Binary relations play an important role in both mathematics and information sciences. In this paper, we focus our discussion on a fuzzy set which is approximated in the sense of the aftersets and foresets. To this end, a soft binary relation has been used. A new approach is being introduced to get two sets of fuzzy soft sets, called the lower approximation and upper approximation regarding the aftersets and foresets. We applied these concepts on semigroups and approximations of fuzzy subsemigroups, fuzzy left (right) ideals, fuzzy interior ideals and fuzzy bi-ideals of semigroups are studied.
In this work, we have proposed a new relationship among rough set, soft set and quantales with the help of soft compatible relation. This typical relationship is used to approximate the fuzzy substructures in quantales in association with soft compatible relations by using aftersets and foresets. This type of approximation is extended notation of rough quantales, rough fuzzy subquantales and soft subquantales. We have corroborated this work by considering some test examples containing soft compatible relations over quantales. Moreover, by using soft compatible relations, we will describe the relationship between upper (lower) generalized rough fuzzy soft substructures of quantale and the upper (lower) approximations of their homomorphic images with the help of weak quantale homomorphism. The comparison of this type approximations and their results affirms the superiority of our new approximation method over current methods on the topic.
The present work is conducted to investigate the relationship among rough sets, soft sets and quantales. The concept of generalized approximation of substructures in quantales by soft relations is introduced, which is an extended notion of a rough quantale and a soft quantale. This paper is focused on studying the rough sets within the context of algebraic structure quantale using soft reflexive and soft compatible relations. Further, we put forward the concepts of aftersets and foresets, which provide a new research idea for soft rough algebraic research. Some basic concepts, operations and related properties with regard to soft binary relations are proposed. Keywords Quantales • Ideals of quantales • Approximation by soft relations • Approximation of ideals in quantales • Quantale homomorphism Mathematics Subject Classification 08Axx • 08A72 Communicated by Anibal Tavares de Azevedo.
The aim of this research article is to derive a new relation between rough sets and soft sets with an algebraic structure quantale by using soft binary relations. The aftersets and foresets are utilized to define lower approximation and upper approximation of soft subsets of quantales. As a consequence of this new relation, different characterization of rough soft substructures of quantales is obtained. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Moreover, this concept generalizes the concept of rough soft substructures of other structures. Furthermore, the algebraic relations between the upper (lower) approximation of soft substructures of quantales and the upper (lower) approximation of their homomorphic images with the help of soft quantales 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.
In this paper, we use an algebraic structure quantale and define the idea of fuzzy soft substructures as a generalization of fuzzy substructures in quantale. These fuzzy soft substructures include fuzzy soft subquantales, fuzzy soft ideals, fuzzy soft prime ideals, fuzzy soft semiprime ideals, and fuzzy soft primary ideals. Furthermore, different characterizations of fuzzy soft substructures in quantales are introduced. Moreover, we extend this ideology to investigate that for each fuzzy soft substructure in quantale, there exists an α-soft substructure in quantales. These fuzzy soft subquantales and fuzzy soft ideals are characterized by their level subquantales and ideals, respectively. Finally, fuzzy soft image and fuzzy soft inverse image of fuzzy soft substructures under quantale homomorphism in quantale are discussed.
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