Homomorphism on Rough Groups

Abstract

Rough set is a mathematical concept to model uncertainty, vagueness, or incomplete information. This research discusses the concept and properties of homomorphisms on rough groups. Homomorphisms on rough groups are formed by mapping two upper approximations of subsets of classical groups, where those approximations are classical subgroups. The results show that a homomorphism on rough groups preserves the binary operation and a classical group homomorphisms is a special case of homomorphisms on rough groups. Furthermore, the properties of homomorphisms on rough groups related to preservation identity element and inverse element, as well as injective and surjective functions, are obtained. The concepts of kernel and image of homomorphisms on rough groups are defined by mapping elements within these approximations. In addition, a necessary and sufficient condition for a homomorphisms on rough groups to be isomorphisms is iff the kernel contains only the identity element of the domain and the image equals the entire codomain.