Rough set theory is an important technique for knowledge discovery in databases, and its algebraic structure is part of the foundation of rough set theory. In this paper, we present the structures of the lower and upper approximations based on arbitrary binary relations. Some existing results concerning the interpretation of belief functions in rough set backgrounds are also extended. Based on the concepts of definable sets in rough set theory, two important Boolean subalgebras in the generalized rough sets are investigated. An algorithm to compute atoms for these two Boolean algebras is presented.
The rough set theory, proposed by Pawlak [9] and [10] as a method for data mining in 1982, has attracted the interest of researchers and practitioners in various fields of science and technology. This technique has led to many practical applications in various areas such as, but not limited to, medicine, economics, finance, engineering, and even arts and culture [16] and [17]. Combined with other complementary concepts such as fuzzy sets, statistics, and logical data analysis, rough sets have been exploited in hybrid approaches to improve the performance of data analysis tools.
The basic structure of rough set theory [11], [12] and [13] is an approximation space consisting of a universe of discourse and an equivalence relation imposed thereon. However, equivalence relations are too restrictive for many applications; for instance, in existing databases the values of attributes could be either symbolic or real-valued. Rough set theory would have difficulty in handling such values. To address this issue, several known generalizations of rough set model have been reported in the literature. For example, rough set model is extended to arbitrary binary relations [7], [14], [23], [24], [26], [27], [29] and [38] and coverings [35], [36] and [37]. Some researchers have even extended rough sets to Boolean algebra [5] and [18], completely distributive lattices [2] and residuated lattices [19].
An important generalization of rough set theory is the generalized rough set based on arbitrary binary relations on a universal set. Numerous papers have been published on rough sets. In comparison, however, relatively few results have been obtained for generalized rough sets based on arbitrary binary relations. In the past, studying the algebraic structure of a mathematical theory has proved itself effective in making the applications in the sciences more efficient. This is the inherent motivation for us to study the algebraic structures of these generalized rough sets. Such research may not only provide more insight into rough set theory, but also hopefully develop methods for applications.
A similar investigation was done by Yao [27], but this investigation focused on binary relation-based rough sets with special properties. It is preferable to establish the algebraic structure for binary relation-based rough sets without any constraint on the binary relations. Our aim in this paper is to explore the algebraic structures of the lower and upper approximations of generalized rough sets for general binary relations. In fact, our results extend those given in [27]. In achieving our aim, we use as core concept the solitary set, which is introduced for the first time in this paper. With this approach, we will determine a clearer algebraic structure for generalized rough sets based on binary relations that will allow researchers to better understand this type of rough set.
Applications of binary relation-based rough sets to practical situations can be found in the literature [23], [24], [32] and [33]. Rough set theory has also been used to interpret belief functions [20], [21] and [28]. Based on arbitrary binary relations, this paper studies the structures of generalized rough sets and interprets the associated belief functions. We propose two concepts of definable sets for generalized rough sets corresponding to definable sets in classical rough sets. Their lattice structures are investigated and an algorithm is presented to compute the particular atoms thereof. Kondo [6] also studied the structure of generalized rough sets based on binary relations from a topological point of view.
The paper is organized as follows. Section 2 introduces relevant definitions pertaining to rough sets and presents the concept of the solitary set to describe the structure of the lower and upper approximations in the generalized rough set environment. Section 3 is concerned with the interpretation of belief functions in generalized rough sets based on arbitrary binary relations. Section 4 presents the definitions of two concepts of definable sets in generalized rough sets based on arbitrary binary relations, and also an algorithm to compute atoms for two important Boolean subalgebras related to these two concepts. Finally, Section 5 concludes the paper.
In this paper, we have proposed the solitary set for an arbitrary binary relation and studied the structures of generalized rough sets based on arbitrary binary relations. Using the solitary set, we have highlighted the basic properties that are interesting and valuable in the theory of rough sets. The relationship between belief functions and generalized rough sets based on an arbitrary binary relation without serial conditions has also been established. The concept of definable sets in classical rough sets has been extended to generalized rough sets based on binary relations, and two important Boolean subalgebras have been discovered through the use of these concepts. Hopefully, the study of the algebraic structure of generalized rough set theory may bring the rough set theory to a new horizon of applications in the real world.
