Fully stable on Gamma Acts

Abstract

Abstract Let M be a Γ-monoid and A a unitary right MΓ-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 MΓ-subact B of A and MΓ- 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 MΓ-act A is equivalent to the following equivalent conditions (1) For each a∈A, (LA(RM(aΓM)))= aΓM. (2) For all a, b ∈ A and RM(aΓM) ⊆ RM(bΓM) implies that b ∈ aΓM. (3) For each a ∈ A, there is a Γ-compatible ρ on M such that (LA(ρ)) = aΓM. (4) [aΓM:bΓM]=[RM(bΓM): RM(aΓM)] for all a, b ∈ A where M is a commutative Γ-monoid. (5) Every MΓ-homomorphism from any essential MΓ-subact B of A into A satisfies f (B) ⊆ B and HomMΓ (B,C) = HomMΓ (B,C) = ϴ, for any MΓ-subacts B, C of A with zero intersection.