpreted by aftersets and foresets. This concept generalizes the concept of rough fuzzy quantale. Finally, using BIRs, quantale homomorphism is used to build a relationship between the approximations of fuzzy substructures of quantale and the approximations of their homomorphic images.

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Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations

Qurashi

Tawfiq

Xin

et al. 2023

MATH

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<abstract> <p>The quantale module introduced by Abramsky and Vickers, engaged a large number of researchers. This research article focuses the combined behavior of rough set, soft set and an algebraic structure quantale module with the left action. In fact, the paper reflects the generalization of rough soft sets. This combined effect is totally dependent on soft binary relation including aftersets and foresets. Different soft substructures in quantale modules are defined. The characterizations of soft substructures in quantale modules based on soft binary relation are presented. Further, in quantale modules, we define soft compatible and soft complete relations in terms of aftersets and foresets. Furthermore, we use soft compatible and soft complete relations to approximate soft substructures of quantale modules and these approximations are interpreted by aftersets and foresets. This concept generalizes the concept of rough soft quantale modules. Additionally, we describe the algebraic relationships between the upper (lower) approximations of soft substructures of quantale modules and the upper (lower) approximations of their homomorphic images using the concept of soft quantale module homomorphism.</p> </abstract>

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Soft Relations Applied to the Substructures of Quantale Module and Their Approximation

Cited by 3 publications

References 29 publications

Rough Fuzzy Substructures of Quantale Module Under Soft Relations and Corresponding Decision-Making Methods

Rough Fuzzy Substructures of Quantale Module Under Soft Relations and Corresponding Decision-Making Methods

Binary relations applied to the fuzzy substructures of quantales under rough environment

Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations

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