On (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices
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 (resp. ) is called a triangular norm (t-norm, for short) (resp. triangular conorm (t-conorm, for short)) [3], [39], [74] on 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
-Fuzzy rough sets based on residuated and co-residuated lattices
Let be two nonempty universes and R be an L-fuzzy relation from X to Y. Then the triple is called an L-fuzzy approximation space. When and R is an L-fuzzy relation on X, we say an L-fuzzy approximation space. Definition 3.1 Let be an L-fuzzy approximation space. Then the following two mappings are defined as follows: for all and . The operators and 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 is called a ⊙-lower L-fuzzy approximation operator if it satisfies the following axioms: for all and any index set Λ, for all and .
Proposition 4.2
A mapping is a ⊙-lower L-fuzzy approximation operator if and only if there exists a unique L-fuzzy relation R such that .
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, and , respectively, for all and , where J is any index set.
Let be two nonempty universes and R be an -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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