The concept of topological group is a simple combination of the concepts of abstract group and topological space. The purpose of this paper is to combine the concepts of topological space and rough groups; called topological rough groups on an approximation space.
In this paper, we introduced the notion of rough semigroups, rough subsemigroups, rough ideals, rough bi-ideals and homomorphisms of rough semigroups in approximation spaces. Then we give some properties of these rough structures.
The groupoid was offered by Brandt (1926). The structure of the topological groupoid was given by Ehresmann (1958). A groupoid action is a significant appliance in algebraic topology which is offered by Ehresmann. Another algebraic notion is a covering given by Brown (1988). The topological group-groupoids (Γ-groupoid) were first provided by İçen & Ozcan (2001). The definition of coverings of topological Γ groupoid and actions of topological Γ-groupoid were also given by İçen et al.(2005). In this paper, we will create a category TΓGpdCov(Γ) of coverings of topological Γ-groupoid and a category TΓGpdOp(Γ) of actions of topological Γ-groupoid. And then we will prove that these categories are equivalent.
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