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;

Some Properties of Rough Ideals on Rough Rings

The concept of rough set was first introduced by Pawlak in 1982. The basic concepts in set theory such as intersections, unions, differences, and complements still apply to rough sets. Furthermore, researchers in the field of mathematics and informatics who study rough sets can relate the concept of rough sets to algebraic structures so that a concept called rough algebraic structures is obtained. Some concepts on rough algebraic structures are rough groups, rough rings, and rough modules. In this paper, the properties related to the ideal of roughness will be given to the rough ring.Konsep himpunan kasar pertama kali diperkenalkan oleh Pawlak pada tahun 1982. Konsep dasar pada teori himpunan seperti irisan, gabungan, selisih, dan komplemen masih berlaku pada himpunan kasar. Selanjutnya, para peneliti bidang matematika dan informatika yang mendalami himpunan kasar dapat mengaitkan konsep himpunan kasar dengan struktur aljabar sehingga diperoleh konsep yang dinamakan struktur aljabar kasar. Beberapa konsep pada struktur aljabar kasar adalah grup kasar, ring kasar, dan modul kasar. Pada paper ini, akan diberikan sifat-sifat terkait ideal kasar pada dari ring kasar

A classification study of rough sets generalization

In the development of rough set theory, many different interpretations and formulations have been proposed and studied. One can classify the studies of rough sets into algebraic and constructive approaches. While algebraic studies focus on the axiomatization of rough set algebras, the constructive studies concern with the construction of rough set algebras from other well known mathematical concepts and structures. The constructive approaches are particularly useful in the real applications of rough set theory. The main objective of this thesis to provide a systematic review existing works on constructive approaches and to present some additional results. Both constructive and algebraic approaches are first discussed with respect to the classical rough set model. In particular, three equivalent constructive definitions of rough set approximation operators are examined. They are the element based, the equivalence class based, and the subsystem based definitions. Based on the element based and subsystem based definitions, generalized rough set models are reviewed and summarized. One can extend the element based definition by using any binary relations instead of equivalence relations in the classical rough set model. Many classes of rough set models can be established based on the properties of binary relations. The subsystem based definition can be extended in the set-theoretical setting, which leads to rough set models based on Pawlak approximation space, topological space, and closure system. Finally, the connections between the algebraic studies, relation based, and subsystem based formulations are established