Rough analysis in lattices

An outline of an algebraie generalization of the rough set theory is presented in the paper. It is shown that the majority of the basic concepts of this theory has an immediate algebraic generalization, and that some rough set facts are true in general algebraic structures. The formalism employed is that of lattice theory. New concepts of rough order, approximation space and rough (quantitative) approximation space are introduced and investigated. It is shown that the original Pawlak's theory of rough sets and information systems is a model of this general approach

Rough analysis in lattices.

An outline of an algebraie generalization of the rough set theory is presented in the paper. It is shown that the majority of the basic concepts of this theory has an immediate algebraic generalization, and that some rough set facts are true in general algebraic structures. The formalism employed is that of lattice theory. New concepts of rough order, approximation space and rough (quantitative) approximation space are introduced and investigated. It is shown that the original Pawlak's theory of rough sets and information systems is a model of this general approach.Rough set; Information system; Rough dependenee; Rough lattiee; Approximation spaee;

Matroidal structure of generalized rough sets based on symmetric and transitive relations

Rough sets are efficient for data pre-process in data mining. Lower and upper approximations are two core concepts of rough sets. This paper studies generalized rough sets based on symmetric and transitive relations from the operator-oriented view by matroidal approaches. We firstly construct a matroidal structure of generalized rough sets based on symmetric and transitive relations, and provide an approach to study the matroid induced by a symmetric and transitive relation. Secondly, this paper establishes a close relationship between matroids and generalized rough sets. Approximation quality and roughness of generalized rough sets can be computed by the circuit of matroid theory. At last, a symmetric and transitive relation can be constructed by a matroid with some special properties.Comment: 5 page

Covering rough sets based on neighborhoods: An approach without using neighborhoods

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin

Some characteristics of matroids through rough sets

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the terminology of linear algebra and graph theory. We can combine rough set theory with matroid theory through using rough sets to study some characteristics of matroids. In this paper, we apply rough sets to matroids through defining a family of sets which are constructed from the upper approximation operator with respect to an equivalence relation. First, we prove the family of sets satisfies the support set axioms of matroids, and then we obtain a matroid. We say the matroids induced by the equivalence relation and a type of matroid, namely support matroid, is induced. Second, through rough sets, some characteristics of matroids such as independent sets, support sets, bases, hyperplanes and closed sets are investigated.Comment: 13 page

"Possible Definitions of an ’A Priori’ Granule\ud in General Rough Set Theory" by A. Mani

We introduce an abstract framework for general rough set theory from a mereological perspective and consider possible concepts of ’a priori’ granules and granulation in the same. The framework is ideal for relaxing many of the\ud relatively superfluous set-theoretic axioms and for improving the semantics of many relation based, cover-based and dialectical rough set theories. This is a\ud relatively simplified presentation of a section in three different recent research papers by the present author.\u