Fuzzy rough sets based on generalized residuated lattices
Introduction
The origins of residuation theory lie in the study of ideal lattices of rings in the 1930s. Ward and Dilworth investigated residuated lattices in a series of important papers [8], [9], [47], [48], [49]. Since that time, there have been substantial researches regarding some specific classes of residuated structures, such as [4], [5], [11], [13], [19], [20]. BL-algebras [19] are the generalization of the residuated structures on the unit interval [0, 1] induced by continuous t-norms. The monoidal t-norms based logic (MTL, for short) was obtained in [11] by considering left-continuous t-norms. When we consider the conjunction to be non-commutative, the corresponding truth structures are weak-pseudo-BL-algebras [13] or pseudo-MTL-algebras [20]. Similarly to BL-algebras and MTL-algebras, weak-pseudo-BL-algebras or pseudo-MTL-algebras are the generalization of the generalized residuated lattices on [0, 1] induced by left-continuous pseudo-t-norms [13], [17].
Rough sets theory was proposed by Pawlak [33], [34] as a formal tool to deal with imprecision and uncertainty in data analysis. Usefulness of rough sets theory has been fully demonstrated by its applications [42], [43].
In recent years, two different approaches have been formed to investigate rough sets theory, i.e., the constructive way and axiomatic way. In the constructive approach, lower and upper approximation operators are constructed and discussed from a binary relation. Diverse forms of rough sets have been proposed and studied with different binary relations [10], [12], [23], [25], [28], [32], [33], [38], [39], [55], [58], [62], [63]. Moreover, some of them are investigated with addition topological or algebraic structures [18], [22], [29], [35], [37], [53]. In contrast to the constructive approach, abstract lower and upper approximation operators are characterized by certain axioms [6], [26], [30], [31], [41], [44], [50], [54] in the axiomatic approach.
Morsi and Yakout studied fuzzy rough sets based on a left-continuous t-norm and its residual implication in [31]. In [38], Radzikowska and Kerre defined a broad family of -fuzzy rough sets which is determined by arbitrary implication and any t-norm , where the work of Morsi and Yakout is a special case of -fuzzy rough sets. Wu et al. discussed -fuzzy rough sets from the axiomatic approach in [51]. However, -fuzzy rough sets still investigate fuzzy rough sets on the unit interval. Since a left-continuous t-norm and its residual implication form an MTL-algebra on [0, 1], Radzikowska and Kerre proposed fuzzy rough sets based on residuated lattices in [39], [40], which were also called L-fuzzy rough sets for short and further studied in [52]. The axiomatic studies for L-fuzzy rough sets were made by She and Wang [41], in which residuated lattices are assumed to be regular. Hao and Li discussed the relationship between L-fuzzy rough sets and L-topologies in [18]. As a further discussion on [18], [41], Ma and Hu investigated topological structures of L-fuzzy rough sets and the relationship between upper (resp. lower) sets and lower (resp. upper) L-fuzzy rough approximation operators in [29]. Although Radzikowska and Kerre defined L-fuzzy rough sets based on generalized residuated lattices in [39], only left implication was considered. Most of properties of L-fuzzy rough sets discussed in [18], [29], [39], [40], [41], [52] are still limited to the commutative and integral generalized residuated lattices (residuated lattices, for short). Thus fuzzy rough sets based on generalized residuated lattices (generalized L-fuzzy rough sets, for short) should be studied. The aim of the present paper is to properly propose generalized L-fuzzy rough sets and then to investigate them from both constructive and axiomatic approaches, where the conjunctions are not always commutative.
