Fuzzy Approximation Operators Research Articles

On the relationship among F-transform, fuzzy rough set and fuzzy topology

The objective of this work is to associate the concepts of fuzzy rough sets and fuzzy topologies/co-topologies with the F-transforms. The notions of the direct \(F^{\uparrow }\) and \(F^{\downarrow }\)-transforms are extended to the case where they are applied to an L-valued function on a space with an L-valued fuzzy partition. It is shown that these F-transforms are particular cases of upper and lower fuzzy approximation operators. Moreover, every F-transform component induces a continuous map between two associated fuzzy topological spaces or fuzzy co-topological spaces.

Categories of fuzzy preorders, approximation operators and Alexandrov topologies1

In this paper, we investigate categories of fuzzy preorders, approximating operators and Alexandrov topologies in complete residuated lattices. In fact, categories of fuzzy preorders, approximating operators and Alexandrov topologies are isomorphic. We give their examples.

On $L$-double fuzzy rough sets

ur aim of this paper is to introduce the concept of $L$-double fuzzy rough sets in whichboth constructive and axiomatic approaches are used. In constructive approach, a pairof $L$-double fuzzy lower (resp. upper) approximation operators is defined and the basic properties of them are studied.From the viewpoint of the axiomatic approach, a set of axioms is constructed to characterize the $L$-double fuzzy upper (resp. lower) approximation of $L$-double fuzzy rough sets. Finally, from $L$-double fuzzy approximation operators, we generated Alexandrov $L$-double fuzzy topology.

Topologies associated to (dual) fuzzy approximation operators on fuzzy finite state automata

Topologies associated to (dual) fuzzy approximation operators on fuzzy finite state automata

L-FUZZY APPROXIMATION SPACES AND L-FUZZY TOPOLOGICAL SPACES

The $L$-fuzzy approximation operator associated with an $L$-fuzzy approximation space $(X,R)$ turns out to be a saturated $L$-fuzzy closure (interior) operator on a set $X$ precisely when the relation $R$ is reflexive and transitive. We investigate the relations between $L$-fuzzy approximation spaces and $L$-(fuzzy) topological spaces.

On Characterization of Rough Type-2 Fuzzy Sets

Rough sets theory and fuzzy sets theory are important mathematical tools to deal with uncertainties. Rough fuzzy sets and fuzzy rough sets as generalizations of rough sets have been introduced. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle high uncertainties. In this paper, the rough type-2 fuzzy set model is proposed by combining the rough set theory with the type-2 fuzzy set theory. The rough type-2 fuzzy approximation operators induced from the Pawlak approximation space are defined. The rough approximations of a type-2 fuzzy set in the generalized Pawlak approximation space are also introduced. Some basic properties of the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators are discussed. The connections between special crisp binary relations and generalized rough type-2 fuzzy approximation operators are further examined. The axiomatic characterization of generalized rough type-2 fuzzy approximation operators is also presented. Finally, the attribute reduction of type-2 fuzzy information systems is investigated.

Generalized Interval-Valued Fuzzy Rough Set and its Application in Decision Making

This paper presents a general study of generalized interval-valued fuzzy rough sets integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. In the constructive approach, by employing an interval-valued fuzzy residual implicator and its dual operator, generalized upper and lower interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy approximation space are first defined. Then properties of generalized interval-valued fuzzy rough approximation operators are discussed. Furthermore, connections between special types of interval-valued fuzzy relations and properties of generalized interval-valued fuzzy approximation operator are also established. In the axiomatic approach, generalized interval-valued fuzzy rough approximation operators are defined by axioms. We prove that different axiom sets can characterize the essential properties of generalized interval-valued fuzzy rough approximation operators. Also the composition of two approximation spaces is explored. Finally, a practical application is provided to illustrate the efficiency of the generalized interval-valued fuzzy rough set model.

A comparison of three types of rough fuzzy sets based on two universal sets

The extension of rough set model is an important research direction in rough set theory. This paper presents two new extensions of the rough set model over two different universes. By means of a binary relation between two universes of discourse, two 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, some interesting properties are investigated, the connections between relations and rough fuzzy approximation operators are examined. Finally, the connections of these approximation operators are made, and conditions under which these approximation operators made equivalent are obtained.

On uniqueness of (fuzzy) relation in some generalized rough set model

In rough set theory, crisp and/or fuzzy binary relations play an important role in both constructive and axiomatic considerations of various generalized rough sets. This paper considers the uniqueness problem of the (fuzzy) relation in some generalized rough set model. Our results show that by using the axiomatic approach, the (fuzzy) relation determined by (fuzzy) approximation operators is unique in some (fuzzy) double-universe model.

