Approximations in a semigroup by using a neighbourhood system

Let S be a semigroup. We consider the relation γ and its transitive closure γ*. The relation γ is the smallest equivalence relation on S so that S/γ* is a commutative semigroup. Based on the relation γ, we define a neighbourhood system for each element of S, and we present a general framework of the study of approximations in semigroups. The connections between semigroups and operators are examined.