Topological and lattice structures of -fuzzy rough sets determined by lower and upper sets
Introduction
The concept of rough set was originally proposed by Pawlak [25], [26] as a mathematical approach to handle imprecision and uncertainty in data analysis. Usefulness and versatility of this theory have amply been demonstrated by successful applications in a variety of problems [33], [34].
At present, there are two different basic approaches in the rough set theory, i.e. the axiomatic approach and constructive approach. In the axiomatic approach, the lower and upper approximation operators are the primitive notions. A set of axioms is used to characterize approximation operators [2], [19], [22], [23], [32], [35], [37], [40]. In contrast to axiomatic approach, the constructive approach takes binary relations on the universe as the primary notions by which the lower and upper approximation operators are constructed. Diverse forms of the constructive approach have been started from the properties of binary relations, for example, reflexivity, (fuzzy) preorder, (fuzzy) equivalence relation, and (fuzzy) coverings, to investigate the essential properties of the lower and upper approximation operators generated by such relations [6], [7], [16], [18], [21], [24], [25], [30], [31], [39], [42], [43], [44].
However, these forms in conjunction with additional topological or algebraic structures on a universe are considered. In [15], [38], it was proven that, under a crisp preorder, the pair of lower and upper approximation operators is just a pair of interior and closure operators of an Alexandrov topology. In [27], it was pointed that inverse serial relations are the weakest relations which can induce topological spaces, and different relations based generalized rough set models can also induce different topological spaces. As the generalizations of rough sets from fuzzy sets point of view, in [29], it was verified that there exists a one-to-one correspondence between the set of all the lower approximation sets based on fuzzy preorder and the set of all fuzzy topologies satisfying so-called (TC) axiom. In [31], -fuzzy rough sets based on residuated lattices were proposed. In [32], the axiomatic characterizations of various -fuzzy rough sets were investigated. Moreover, it was examined that an -interior (resp. closure) operator of an -topological space could associate with an -preorder such that the corresponding lower (resp. upper) -fuzzy approximation operator was the -interior (resp. closure) operator. In [10], [17], under the context of left continuous t-norms and residuated lattices, a one-to-one correspondence between the set of all fuzzy preorders and the set of all Alexandrov fuzzy topologies was obtained, where the fuzzy preorders and fuzzy topologies were connected by the upper sets and lower -fuzzy approximation operators, respectively. However, it has not been studied on the relationships between upper sets and lower -fuzzy approximation operators and applying the upper set to study -fuzzy rough set theory, and the aim of the present paper is to investigate and solve these questions.
In this paper, we focus on a complete study on the topological and lattice structures in -fuzzy approximation spaces by lower and upper sets. In Section 2, we recall some fundamental concepts and related properties. In Section 3, we propose some new properties of the -fuzzy approximation operators. In Section 4, the main part of this paper, we prove that the upper (resp. lower) set is equivalent to the lower (resp. upper) -fuzzy approximation set under a reflexive -relation. Then by the upper (resp. lower) set, we point out that an -preorder is the equivalence condition under which the set of all the lower (resp. upper) -fuzzy approximation sets and the Alexandrov -topology coincide. At the same time, associating with an -preorder, we verify that the equivalence condition that -interior (resp. closure) operator accords with the lower (resp. upper) -fuzzy approximation operator. At last, when the -relation is reflexive, we construct a complete lattice by the set of all the lower (resp. upper) -fuzzy approximation sets.
Section snippets
Residuated lattices
A residuated lattice is an algebra such that is a bounded lattice with the least element 0 and the greatest element 1, is a communicative monoid, and form an adjoint pair, i.e. if and only if , for each . A residuated lattice is complete if the underlying lattice is complete. will be reserved as , for all . Throughout this paper, always denotes a complete residuated lattice.
Some basic properties of residuated
New properties of -fuzzy approximation operators
In this section, associated with related operations of -sets, some new properties of -fuzzy approximation operators are obtained.
The theorem below generalizes the results in Theorem 2.8 (2) and (3) by replacing with . Theorem 3.1 Let be an -fuzzy approximation space. Then for all and , . . . .
Proof
- (1)
By Lemma 2.1 (6)–(9), we get
Topological and lattice structures determined by lower and upper sets
In this section, the notions of lower and upper sets are introduced, by which the topological and lattice structures in an -fuzzy approximation space are investigated.
Conclusion
In this paper, the topological and lattice structures of -fuzzy rough sets by lower and upper sets were determined. Our main conclusions are list as follows:
- (1)
The upper (resp. lower) set is equivalent to the lower (resp. upper) -fuzzy approximation set under a reflexive -relation.
- (2)
An -preorder is the equivalece condition under which the set of all the lower (resp. upper) -fuzzy approximation sets and the Alexandrov -topology coincide.
- (3)
Associating with an -preorder, the equivalence condition
Acknowledgements
The authors thank the anonymous reviewers and the Editor-in-Chief, Professor Witold Pedrycz for their valuable suggestions in improving this paper. This research was supported by the Natural Science Foundation of Shandong Province (Grand Nos. ZR2010AL004 and ZR2011FL017), the National Natural Science Foundation of China (Grand Nos. 61179038 and 70771081).
References (44)
Fuzzy functions and their fundamental properties
Fuzzy Sets and Systems
(1999)Fuzzy groups, fuzzy functions and fuzzy equivalence relations
Fuzzy Sets and Systems
(2004)Topological properties of the class of generators of an indistinguishability operator
Fuzzy Sets and Systems
(2004)- et al.
On rough set and fuzzy sublattice
Information Sciences
(2011) L-fuzzy sets
Journal of Mathematical Analysis and Applications
(1967)- et al.
The relationship between L-fuzzy rough set and L-topology
Fuzzy Sets and Systems
(2011) On the fundamentals of fuzzy set theory
Journal of Mathematical Analysis and Applications
(1996)On the relationship between modified sets, topological spaces and rough sets
Fuzzy Sets and Systems
(1994)Rough set approach to incomplete information systems
Information Sciences
(1998)- et al.
Fuzzy preorder and fuzzy topology
Fuzzy Sets and Systems
(2006)
Fuzzy topological spaces and fuzzy compactness
Journal of Mathematical Analysis and Applications
An axiomatic characterization of a fuzzy generalization of rough sets
Information Sciences
Axiomatics for fuzzy rough sets
Fuzzy Sets and Systems
Topology vs generalized rough sets
International Journal of Approximate Reasoning
On the topological properties of fuzzy rough sets
Fuzzy Sets and Systems
A comparative study of fuzzy rough sets
Fuzzy Sets and Systems
An axiomatic approach of fuzzy rough sets based on residuated lattices
Computers and Mathematics with Applications
On axiomatic characterizations of crisp approximation operators
Information Sciences
Two views of the theory of rough sets in finite universes
International Journal of Approximation Reasoning
Constructive study of fuzzy sets and rough sets
Information Sciences
Constructive and algebraic methods of the theory of rough sets
Information Sciences
Reduction and axiomization of covering generalized rough sets
Information Sciences
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