UPPER APPROXIMATION OPERATORS INDUCED BY MAPS

In this paper, we investigate the properties of Alexandrov fuzzy topologies and upper approximation operators induced by maps. We give their examples. Hohle (3) introduced L-fuzzy topologies and L-fuzzy interior operators. The relationship between rough set theory and topological spaces was investigated (4-12). Hajek (2) introduced a complete residuated lattice which is an algebraic structure for many valued logic. Kim (5-7) investigated the properties of join (resp. meet, meet join, join meet) preserving operators and Alexandrov fuzzy topology in complete residuated lattices. In this paper, we investigate the properties of Alexandrov fuzzy topologies and upper approximation operators induced by maps. We give their examples.

Fuzzy information systems and their homomorphisms

With the arrival of the information age, information acquisition and communication have become more and more important in the field of information technology. This paper uses the concept of homomorphism as a basic tool to study the communication between fuzzy information systems. The concepts of consistent and compatible mappings with respect to fuzzy sets are firstly defined and their basic properties are studied. Then, a pair of lower and upper rough fuzzy approximation operators is constructed by means of the concept of fuzzy mappings. Basic invariant properties of the approximation operators are investigated. Finally, the concepts of fuzzy information system and its homomorphism are introduced, and some invariant properties of fuzzy information systems under homomorphisms are examined. It is proved that the attribute reductions of an original information system and its image system are equivalent to each other in the context of fuzzy attributes. These results may have potential applications in attribute reduction and classification issues.

Intuitionistic Fuzzy Soft Rough Set and Its Application in Decision Making

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. In this paper, we present concepts of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets, and investigate some properties of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets in detail. Furthermore, classical representations of intuitionistic fuzzy soft rough approximation operators are presented. Finally, we develop an approach to intuitionistic fuzzy soft rough sets based on decision making and a numerical example is provided to illustrate the developed approach.

Covering-based rough fuzzy sets and binary relation

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. In this paper we study covering-based rough fuzzy sets in which a fuzzy set can be approximated by the intersection of some elements in a covering of the universe of discourse. Some properties of the covering-based fuzzy lower and upper approximation operators are examined. We present the conditions under which two coverings generate the same covering-based fuzzy lower and upper approximation. We approximate fuzzy sets based on a binary relation and its properties are introduced. Finally, we establish the equivalency between rough fuzzy sets generated by a covering and rough fuzzy sets generated by a binary relation.

On relationships between revised rough fuzzy approximation operators and fuzzy topological spaces

In this paper, revised rough fuzzy lower and revised rough fuzzy upper approximations of fuzzy sets are defined. Properties of revised rough fuzzy approximation operators are examined. We first show that an inverse serial revised rough fuzzy approximation space can induce a fuzzy Alexandrov space. Relationships between revised rough fuzzy approximation operators and fuzzy topological spaces are then discussed. We then prove that the revised lower and upper rough fuzzy approximation operators are, respectively, the fuzzy interior operator and fuzzy closure operator if and only if the binary relation in the crisp approximation space is inverse serial. Finally, we prove that a strong inverse serial revised rough fuzzy approximation space can produce a fuzzy clopen topological space.

On interval type-2 rough fuzzy sets

In this paper, we present a general framework for the study of interval type-2 rough fuzzy sets by using both constructive and axiomatic approaches. First, several concepts and properties of interval type-2 fuzzy sets are introduced. Then, a pair of lower and upper interval type-2 rough fuzzy approximation operators with respect to a crisp binary relation is proposed. Classical representations of the interval type-2 rough fuzzy approximation operators are then constructed, and the connections between the special binary relations and the interval type-2 rough fuzzy approximation operators are investigated. Furthermore, an operator-oriented characterization of interval type-2 rough fuzzy sets is proposed; that is, interval type-2 rough fuzzy approximation operators are characterized by axioms. Different axiom sets of interval type-2 fuzzy set-theoretic operators guarantee the existence of different types of crisp binary relations, which produce the same operators. Furthermore, the relationship between interval type-2 rough fuzzy sets and interval type-2 fuzzy topological spaces is obtained. The sufficient and necessary condition for the conjecture that an interval type-2 fuzzy interior (closure) operator derived from an interval type-2 fuzzy topological space can be associated with a reflexive and transitive binary relation such that the corresponding lower (upper) interval type-2 rough fuzzy approximation operator is the interval type-2 fuzzy interior (closure) operator is examined. Finally, we provide a practical application to illustrate the usefulness of the interval type-2 rough fuzzy sets model.

