The necessary and sufficient conditions for a fuzzy relation being ⊤-Euclidean
Introduction
The rough set theory, initiated by Pawlak in 1982 [16], is a tool to study intelligent systems characterized by insufficient and incomplete information [17]. A key and primitive notion in Pawlak’s rough set model is an equivalence relation. However, the requirement of an equivalence relation seems to be a very restrictive condition in the applications of rough set theory. To address this issue, many authors have generalized the notion of approximation operators by non-equivalence binary relations [1], [4], [15], [23], [24], [34], [35], [40].
Dubois and Prade presented the concept of rough fuzzy sets and fuzzy rough sets [5], [6] by means of integrating the rough set theory with the fuzzy set theory [36]. Since then, a number of approximation operators have been studied on various relations with respect to Zadeh’s fuzzy sets and their extensions in the literature [3], [8], [9], [12], [19], [22], [26], [37], [38], [39].
There are different definitions of generalized fuzzy rough sets, for instance, (⊥, ⊤)-generalized fuzzy rough sets, (θ, σ)-generalized fuzzy rough sets and (I, ⊤)-fuzzy rough sets, etc. Hu and Huang introduced the concept of (⊥, ⊤)-generalized fuzzy rough sets [10] on the basis of dual triangular norms. (θ, σ)-generalized fuzzy rough sets were proposed by Mi and Zhang [13] based on a residual implication and its dual. Radzikowska and Kerre defined a broad class of fuzzy rough sets, each one of which, called (I, ⊤)-fuzzy rough sets [18], is represented by a fuzzy implication I and a t-norm ⊤, while the properties and axiomatic characterization of (I, ⊤)-fuzzy rough sets corresponding to an arbitrary fuzzy relation or a special relation were examined by Wu et al. [29].
A general framework for the study of rough set and fuzzy rough set was established by Yao [33], [34], [35] and Wu et al. [30], [31], respectively. They started from the investigation of the properties of the lower and upper approximation operators generated by some binary relations, for instance, serial, reflexive, symmetric, and transitive ones. However, the research on the necessary and sufficient conditions for a fuzzy relation being Euclidean in fuzzy rough approximation space has not been faced, that is, the essential properties of lower and upper approximation operators generated by Euclidean fuzzy relation have not been investigated so far. This problem was presented by Wu and Zhang [31] and will be solved in this paper.
The other parts of this paper are organized as follows. In Section 2, we recall basic concepts of fuzzy sets and fuzzy logic to be used in the following sections. In Section 3, we study the necessary and sufficient conditions when the fuzzy relation is said to be ⊤-Euclidean in (⊥, ⊤)-fuzzy rough approximation space. While, in Section 4, the necessary and sufficient conditions are in discussion when the fuzzy relation is referred to as ⊤-Euclidean with respect to fuzzy rough approximation operators based on residual operators. In Section 5, we study the necessary and sufficient conditions for a fuzzy relation to be ⊤-Euclidean with respect to (I, ⊤)-fuzzy rough approximation operators. Finally, some concluding remarks and suggestions for further work are made in Section 6.
Section snippets
Preliminaries
In this section, we briefly recall basic concepts of fuzzy sets as well as fuzzy logic.
A binary operation ⊤(⊥) on the closed unit interval [0, 1] is a t-norm (t-conorm), if it satisfies (T1) (x⊤y)⊤z = x⊤(y⊤z), (T2) x⊤y = y⊤x, (T3) , (T4) x⊤1 = x(x⊥0 = x), where x, y, z ∈ [0, 1]. If t-norm ⊤ and t-conorm ⊥ satisfy 1 − x⊥y = (1 − x)⊤(1 − y) for all x, y ∈ [0, 1], then they are called dual. And in what follows, unless otherwise stated, we always assume that t-norm ⊤ and t-conorm ⊥ are dual. Definition 2.1 A function N:[0,
The necessary and sufficient conditions of ⊤-Euclidean fuzzy relation in (⊥, ⊤)-fuzzy rough approximation space
If and , then we define R∘⊤A(x) = ∨y∈Y{R(x, y)⊤A(y)} and for all x ∈ X. Definition 3.1 Let X and Y be two finite and nonempty universes of discourse and R a fuzzy relation from X to Y. For an arbitrary fuzzy set , the lower and upper approximation of A, denoted respectively by R(A) and , with respect to the approximation space (X, Y, R) are fuzzy sets of X whose membership functions are defined pointwise by the formulas[31]
The necessary and sufficient conditions of ⊤-Euclidean fuzzy relation in (θ, σ)-fuzzy approximation space based on residual operators
In this section we will investigate the corresponding results of (θ, σ)-fuzzy rough sets. Firstly, we recall here the definition of (θ, σ)-fuzzy rough sets. And throughout this section, ⊤ will be a left-continuous t-norm, therefore, its dual t-conorm ⊥ is right-continuous. Definition 4.1 Let X and Y be two finite and nonempty universes of discourse, and R a fuzzy relation from X to Y. For a fuzzy set , the θ-lower and σ-upper approximation of A, denoted respectively by Rθ (A) and , are fuzzy sets[13]
The necessary and sufficient conditions of ⊤-Euclidean fuzzy relation in (I, ⊤)-fuzzy approximation space
In this section we characterize the relationship of ⊤-Euclidean fuzzy relation in terms of the properties of (I, ⊤)-fuzzy approximation operators. And throughout this section, ⊤ will be a left-continuous t-norm. A special case of I, an R-implication, is also investigated, and thus an interesting result can be obtained. Definition 5.1 Let ⊤ and I be a t-norm and an implication on [0, 1], respectively, X and Y be two finite and nonempty universes of discourse and R be a fuzzy relation from X to Y. For a fuzzy[18], [29]
Conclusions
Rough set theory can be developed in at least two ways, namely, the constructive and the axiomatic approaches. And it seems that the constructive approach is more useful for practical applications of the rough set theory in the fuzzy environment. In this paper, we have worked out the necessary and sufficient conditions for a fuzzy relation being ⊤-Euclidean in different fuzzy rough approximation spaces which can be regarded as the completion of the works in constructive approaches by Yao [33],
Acknowledgements
The authors are extremely grateful to the anonymous referees and professor Witold Pedrycz, Editor-in-Chief, for their critical suggestions for improvements. This research was supported by the National Natural Science Foundation of China (Grand Nos. 61179038, 70771081).
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