Elsevier

Fuzzy Sets and Systems

Volume 336, 1 April 2018, Pages 54-86
Fuzzy Sets and Systems

On (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices

https://doi.org/10.1016/j.fss.2017.07.010Get rights and content

Highlights

  • A generalized rough set model, (,&)-fuzzy rough set, is proposed from both constructive and axiomatic approaches.

  • The topological properties of (,&)-fuzzy rough sets are discussed, especially from the categorical point of view.

  • A new model of fuzzy rough set, (,)-fuzzy rough set, is introduced and some related properties are investigated.

Abstract

Rough sets were introduced by Pawlak as a formal tool for dealing with imprecision and uncertainty in data analysis. After that time, varieties of fuzzy generalizations of rough approximations have been investigated in the literature. In particular, generalized fuzzy rough sets determined by a triangular norm have been considered. On the other hand, binary operations & and ⊙ on complete residuated and co-residuated lattice L can be regarded as two more extensive operations than left continuous triangular norms and right continuous triangular conorms, respectively. Therefore, as a further generalization of the notion of rough sets, this paper is devoted to proposing (,&)-fuzzy rough sets based on residuated and co-residuated lattices from both constructive and axiomatic approaches. In the constructive approach, we define a pair of lower and upper L-fuzzy rough approximation operators determined by ⊙ and &, respectively. Meanwhile, various classes of (,&)-fuzzy rough sets are investigated. In the axiomatic approach, the axiomatic characterizations of different (,&)-fuzzy rough approximation operators are studied. Furthermore, the topological properties of (,&)-fuzzy rough sets are discussed, especially from the categorical point of view. At the end of this paper, we give a brief introduction to a new model of fuzzy rough sets and discuss some properties of it.

Section snippets

A brief review on the development of rough sets theory

The theory of rough sets was developed by Pawlak [64], [65]. For the past few years, there are two approaches (i.e., the constructive approach and the axiomatic approach) to propose it. In the constructive approach, two approximation operators are constructed and investigated based on a binary relation (see, e.g., [23], [44], [63], [68], [71], [96], [110]). Meanwhile, their topological or algebraic properties are discussed in the literature (see, e.g., [30], [57], [66], [67], [69], [95]). By

Preliminaries

In this section, we give a brief introduction to some basic concepts and related properties, which are used throughout the paper.

A binary operation :[0,1]×[0,1][0,1] (resp. :[0,1]×[0,1][0,1]) is called a triangular norm (t-norm, for short) (resp. triangular conorm (t-conorm, for short)) [3], [39], [74] on [0,1] if it is commutative, associative, increasing in each argument and has a unit element 1 (resp. 0).

By a complete residuated lattice [16], [29], [33], [86], [87], we mean an algebra (L,

(,&)-Fuzzy rough sets based on residuated and co-residuated lattices

Let X,Y be two nonempty universes and R be an L-fuzzy relation from X to Y. Then the triple (X,Y,R) is called an L-fuzzy approximation space. When X=Y and R is an L-fuzzy relation on X, we say (X,R) an L-fuzzy approximation space.

Definition 3.1

Let (X,Y,R) be an L-fuzzy approximation space. Then the following two mappings R_,R&:LYLX are defined as follows:R_(A)(x)=yYRN(x,y)A(y),R&(A)(x)=yYR(x,y)&A(y), for all ALY and xX. The operators R_ and R& are called ⊙-lower and &-upper L-fuzzy rough

Axiomatic characterizations of (,&)-fuzzy rough sets

In this section, we give the axiomatic characterizations of ⊙-lower and &-upper L-fuzzy rough approximation operators.

Definition 4.1

A mapping Ψ:LYLX is called a ⊙-lower L-fuzzy approximation operator if it satisfies the following axioms:

  • (f1)

    Ψ(iΛAi)=iΛΨ(Ai) for all {Ai:iΛ}LY and any index set Λ,

  • (f2)

    Ψ(αYA)=αXΨ(A) for all ALY and αL.

Proposition 4.2

A mapping Ψ:LYLX is a-lower L-fuzzy approximation operator if and only if there exists a unique L-fuzzy relation R such that Ψ=R_.

Proof

Sufficiency. It follows immediately from

Topological properties of (,&)-fuzzy rough sets

In this section, firstly, we introduce the concepts of L-semi-topology and L-semi-co-topology to study the topological properties of (,&)-fuzzy rough sets. And then, similarly to the discussions of [67], we investigate the relationship among L-fuzzy approximation spaces, L-Čech closure spaces and L-pretopological spaces from the categorical point of view.

(,)-Fuzzy rough sets

In this section, we give a new model for fuzzy rough set theory, (,)-fuzzy rough sets, based on the unit interval and investigate some properties of it. It should be pointed out that, in this section, we always consider left continuous t-norms and right continuous t-conorms, that is, (a,jJbj)=jJ(a,bj) and (a,jJbj)=jJ(a,bj), respectively, for all a[0,1] and {bj:jJ}[0,1], where J is any index set.

Let X,Y be two nonempty universes and R be an [0,1]-fuzzy relation from X to Y.

Conclusions

In this paper, we continue to investigate two rough set models from the mathematical point of view. The main contributions of this paper are listed as follows.

  • As an extension of the works [35], [38], [60], we give ⊙-lower and &-upper L-fuzzy rough approximation operators on the basis of complete residuated and co-residuated lattices. In addition, we list some basic properties of those two L-fuzzy rough approximation operators and characterize them on diverse L-fuzzy relations, such as serial,

Acknowledgements

The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).

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