On Gamma Acts over Gamma Semigroups

For a semigroup S, actions of S on non-empty sets, namely S-acts, are of interest to consider for their applications in many branches of science. The well known generalization of a semigroup is the Γ-semigroup. The notion of a Γ-act over a Γ-semigroup is a generalization of actions over semigroups. In this paper, certain intrinsic and basic properties of Γ-acts including cyclic, indecomposable and free are studied as well. Among other results, it is also shown that a Γ-act is free only if |Γ| = 1.