Fully stable on Gamma Acts

Let M be a Γ-monoid and A a unitary right M Γ -act. We have introduced and studied the notion of full stability on gamma acts. We say that A is fully stable if f(B)⊆ B, for each M Γ -subact B of A and M Γ - homomorphism f: B→A. This is equivalent to saying that f(aΓM) ⊆ aΓM for each element a in A. Many properties and characterizations of this class of gamma acts have been considered. In fact we show that full stability of M Γ -act A is equivalent to the following equivalent conditions (1) For each a∈A, (L A (R M (aΓM)))= aΓM. (2) For all a, b ∈ A and R M (aΓM) ⊆ R M (bΓM) implies that b ∈ aΓM. (3) For each a ∈ A, there is a Γ-compatible ρ on M such that (L A (ρ)) = aΓM. (4) [aΓM:bΓM]=[R M (bΓM): R M (aΓM)] for all a, b ∈ A where M is a commutative Γ-monoid. (5) Every M Γ -homomorphism from any essential M Γ -subact B of A into A satisfies f (B) ⊆ B and Hom M Γ (B,C) = Hom M Γ (B,C) = ϴ, for any M Γ -subacts B, C of A with zero intersection.