Affiliations: [a] Departament of Mathematics, Faculty of Science, Naresuan University, Mueang, Phitsanulok, Thailand
| [b] Research Center for Academic Excellence in Mathematics, Faculty of Science, Naresuan University, Thailand
Abstract: The Mareay’s rough set has been regarded as an approximation processing model in an approximation space induced by an arbitrary binary relation on the single universe. This is an effective mathematical tool for dealing with uncertain knowledge and vagueness in data for the single universe. Based on this induced notion, we firstly establish and verify two new classes, called a successor class and a core of a successor class induced by an arbitrary binary relation between two universes. Then we propose a generalization of the Mareay’s rough set in an approximation space based on cores of successor classes induced by an arbitrary binary relation between two universes, with a corresponding example. Some interesting algebraic properties of the new approximation processing model are investigated. We develop the use of the novel rough set in a semigroup under a preorder and compatible relation. We establish the notions of rough semigroups, rough ideals and rough completely prime ideals. Then we provide sufficient conditions of rough semigroups, rough ideals and rough completely prime ideals. Under homomorphism problems in semigroups, the relationship between the rough semigroup (resp. rough ideal and rough completely prime ideal) and the homomorphic image of the rough semigroup (resp. rough ideal and rough completely prime ideal) is demonstrated.
