Research On The Property Of Group And Ring In Rough Set Algebra

Rough set theory is not only a new mathematical tool dealing with vagueness and uncertainty but also a new and effective soft computing method. It was been widely used in the area of machine learning, knowledge discovery, decision analysis, artificial intelligence, data mining, Pattern recognition, fault diagnosis, etc. At the same time, some articles that have been studied about the theory combined pure mathematics with rough sets have been emerged, and some new mathematical notions. Rough ideals been first introduced by Kuroki N. Under the condition of the congruence relation, a rough set of a subsemigroup was proved to be its subsemigroup, while that of a left(right, bisides)ideal was also proved to be its left(right, bisides)ideal. Next, rough sets in a group have been studied and the concepts of Rough subgroups, Rough normal subgroups been first introduced. Under the condition of the congruence relation determined by a given normal subgroup in a group, rough sets of a subgroup was proved to be its subgroup, while that of a normal one was also proved to be its normal subgroup. Certainly, with the integration of rough structure and algebra structure, topology structure, order structure and the other structures, some new vital mathematical branches will be emerged. This article continues to study the application of rough set theory rough and algebra system -group, rings to combine research. So that rough algebra system is built better perfectly. The general process as follows:There have seven chapters in this paper. The first chapter is Introduction, the theory of Rough set .The second chapter is preparation, the knowledge about rough set that we must know before read this paper is given. The third charter is the Roughness of Group based on Normal subgroup, a rough subgroup with respect to a normal subgroup of a group is continued discussed. At the same time, the concept and properties of rough subgroup are given. Finally, some properties of rough set in quotient group are given and proved. The fourth chapter is the Roughness of Ring based on Ideal, a rough subring with respect to a ideal of a ring is discussed, and some properties of the lower and the upper approximations in a ring are studied. At the same time, some properties of rough Set in quotient ring are given and proved. The fifth chapter is Rough Primary Ideals and Rough Fuzzy Primary Ideals in Semigroups, the concept of rough primary ideals and fuzzy rough primary ideals in semigroups are introduced. Under the condition of the complete congruence relation, a rough set of a primary ideal in semigroups is proved to be its primary ideal. In the sense of cutset, the relation between the rough set of a cutset and the cutset of a rough set, a fuzzy rough set of a fuzzy primary ideal in a semigroup is proved to be its fuzzy primary ideal. The sixed chapter is Rough Primary Ideals and Rough Fuzzy Primary Ideals in rings, the corresponding results of a semigroup are generalized in a ring. The last chapter is Rough-fuzzy Subsemirings in Semirings, First, the lower and upper rough-fuzzy semirings together with the left and the right ideal are defined. Then the properties of these mathematical structures are researched. We prove that the upper rough-fuzzy semirings and the left(right,two-side) ideal are the generalization of corresponding notions defined on the fuzzy semirings.