Single axioms for lower fuzzy rough approximation operators determined by fuzzy implications
Introduction
Rough set theory was proposed by Pawlak [17], [18] to deal with uncertainty due to incompleteness and indiscernibility in information systems. However, Pawlak introduced rough sets with equivalence relations, which are too strict to limit the applicabilities of rough sets. The equivalence relation in rough sets was replaced with a general binary relation or a neighborhood operator [3], [33] to expand rough sets. Dubois and Prade [6] combined fuzzy sets and rough sets to further expand rough sets in the fuzzy environment. Moreover, the combinations of fuzzy sets and rough sets were investigated with different fuzzy logic operations and binary fuzzy relations in [2], [15], [16], [19], [20], [26], [28], [34], where fuzzy implications play an important role in the extensions of fuzzy rough sets.
In recent years, there are mainly two different approaches for the study of rough set theory, i.e., the constructive way and axiomatic way. In the constructive approach, lower and upper approximation operators are constructed and investigated on a binary relation. On the contrary, abstract lower and upper approximation operators are characterized by certain axioms in the axiomatic approach. The axiomatic approach in rough set theory has been discussed in many published works [11], [22], [23], [32], [35], [37]. Especially, the minimal independent axiom sets characterizing rough sets have been studied in [12], [13], [31], [36]. Considering rough approximation operators in the fuzzy environment, Liu [12] sought for one axiom for fuzzy rough approximation operators. In [30], Wu et al. characterized -fuzzy rough approximation operators [15] by only one axiom, where S and T denote a t-conorm and a t-norm on the unit interval , respectively. However, t-(co)norms are assumed to be continuous and, t-norms and t-conorms are assumed to be dual with respect to the standard negation in [30], which seem to be too strict to limit the applicabilities of those axiomatic characterizations.
The axiomatic characterizations of lower fuzzy rough approximation operators determined by fuzzy implications have been studied in [16], [27]. For example, Morsi and Yakout [16] investigated the axiomatic characterizations of fuzzy rough sets based on left-continuous t-norms and R-implications. As a left-continuous t-norm and its R-implication form a special residuated lattice, i.e., an MTL-algebra [7], -fuzzy rough sets were studied in [14], [21], [25] in the axiomatic approach, where denotes a residuated lattice. Wu et al. [27] investigated the axiomatic characterizations of -fuzzy rough sets [19], where I denotes a fuzzy implication. However, the axiomatic systems mentioned above consist of at least two axioms. Wu et al. [29] used one axiom to characterize fuzzy rough approximation operators determined by fuzzy implications, where fuzzy implication I is always assumed to be continuous and satisfy both exchange principle and ordering property. Meanwhile, fuzzy rough approximation operators determined by fuzzy implications cannot be characterized by only one axiom in [29], when fuzzy relation is symmetric. In fact, Baczyński and Jayaram [1] pointed out that a fuzzy implication satisfying continuity and exchange principle is an S-implication with a continuous fuzzy negation. Moreover, an S-implication satisfying ordering property must be an S-implication with a strong fuzzy negation. Hence the axiomatic characterizations of I-lower fuzzy rough approximation operators by a single axiom in [12], [29], [30] are limited into a special case, where fuzzy implication is a continuous S-implication. In [10], lower fuzzy rough approximation operator of an -fuzzy rough set was characterized by a single axiom, when fuzzy implication in satisfies regular property. However, that assumption is too strict for a general R-implication. For example, the R-implication based on minimum does not satisfy regular property. These suggest that single axioms for I-lower fuzzy rough approximation operators should be studied more thoroughly in this context, which further studies the axiomatic characterizations of -fuzzy rough sets [19], I-fuzzy rough sets [28] and -fuzzy rough sets. Moreover, the single axioms for I-lower fuzzy rough approximation operators have axiomatic characterizations of both fuzzy rough sets in [12] and -fuzzy rough sets in [30] as special cases.
