Since upper and lower approximations could be induced from the rough set structures, rough sets are considered as approximations. The concept of fuzzy rough sets was proposed by replacing crisp binary relations with fuzzy relations by Dubois and Prade. In this paper, we introduce and investigate some properties of intuitionistic fuzzy rough approximation operators and intuitionistic fuzzy relat...

Soft set theory, fuzzy set theory and rough set theory are all mathematical tools for dealing with uncertainties and are closely related. Feng et al. introduced the notions of rough soft set, soft rough set and soft rough fuzzy set by combining fuzzy set, rough set and soft set all together. This paper is devoted to the discussions of the combinations of intervalvalued intuitionistic fuzzy set,...

In this paper, the concept of semiprimary fuzzy ideals of an ordered semigroup S is introduced. Some characterizations for an ordered semigroup S to be a semilattice of archimedean (rarchimedean, t-archimedean) ordered subsemigroups are given by some binary relations on S and the fuzzy radical of fuzzy ideals (fuzzy right ideals, fuzzy bi-ideals) of S. Furthermore, some characterizations for an...

We introduce the notion of sensible fuzzy ideals of BCK-algebras with respect to a t-conorm and investigate some of their properties. We give the conditions for a sensible fuzzy subalgebra with respect to a t-conorm to be a sensible fuzzy ideal with respect to a t-conorm. Some properties of the direct product and S-product of fuzzy ideals of BCK-algebras with respect to a t-conorm are also disc...

let $s$ be an ordered semigroup. a fuzzy subset of $s$ is anarbitrary mapping   from $s$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. in this paper,  the concept of fuzzygeneralized bi-ideals of an ordered semigroup $s$ is introduced.regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.finally, two m...

This article introduces a general framework of multi-granulation fuzzy probabilistic roughsets (MG-FPRSs) models in multi-granulation fuzzy probabilistic approximation space over twouniverses. Four types of MG-FPRSs are established, by the four different conditional probabilitiesof fuzzy event. For different constraints on parameters, we obtain four kinds of each type MG-FPRSs...

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic ...

Fuzzy rough sets, generalized from Pawlak’s rough sets, were introduced for dealing with continuous or fuzzy data. This model has been widely discussed and applied these years. It is shown that the model of fuzzy rough sets is sensitive to noisy samples, especially sensitive to mislabeled samples. As data are usually contaminated with noise in practice, a robust model is desirable. We introduce...

The concept of the rough set was originally proposed by Pawlak as a formal tool for modelling and processing incomplete information in information systems, then in 1990, Dubois and Prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets. The aim of this paper is to present a new extension of the rough set theory by means of integrating the classic...

A ternary semigroup is a nonempty set together with a ternary multiplication which is associative. Any semigroup can be reduced to a ternary semigroup but a ternary semigroup does not necessarily reduce to a semigroup. The notion of fuzzy sets was introduced by Zadeh in 1965 and that of rough sets by Pawlak in 1982. Applications of the fuzzy set theory and rough set theory have been found in va...