Fuzzy Approximation Operators Research Articles

Rough fuzzy approximations on two universes of discourse

In rough set theory, the lower and upper approximation operators can be constructed via a variety of approaches. Various fuzzy generalizations of rough approximation operators have been made over the years. This paper presents a framework for the study of rough fuzzy sets on two universes of discourse. By means of a binary relation between two universes of discourse, a covering and three relations are induced to a single universe of discourse. Based on the induced notions, four pairs of rough fuzzy approximation operators are proposed. These models guarantee that the approximating sets and the approximated sets are on the same universes of discourse. Furthermore, the relationship between the new approximation operators and the existing rough fuzzy approximation operators on two universes of discourse are scrutinized, and some interesting properties are investigated. Finally, the connections of these approximation operators are made, and conditions under which some of these approximation operators are equivalent are obtained.

Set approximations in fuzzy formal concept analysis

Formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems. The purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment. Properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined. Based on both lattice-theoretic and fuzzy set-theoretic operators, two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined. Finally, properties of the rough fuzzy set approximation operators are presented in detail.

Minimization of axiom sets on fuzzy approximation operators

Axiomatic characterization of approximation operators is an important aspect in the study of rough set theory. In this paper, we examine the independence of axioms and present the minimal axiom sets characterizing fuzzy rough approximation operators and rough fuzzy approximation operators.

Fuzzy modal-like approximation operators based on double residuated lattices

In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of these operators are presented.

On characterizations of [formula omitted]-fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Various fuzzy generalizations of rough approximations have been made in the literature. This paper proposes a general framework for the study of ( I , T ) -fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper generalized fuzzy rough approximation operators, determined by an implicator I and a triangular norm T , is first defined. Basic properties of ( I , T ) -fuzzy rough approximation operators are investigated. The connections between fuzzy relations and fuzzy rough approximation operators are further established. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, ( I , T ) -fuzzy approximation operators are defined by axioms. Different axiom sets of T -upper and I -lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. Finally, an open problem proposed by Radzikowska and Kerre in (Fuzzy Sets and Systems 126 (2002) 137) is solved.

An uncertainty measure in partition-based fuzzy rough sets

This paper extends Pawlak's rough set onto the basis of a fuzzy partition of the universe of discourse. Some basic properties of partition-based fuzzy approximation operators are examined. To measure uncertainty in generalized fuzzy rough sets, a new notion of entropy of a fuzzy set is introduced. The notion is demonstrated to be adequate for measuring the fuzziness of a fuzzy event. The entropy of a fuzzy partition and conditional entropy are also proposed. These kinds of entropy satisfy some basic properties similar to those of Shannon's entropy. It is proved that the measure of fuzziness of a partition-based fuzzy rough set, FR(A), is equal to zero if and only if the set A is crisp and definable.

Constructive and axiomatic approaches of fuzzy approximation operators

This paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used. In constructive approach, a pair of lower and upper generalized fuzzy rough (and rough fuzzy, respectively) approximation operators is first defined. The representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented. The connections between fuzzy (and crisp, respectively) relations and fuzzy rough (and rough fuzzy, respectively) approximation operators are further established. In axiomatic approach, various classes of fuzzy approximation operators are characterized by different sets of axioms. The minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators.

Generalized fuzzy rough sets

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.

On relationships among fuzzy approximation operators, fuzzy topology, and fuzzy automata

The fuzzy approximation operator associated with an approximation space ( X, R) turns out to be a (saturated) Kuratowski fuzzy closure operator on X precisely when the relation R on X is reflexive and transitive. It is shown that all the level topologies of the fuzzy topology so obtained on X coincide. These observations are then applied to fuzzy automata.