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

Saqib Mazher Qurashi; Ferdous Tawfiq; Qin Xin; Rani Sumaira Kanwal; Khushboo Zahra Gilani; Saqib Mazher Qurashi; Ferdous Tawfiq; Qin Xin; Rani Sumaira Kanwal; Khushboo Zahra Gilani

doi:10.3934/math.2023592

AIMS Mathematics

2023, Volume 8, Issue 5: 11684-11708. doi: 10.3934/math.2023592

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

1.
Department of Mathematics, Government College University, Faisalabad, Pakistan
2.
Department of Mathematics, College of Science, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia
3.
Faculty of Science and Technology, University of the Faroe Islands, Faroe Islands, Denmark
4.
Department of Mathematics, Government College Women University, Faisalabad, Pakistan
5.
Department of Mathematics, Government College University, Faisalabad, Pakistan

Received: 27 September 2022 Revised: 27 February 2023 Accepted: 06 March 2023 Published: 17 March 2023
MSC : 06D72, 60Lxx, 60L70

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.
- quantale module and its substructures,
- rough set,
- soft set,
- aftersets and foresets,
- compatible relations
Citation: Saqib Mazher Qurashi, Ferdous Tawfiq, Qin Xin, Rani Sumaira Kanwal, Khushboo Zahra Gilani. Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations[J]. AIMS Mathematics, 2023, 8(5): 11684-11708. doi: 10.3934/math.2023592

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Abstract

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.

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