Axiomatic approaches to rough approximation operators on complete completely distributive lattices
Introduction
Rough set theory is an important tool for dealing with the vagueness and granularity in information systems [27]. It was first proposed by Pawlak in 1982 [26]. Now it has been successfully used in many fields including data mining, decision making, artificial intelligence, etc. [13], [25], [31], [32].
There are two basic approaches to investigate rough sets, namely the constructive approach and axiomatic approach. From the perspective of the constructive approach, binary relations and coverings on the universe are the primitive notions, which construct the pair of lower and upper approximation operators. The equivalence relation or the partition is essential for Pawlak’s rough set model. However, the equivalence relation is too restrictive for practical applications. It is hereby replaced by a tolerance relation [24], [34], a similarity relation [33], [35], or even an arbitrary binary relation [46]. Similarly, partition is generalized to covering [2], [3], [50]. By combing fuzzy set theory with rough set theory, Dubois and Prade introduced the fuzzy rough set model which approximates a fuzzy set by a fuzzy equivalence relation [7], [8]. Hereafter, the fuzzy rough set has been widely concerned [20], [21], [23], [30], [38], [41].
From the perspective of the axiomatic approach, the lower and upper rough approximation operators are the primitive notions which are characterized by sets of axioms. Zakowski [48] and Comer [4] studied a set of axioms on approximation operators within the context of Pawlak information systems. Yao [44], [45] extended axiomatic approaches to rough set algebras constructed by arbitrary binary relations. Lin and Liu [18] suggested six axioms on rough approximation operators in the framework of topological spaces. Zhu et al. discussed the axiomatic approaches of covering rough sets [50], [51]. Morsi and Yakout investigated the axiomatic approaches of fuzzy rough sets [21] and Mi and Zhang [20] and Wu and Zhang [40], [41] researched further more. Li [17] studied an axiomatic characterization of probabilistic rough sets and Yang et al. [43] introduced constructive and axiomatic approaches to the hesitant fuzzy rough set. Based on these researches of axiomatic approaches, algebra formalisms for information systems were discussed [1], [14] and modal logics for rough set theory were proposed [15], [16], [37].
Pawlak’s rough set model has been generalized from different aspects due to demands from different practical situations. It becomes an interesting area to study the common properties of existing rough set models and bring them into a framework. This work was first started over the power lattice [6], [12], [22], [28]. In 2006, Chen et al. initiated the study of rough sets on a CCD lattice [5], which is an extension of power lattice. They proposed a pair of rough approximation operators based on a covering of a CCD lattice, which brings rough sets based on tolerance relation, similarity relation and fuzzy equivalence relation into a unified framework. Gao et al. researched the topological properties of the pair of rough approximation operators [10]. Qin et al. further discussed the rough approximation operators on a CCD lattice based on the concept of neighborhood [29]. Rough sets on a CCD lattice based on ordinary binary relations were introduced in [49], which unifies rough sets based on ordinary binary relations, rough fuzzy sets and the interval-valued rough fuzzy sets. Tantawy and Mustafa proposed rough approximations via the ideal of a complete atomic Boolean lattice [36] and Xiao et al. studied the properties of rough set constructed by the ideals of a CCD lattice [42]. Dzik et al. represented expansions of bounded distributive lattices by rough sets on it [9].
In this paper we further discuss the properties of the rough approximation operators in [49] and the axiomatic approaches of these operators, including the axiomatic approaches of selected classes of rough approximation operators. These axiomatic approaches are the generalization of the axiomatic approaches of rough sets, rough fuzzy sets and fuzzy rough sets. Usually, the axiomatic approaches of lower rough approximation are proposed. Then the upper approximation operators are directly obtained by the duality of the lower and upper rough approximation operators [20], [40], [41]. But there exists no proper inverse on a CCD lattice to define the duality of the lower and upper rough approximation operators. Therefore, in this study we exploit the concept of Galois connection to discuss the relationship between the lower and the upper approximation operators. Through this study, it is concluded that fuzzy rough approximation operators based on a similarity relation can be seen as special cases of rough approximation operators on a CCD lattice.
This paper is organized as follows. In Section 2, we briefly introduce the concept of CCD lattice and the relation over a CCD lattice. Section 3 introduces the definition and discusses the properties of the pair of rough approximation operators on an arbitrary binary relation over a CCD lattice. Furthermore, the connections between rough approximation operators based on a relation and based on a covering of a CCD lattice are studied. In Section 4, we research the axiomatic approaches of rough approximation operators based on relations. The axiomatic approaches of generalized rough approximation operators based on two universes are studied in Section 5. Section 6 concludes this paper.
Section snippets
Preliminaries
Throughout this paper, the complete completely distributive lattice, called CCD lattice for short, plays a fundamental role. In this section, we recall some basic concepts on a CCD lattice.
A CCD lattice (L, ⩽, ∨, ∧) is a complete lattice which satisfies the completely distributivity laws as follows: where I and Ji are two nonempty index sets, aij ∈ L and ∏i ∈ IJi is the Cartesian product of Ji, i ∈ I. In the following
Rough approximation operators on a CCD lattice and their properties
In this section, we first recall the concept of rough approximation operators based on a relation over a CCD lattice and discuss their properties. Then we review the rough approximation operators based on a covering of a CCD lattice and discuss the relationship between two kinds of rough approximation operators. Finally, some properties of selected classes of rough approximation operators based on a relation are introduced.
The axiomatic approaches of rough approximation operators on a CCD lattice
In this section, we briefly review the Galois connection and its properties. Then the axiomatic approaches of the lower and upper rough approximation operators are discussed.
The axiomatic approaches of generalized rough approximation operators on CCD lattices
In this section, we first review the concepts of binary relation on two CCD lattices and propose the concept of ordered equivalence relation on two CCD lattices, which is a generalization of ordered equivalence relation on a CCD lattice. Then two pairs of generalized rough approximation operators on two CCD lattices are discussed, which are the generalizations of rough approximation operators on a CCD lattice. Finally, we consider the axiomatic approaches of these operators. In the following
Conclusion
In this paper we further research the properties of rough approximation operators based on an arbitrary binary relation over a CCD lattice [49] and the connection between the rough approximation operators based on a relation and a covering of a CCD lattice. The axiomatic approaches of the pair of rough approximation operator based on a relation are presented. According to these axiomatic approaches, we prove that the fuzzy rough approximation operators based on similarity relations can be seen
Acknowledgment
The authors are extremely grateful to the anonymous referees for their critical suggestions to improve the quality of this paper. This research is supported by the National Natural Science Foundation of China (Grant nos. 11571010, 61179038), the Research Project of Hubei Provincial Department of Education (Grant no. Q20141407) and Dr. Scientific Research Foundation of Hubei University of Technology (Grant no. BSQD14065).
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