Rough Subgroup Of Rough Subring And Fuzzy Rough Subgroup And Fuzzy Rough Subring

Connected Rough structure with Algebra structure, Topology structure and Order structure, many new prosperous mathematics branches will appear Currently, there have been some articles on connecting Rough structure with Algebra structure . Rough ideals in a semigroup have been studied and the concepts of Rough subsemigroups , Rough ideals been first introduced by Kuroki N .Under the condition of the congruence relation, rough sets 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 nornal subgroups been first introduced. Under the condition of the congruence relation determined by a given normal sungroup in a group , rough sets of a subsgroup was proved to be its subgroup, while that of a normal one was also proved to be its normal subgroup.In this paper, homomorphism issues on rough sets are discussed.lt is proved that the image of homomorphism of the lower approximate set included in the lower approximate set of the corresponding image of homomorphism, and that of upper one equals to that of the upper one. And, isomorphism andf homomorphism issues between the quotient groups determined by yough normal subgroups are further discussed. Moreover, rough sets in a ring is studied and the concepts of Rough subrings , Rough ideals are been first introduced. Under the condition of the congruence relation determined by a given ideal in a ring , rough sets of a subring is proved to be its subring, while that of a ideal is also proved to be its ideal.Next, in the paper the concepts of Fuzzy rough subsemigroups and Fuzzy rough ideals in a semigroup are first given. In the sense of intersecting sets, Fuzzy rough sets of a fuzzy subsemigroup is proved to be its fuzzy subsemigroup, while that of fuzzy ideal is also proved to be its fuzzy ideal. Moreover, in a group, the concepts of Fuzzy rough subgroups and Fuzzy rough normai subgroups are first given. Under the condition of the congruence relation determined by a given normal sungroup in a group ,fuzzy rough sets of a fuzzy subsgroup is proved to be its fuzzy subgroup, while that of a fuzzy normal one is also proved to be its fuzzy normal subgroup. Fonally,in a ring ,the corresponding results of a semigroup and a group are generalized. Fuzzy rough sets in a ring is studied and the concepts of r Fuzzy rough subrings ,Fuzzy rough ideals are been first introduced. Under the condition of the congruence relation determined by a given ideal in a ring , Fuzzy rough sets of a fuzzy subring is proved to be its fuzzy subring, while that of a fuzzyideal is also proved to be its fuzzy ideal.