Characterization of rough set approximations in Atanassov intuitionistic fuzzy set theory

The primitive notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation and satisfying many interesting properties. Many types of generalized rough set models have been proposed in the literature. This paper discusses the rough approximations of Atanassov intuitionistic fuzzy sets in crisp and fuzzy approximation spaces in which both constructive and axiomatic approaches are used. In the constructive approach, concepts of rough intuitionistic fuzzy sets and intuitionistic fuzzy rough sets are defined, properties of rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are examined. Different classes of rough intuitionistic fuzzy set algebras and intuitionistic fuzzy rough set algebras are obtained from different types of fuzzy relations. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, rough intuitionistic fuzzy approximation operators and intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of upper and lower intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of crisp/fuzzy relations which produce the same operators.

On intuitionistic fuzzy rough sets and their topological structures

In this paper, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of intuitionistic fuzzy approximation operators are examined. Relationships between intuitionistic fuzzy rough set approximations and intuitionistic fuzzy topologies are then discussed. It is proved that the set of all lower approximation sets based on an intuitionistic fuzzy reflexive and transitive approximation space forms an intuitionistic fuzzy topology; and conversely, for an intuitionistic fuzzy rough topological space, there exists an intuitionistic fuzzy reflexive and transitive approximation space such that the topology in the intuitionistic fuzzy rough topological space is just the set of all lower approximation sets in the intuitionistic fuzzy reflexive and transitive approximation space. That is to say, there exists an one-to-one correspondence between the set of all intuitionistic fuzzy reflexive and transitive approximation spaces and the set of all intuitionistic fuzzy rough topological spaces. Finally, intuitionistic fuzzy pseudo-closure operators in the framework of intuitionistic fuzzy rough approximations are investigated.

An axiomatic approach of fuzzy rough sets based on residuated lattices

Rough set theory was developed by Pawlak as a formal tool for approximate reasoning about data. Various fuzzy generalizations of rough approximations have been proposed in the literature. As a further generalization of the notion of rough sets, L -fuzzy rough sets were proposed by Radzikowska and Kerre. In this paper, we present an operator-oriented characterization of L -fuzzy rough sets, that is, L -fuzzy approximation operators are defined by axioms. The methods of axiomatization of L -fuzzy upper and L -fuzzy lower set-theoretic operators guarantee the existence of corresponding L -fuzzy relations which produce the operators. Moreover, the relationship between L -fuzzy rough sets and L -topological spaces is obtained. The sufficient and necessary condition for the conjecture that an L -fuzzy interior (closure) operator derived from an L -fuzzy topological space can associate with an L -fuzzy reflexive and transitive relation such that the corresponding L -fuzzy lower (upper) approximation operator is the L -fuzzy interior (closure) operator is examined.

On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators

This paper presents a general framework for the study of relation-based ( I , T ) -intuitionistic fuzzy rough sets by using constructive and axiomatic approaches. In the constructive approach, by employing an intuitionistic fuzzy implicator I and an intuitionistic fuzzy triangle norm T , lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of ( I , T ) -intuitionistic fuzzy rough approximation operators are examined. The connections between special types of intuitionistic fuzzy relations and properties of intuitionistic fuzzy approximation operators are established. In the axiomatic approach, an operator-oriented characterization of ( I , T ) -intuitionistic fuzzy rough sets is proposed. Different axiom sets characterizing the essential properties of intuitionistic fuzzy approximation operators associated with various intuitionistic fuzzy relations are explored.

On fuzzy approximation operators in attribute reduction with fuzzy rough sets

Generally speaking, there are four fuzzy approximation operators defined on a general triangular norm ( t-norm) framework in fuzzy rough sets. Different types of t-norms specify various approximation operators. One issue whether and how the different fuzzy approximation operators affect the result of attribute reduction is then arisen. This paper addresses this issue from the theoretical viewpoint by reviewing attribute reduction with fuzzy rough sets and then describing and proving some theorems which demonstrate the effects of the fuzzy approximation operators on the results of attribute reduction. First, we review some notions of attribute reduction with fuzzy rough sets, such as positive region, dependency degree and attribute reduction. We then present and prove some theorems which describe how and to what degree fuzzy approximation operators impact the performance of attribute reduction. Finally, we report some experimental simulation results which demonstrate the effectiveness and correctness of the theoretical contributions. One main contribution in this paper is that we have described and proven that each attribute reduction obtained using one type of fuzzy lower approximation operator always contains one reduction obtained using the other type of fuzzy lower approximation operator.