This research article offers a study on a new relation of rough sets and soft sets with an algebraic structure quantale module by using soft reflexive and soft compatible relations. The lower approximation and upper approximation of subsets of quantale module are utilized by aftersets and foresets. As a sequel of this relation, different characterizations of rough soft substructures of quantale modules are obtained. To ensure the results, soft reflective and soft compatible relations are focused and these are interpreted by aftersets and foresets. Furthermore, the algebraic relations between upper (lower) approximation of substructures of quantale module and the upper (lower) approximations of their homomorphic images with the help of quantale module homomorphism are examined. In comparison with the different type of approximations in different type of algebraic structures, it is concluded that this new study is much better.
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<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>
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