Fuzzy roughness via ideals

Journal of the Egyptian Mathematical Society

In this paper, we defined the fuzzy upper, fuzzy lower, and fuzzy boundary sets of a rough fuzzy set λ in a fuzzy approximation space (X, R). Based on λ and R, we introduced the fuzzy ideal approximation interior operator intlambda R and the fuzzy ideal approximation closure operator cl R λ. We joined the fuzzy ideal notion with the fuzzy approximation spaces, and then introduced the fuzzy ideal approximation closure and interior operators associated to a rough fuzzy set λ. Fuzzy ideal approximation connectedness and the fuzzy ideal approximation continuity between fuzzy ideal approximation spaces are introduced.

Iranian Journal of Fuzzy Systems, 2022

Since Pawlak defined the notion of rough sets in 1982, many authors made wide research studying rough sets in the ordinary case and the fuzzy case. This paper introduced a new style of rough fuzzy sets based on a fuzzy ideal ℓ on a universal finite set X. New lower and new upper fuzzy sets are introduced, and consequently, fuzzy interior and fuzzy closure operators of a rough fuzzy set are discussed. These definitions, if ℓ is restricted to ℓ • = {0}, imply the fuzzification of previous definitions given in the ordinary case, and moreover in the crisp case, we get exactly these previous definitions. The new style gives us a better accuracy value of roughness than the previous styles. Rough fuzzy connectedness is introduced as a sample of applications on the recent style of roughness.

viXra, 2020

In this paper, we initiate the concept of intuitionistic fuzzy ideals on rough sets. Using a new relation we discuss some of the algebraic nature of intuitionistic fuzzy ideals of a ring.

International Journal of Rough Sets and Data Analysis, 2019

One of the extensions of the basic rough set model introduced by Pawlak in 1982 is the notion of rough sets on fuzzy approximation spaces. It is based upon a fuzzy proximity relation defined over a Universe. As is well known, an equivalence relation provides a granularization of the universe on which it is defined. However, a single relation defines only single granularization and as such to handle multiple granularity over a universe simultaneously, two notions of multigranulations have been introduced. These are the optimistic and pessimistic multigranulation. The notion of multigranulation over fuzzy approximation spaces were introduced recently in 2018. Topological properties of rough sets are an important characteristic, which along with accuracy measure forms the two facets of rough set application as mentioned by Pawlak. In this article, the authors introduce the concept of topological property of multigranular rough sets on fuzzy approximation spaces and study its properties.

Fuzzy Sets and Systems, 2014

The roughness of a set (according to the notion introduced by Pawlak in 1991) can be regarded as the MZ-distance between its upper and the lower approximations. With this idea in mind, we have generalized Pawlak's definition, by replacing the MZ-distance by a general "distance" measure. We also generalize the notion of roughness of fuzzy sets introduced by Huyhn and Nakamori in 2005.

Information Sciences, 2016

In this paper we deal with the theory of rough ideals started in [4]. We show that the approximation spaces built from an equivalence relation compatible with the ring structure, i.e. associated to a two-sided ideal, are too naive in order to develop practical applications. We propose the use of certain crisp equivalence relations obtained from fuzzy ideals. These relations make available more flexible approximation spaces since they are enriched with a wider class of rough ideals. Furthermore, these are fully compatible with the notion of primeness (semiprimeness). The theory is illustrated by several examples of interest in Engineering and Mathematics.

The concept of fuzzy approximation space that depends on a fuzzy proximity relation is a generalization of the concept of the knowledge base. But intuitionistic fuzzy approximation space that depends on an intuitionistic fuzzy proximity relation is a better generalization of the concept of knowledge base than fuzzy approximation space. Therefore, rough sets defined on intuitionistic fuzzy approximation spaces extend the concept of rough sets on fuzzy approximation spaces. This paper presents how rough sets on intuitionistic fuzzy approximation spaces provides better result over rough sets on fuzzy approximation spaces on knowledge representation. Index Terms: Fuzzy relation, fuzzy proximity relation, fuzzy approximation space, intuitionistic fuzzy approximation space and rough set.

2006 3rd International IEEE Conference Intelligent Systems, 2006

The notion of intuitionistic fuzzy approximation space is introduced. Rough sets on such spaces are defined and some of their properties are studied.

Information Sciences, 2003

This paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized approximation operators is defined. The connections between fuzzy relations and fuzzy rough approximation operators are examined. In axiomatic approach, various classes of fuzzy rough approximation operators are characterized by different sets of axioms. Axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators.

Journal of Computer Science and Cybernetics, 2015

Approximation of a picture fuzzy set on a crisp approximation space gives a rough picture fuzzy set. In this paper, the concept of a rough picture set is introduced, besides we also investigate some topological structures of a rough picture fuzzy set are investigated, such are lower and upper rough picture fuzzy approximation operators.