Narrow Results

In this paper, we study the concept of nearness between $L$-fuzzy rough sets and the concepts of $L$-fuzzy rough grills and $L$-fuzzy rough ${\delta }_{{ℱ}}$-clusters on a Čech $L$-fuzzy rough proximity space $(X,{\delta }_{{ℱ}})$. Also, we investigate the relationship among $L$-fuzzy rough grills, $L$-fuzzy rough ${\delta }_{{ℱ}}$-clan and $L$-fuzzy rough ${\delta }_{{ℱ}}$-clusters. Moreover, we introduce the $L$-fuzzy rough LO-proximity spaces and study some of its basic properties. An adequate number of examples support the study.

In this paper, we study the concept of nearness between $L$-fuzzy rough sets using $L$-fuzzy rough grills. We introduce the concept of Čech $L$-fuzzy rough $G$-proximity spaces and $L$-fuzzy rough $G$-LO-proximity spaces. On this new nearness structure, we prove some basic topological results and to support the proposed approach, examples are given.

In this paper, deriving the type-m fuzzy sets, intuitionistic fuzzy sets, Φ-fuzzy sets, rough sets, fuzzy rough sets and rough fuzzy sets as particular genuine sets, and establishing their connections with genuine sets, it is demonstrated that the theory of genuine sets provides a powerful tool to model various different kinds of uncertainty in a mathematical way. Furthermore, it is also shown that the genuine set theoretic descriptions of type-m fuzzy sets, intuitionistic fuzzy sets and fuzzy rough sets point out new features of these set notations, originated from the peculiar characteristics of genuine sets.

The axiomatic approach is more appropriate than constructive approach for studying the algebraic structure of rough sets. In this paper, the more simple axiomatic characterizations of (υ σ)-fuzzy rough approximation operators are explored where υ is a residuated implicator and σis its dual implicator. Firstly, we review the existing independent axiomatic sets to characterize various types of υ-lower and σ-upper fuzzy rough approximation operators. Secondly, we present one-axiom characterizations of (υ σ)-fuzzy rough approximation operators constructed by a serial fuzzy relation on two universes. Furthermore, we show that (υ σ)-fuzzy rough approximation operators, corresponding to reexive, symmetric and T-transitive fuzzy relations, can be presented by only two axioms respectively. We conclude the paper by introducing some potential applications and future works.

In this paper, we provide a definition of α-fuzzified lower and upper approximations for fuzzy sets based on the α-cut of fuzzy binary relations. We show that the definition is a proper generalization of the previous one for approximations of crisp sets and compare it with an existing definition in the context of fuzzy tolerance relation.

In order to define the concept of nearness between $L$-fuzzy rough sets, we introduce the concept of Čech $L$-fuzzy rough proximity spaces. On this new nearness structure, we prove some basic topological results. To support the proposed approach, examples are given. Further, we introduce $L$-fuzzy rough grills, $L$-fuzzy rough filters and $L$-fuzzy rough clusters and on Čech $L$-fuzzy rough proximity spaces, and find the relationship between them.