In this paper, we discuss the axiomatic characterizations of I-lower fuzzy rough approximation operators by a single axiom. Firstly, we investigate single axioms for I-lower fuzzy rough approximation operators determined by specific fuzzy implications, such as S-implications and R-implications. Secondly, we further study single axioms for I-lower fuzzy rough approximation operators determined by fuzzy implications satisfying left neutrality property and regular property instead of specific fuzzy implications. Since both fuzzy rough sets and -fuzzy rough sets are special cases of -fuzzy rough sets, we derive the conclusions in [12], [29], [30] as special cases of our research.
The content of the paper is organized as follows. In Section 2, we recall some fundamental concepts and related properties of -fuzzy rough sets. In Section 3, we characterize I-lower fuzzy rough approximation operators by only one axiom, where fuzzy implication I is an S-implication generated from a right-continuous t-conorm. Section 4 studies single axioms for I-lower fuzzy rough approximation operators determined by R-implications. In Section 5, we discuss the axiomatic characterizations of I-lower fuzzy rough approximation operators determined by fuzzy implications satisfying left neutrality property and regular property. Moreover, fuzzy implication is assumed to satisfy ordering property, when we discuss single axioms for I-lower fuzzy rough approximation operators on a reflexive, T-transitive or T-Euclidean fuzzy relation. In the final section, we present some conclusions of our research.
Section snippets
Preliminaries
Let be the unit interval and X be a universe. Then a mapping is called a fuzzy set on X. The family of all fuzzy sets on X is denoted as . Let . Then a fuzzy set is a constant, while for all , denoted as . For convenience, we also denote the empty set ∅ and the universe X as and , respectively. Especially, if for all , then A is denoted as . A decreasing function is called a fuzzy negation, if it satisfies
Axioms for I-lower fuzzy rough approximation operators determined by S-implications
From the view of S-implications, Wu et al. [30] characterized I-lower fuzzy rough approximation operators by only one axiom, where I is an S-implication based on a continuous t-conorm and the standard negation. Moreover, Wu et al. [29] characterized I-lower fuzzy rough approximation operators by a single axiom, when fuzzy implication is continuous and satisfies both exchange principle and ordering property. In [29], I-lower fuzzy rough approximation operators cannot be characterized by one
Axioms for I-lower fuzzy rough approximation operators determined by R-implications based on left-continuous t-norms
Although an R-implication based on a left-continuous t-norm satisfies (EP), (NP) and (OP) [1], Example 2.2 shows that an R-implication based on minimum does not satisfy continuity or regular property. Hence neither the conclusions in [29], [30] nor Section 3 can characterize I-lower fuzzy rough approximation operator determined by a general R-implication. In this section, we study the axiomatic characterizations of I-lower fuzzy rough approximation operators determined by general R-implications
Axioms for I-lower fuzzy rough approximation operators determined by fuzzy implications satisfying left neutrality property and regular property
By Note 2.3, a QL-implication satisfies regular property, when it is generated by a strong fuzzy negation. It is obvious that a QL-implication satisfies (NP). Moreover, Dimuro et al. [4] pointed out that if 0 is the neutral element of grouping function G and N is a strong fuzzy negation, then a G-implication satisfies both (NP) and regular property. From a more general form, we study the single axioms for I-lower fuzzy rough approximation operators determined by fuzzy implications satisfying
Conclusion
Since the axiomatic characterizations of I-lower fuzzy rough approximation operators by only one axiom in [12], [29], [30] focus on continuous S-implications, we further investigate single axioms for I-lower fuzzy rough approximation operators determined by semicontinuous S-implications with continuous fuzzy negations and strict fuzzy negations, respectively. We propose two types of axiomatic characterizations of I-lower fuzzy rough approximation operators. We also study single axioms for I
Acknowledgements
The author would like to thank the Editors and the anonymous reviewers for their valuable comments and suggestions in improving this paper. This research was supported by the National Natural Science Foundation of China (Grant No. 11501334) and the Natural Science Foundation of Shandong Province (Grant No. ZR2015FQ014).
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