This paper gives a complete study on the generalized L-fuzzy rough sets. In Section 2, we recall some fundamental concepts and related properties, especially the properties of generalized residuated lattices with examples to show the differences between generalized residuated lattices and residuated lattices. In Section 3, we propose generalized L-fuzzy rough sets, and investigate their properties and several classes of generalized L-fuzzy rough sets. Section 4 characterizes generalized L-fuzzy rough sets from the axiomatic approach. In Section 5, we discuss the relationship between generalized L-fuzzy rough sets and L-topologies. We construct a one-to-one correspondence between the set of L-fuzzy preorders and the set of left (resp. right) Alexandrov L-topologies on an arbitrary universe. In Section 6, we propose and study generalized I-fuzzy rough sets, which are based on generalized residuated lattices induced by left-continuous pseudo-t-norms. When the fuzzy relation is an I-fuzzy equivalence relation, generalized I-fuzzy rough sets are proved to be complete completely distributive (CCD, for short) rough sets [7]. In the final section, our researches are concluded.
Section snippets
Preliminaries
Definition 2.1 A generalized residuated lattice is an algebra (L, ∧, ∨, ⊗, →, ⇝, 0, 1, ⊤) such that the following conditions hold. (L, ∧, ∨, 0, 1) is a bounded lattice with the least element 0 and the greatest element 1. (L, ⊗, ⊤) is a monoid. For all a, b, c ∈ L we have the equivalence:→and ⇝ are called the left and right implication of ⊗, respectively.
Definition 2.2
A generalized residuated lattice (L, ∧, ∨, ⊗, →, ⇝, 0, 1, ⊤) is said to be
- (1)
commutative (resp. non-commutative) if ⊗ is commutative (resp. non-commutative),
- (2)
integral
Generalized L-fuzzy rough sets
In this section, we propose generalized L-fuzzy rough sets and investigate their properties and several classes of generalized L-fuzzy rough sets. When generalized residuated lattices are non-commutative, we have two implications, but not one. As a consequence, a generalized L-fuzzy rough set is not a pair, but a quadruple of approximation operators, which is in fact two pairs of approximation operators.
Assume that X is a nonempty universe and R is an arbitrary L-fuzzy relation on X, then the
Axiomatic characterizations of generalized L-fuzzy rough approximation operators
In this section, we study the axiomatic characterizations of generalized L-fuzzy rough approximation operators. It is shown that generalized L-fuzzy rough approximation operators can be characterized by abstract L-fuzzy sets operators, which guarantees the existence of certain types of L-fuzzy binary relations producing the same generalized L-fuzzy rough approximation operators. Definition 4.1 Let a mapping ψ:LX → LX. Then ψ is called a left lower L-fuzzy rough approximation operator if it satisfies the following
Topological structures of L-fuzzy approximation space
In this section, the notions of lower and upper sets are extended and the topological structures of L-fuzzy approximation space are investigated.
In Pawlak rough set theory, both and are Galois connections. In [1], [16], [36], [56], the fuzzy Galois connection was also studied. In [18], [29], it has been pointed out that the pairs and are fuzzy Galois connections using the subsethood operator [14], when generalized residuated lattices are commutative.
Generalized I-fuzzy rough sets
In this section, we investigate generalized I-fuzzy rough sets as an application of generalized L-fuzzy rough sets, where I denotes the unit interval [0,1].
A binary operation ∗:I × I → I is called a pseudo-t-norm [13], [17] on I if it is associative, increasing in each variable and has a unit element 1. In [13], it was pointed out that all the continuous pseudo-t-norms are commutative, so there does not exist any continuous non-commutative conjunction on I. A pseudo-t-norm ∗ is left-continuous if
Conclusion
In this paper, generalized L-fuzzy rough set was proposed with a quadruple of approximation operators. Generalized L-fuzzy rough sets were investigated from both constructive and axiomatic approaches. Moreover, generalized lower and upper sets were proposed to investigate the relationship between generalized L-fuzzy rough sets and L-topologies, and the connections between L-interior (resp. L-closure) operators and generalized approximation operators. Especially, the one-to-one correspondence
Acknowledgements
The authors thank the anonymous referees and the Editor-in-Chief, Professor Witold Pedrycz for their valuable comments and suggestions in improving this paper. This research was supported by the National Nature Science Foundation of China (Grant No. 61179038).
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