Intuitionistic Fuzzy Rough Approximation Operators Research Articles

Representations of L-fuzzy rough approximation operators

In recent years, there has been tremendous interest in developing lattice-valued rough set theory. In this framework, we primarily take into account the following two problems. Firstly, we include the well-known intuitionistic, interval-valued, neutrosophic and Pythagorean fuzzy rough sets into the framework of lattice-valued rough sets. Specifically, the four kinds of rough sets can be considered as special lattice-valued rough sets. Secondly, based on a completely distributive lattice L, we provide representations of the upper and lower L-fuzzy rough approximation operators by using four kinds of cut sets of an L-fuzzy set and an L-fuzzy relation, which generalizes the existing results in the case that L=[0,1]. In particular, we show that representations of intuitionistic and interval-valued fuzzy rough approximation operators provided by Zhou and Sun are special examples of our proposed representations.

Using single axioms to characterize (S, T)-intuitionistic fuzzy rough approximation operators

In this paper axiomatic characterizations of relation-based intuitionistic fuzzy rough approximation operators determined by an intuitionistic fuzzy triangular norm T and its dual intuitionistic fuzzy triangular conorm S on \([0, 1]\times [0, 1]\) are proposed. The constructive definitions and properties of S-lower and T-upper intuitionistic fuzzy rough approximation operators are first introduced. Operator-oriented characterizations of (S, T)-intuitionistic fuzzy rough approximation operators are then explored. Different sets of independent axioms for characterizing the essential properties of (S, T)-intuitionistic fuzzy rough approximation operators generated by various intuitionistic fuzzy relations are presented. Finally, it is examined that these sets of axioms can all be replaced by single axioms.

Intuitionistic Fuzzy Rough Approximation Operators

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 relations by means of topology.

Independence of axiom sets on intuitionistic fuzzy rough approximation operators

Algebraic approach is widely used in the study of rough sets. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. Axiomatic approaches have been used in the study of intuitionistic fuzzy rough sets. Intuitionistic fuzzy rough approximation operators are defined by axioms, and different axiom sets of lower and upper intuitionistic fuzzy rough approximation operators guarantee the existence of different types of intuitionistic fuzzy relations. However, the independence of the axioms in each of those axiom sets has not been proved. If the axioms are not independent, one of the axioms is redundant. In this paper, we prove that no axioms can be derived from other axioms, which means the axioms in the axiom set are independent. For each axiom set, we examine the independence of axioms and present the independent axiom sets characterizing intuitionistic fuzzy rough approximation operators. Thus, we improve the intuitionistic fuzzy rough set theory.

A general frame for intuitionistic fuzzy rough sets

Intuitionistic fuzzy rough sets are investigated in a general framework which includes generalizations of many related results in early literatures. A new definition of intuitionistic fuzzy rough sets is given with the analysis of its basic properties based on the notion of two universes, general binary relations, and a pair (T, I) of intuitionistic fuzzy t-norm T and intuitionistic fuzzy implicator I. For t-representable intuitionistic fuzzy t-norms and its residual implicators, intuitionistic fuzzy rough approximation operators are defined by axioms, and its connections with special intuitionistic fuzzy relations are investigated.

Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings

Study mainly investigates generalized intuitionistic fuzzy (IF) rough sets based on IF coverings. By using an IF covering, an IF triangular norm, and an IF implicator, two pairs of generalized lower and upper IF rough approximation operators have been constructed, and some fundamental properties are examined. Then, we give some conditions under which the generalized lower IF rough approximation operator is an IF interior operator and the generalized upper IF rough approximation operator is an IF closure operator. Furthermore, the duality of the generalized IF rough approximation operators is discussed. In addition, we propose some concepts and conditions for two intuitionistic coverings to generate an identical lower IF rough approximation operator and an identical upper IF rough approximation operator with the purpose of removing the redundancy in an IF covering. Finally, we compare the IF-neighborhood-oriented IF rough approximation operators with IF-neighborhood-operator-oriented IF rough approximation operators and obtain the conditions under which some or all of these approximation operators are equivalent.

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 topologies based on intuitionistic fuzzy reflexive and transitive relations

Topologies and rough set theory are widely used in the research field of machine learning and cybernetics. An intuitionistic fuzzy rough set, which is the result of approximation of an intuitionistic fuzzy set with respect to an intuitionistic fuzzy approximation space, is an extension of fuzzy rough sets. For further studying the theories and applications of intuitionistic fuzzy rough sets, in this paper, we investigate the topological structures of intuitionistic fuzzy rough sets. We show that an intuitionistic fuzzy rough approximation space can induce an intuitionistic fuzzy topological space in the sense of Lowen if and only if the intuitionistic fuzzy relation in the approximation space is reflexive and transitive. We also examine the sufficient and necessary conditions that an intuitionistic fuzzy topological space can be associated with an intuitionistic fuzzy reflexive and transitive relation such that the induced lower and upper intuitionistic fuzzy rough approximation operators are, respectively, the intuitionistic fuzzy interior and closure operators of the given topology.

An Interval-Valued Intuitionistic Fuzzy Rough Set Model

Given a widespread interest in rough sets as being applied to various tasks of data analysis it is not surprising at all that we have witnessed a wave of further generalizations and algorithmic enhancements of this original concept. This paper proposes an interval-valued intuitionistic fuzzy rough model by means of integrating the classical Pawlak rough set theory with the interval-valued intuitionistic fuzzy set theory. Firstly, some concepts and properties of interval-valued intuitionistic fuzzy set and interval-valued intuitionistic fuzzy relation are introduced. Secondly, a pair of lower and upper interval-valued intuitionistic fuzzy rough approximation operators induced from an interval-valued intuitionistic fuzzy relation is defined, and some properties of approximation operators are investigated in detail. Furthermore, by introducing cut sets of interval-valued intuitionistic fuzzy sets, classical representations of interval-valued intuitionistic fuzzy rough approximation operators are presented. Finally, the connections between special interval-valued intuitionistic fuzzy relations and interval-valued intuitionistic fuzzy rough approximation operators are constructed, and the relationships of this model and the others rough set models are also 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 generalized intuitionistic fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by binary relations satisfy many interesting properties. Various generalizations of Pawlak’s rough approximations have been made in the literature over the years. This paper proposes a general framework for the study of relation-based intuitionistic fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper intuitionistic fuzzy rough approximation operators induced from an arbitrary intuitionistic fuzzy relation are defined. Basic properties of the intuitionistic fuzzy rough approximation operators are then examined. By introducing cut sets of intuitionistic fuzzy sets, classical representations of intuitionistic fuzzy rough approximation operators are presented. The connections between special intuitionistic fuzzy relations and intuitionistic fuzzy rough approximation operators are further established. Finally, an operator-oriented characterization of intuitionistic fuzzy rough sets is proposed, that is, intuitionistic fuzzy rough approximation operators are defined by axioms. Different axiom sets of lower and upper intuitionistic fuzzy set-theoretic operators guarantee the existence of different types of intuitionistic fuzzy relations which produce the same operators.