Topological S-act congruence

In this paper, we establish the necessary and sufficient condition for an equivalence relation ρ on an S-act A endowed with a topology such that A/ρ becomes a Hausdorff topological S-act. Also, we show that if A1 and A2 be two topological S-acts, then for any homomorphism ϕ : A1 → A2, A1/ ker ϕ is a topological S-act if and only if ϕ is ϕ-saturated continuous. Moreover, we establish for any two congruences θ1 and θ2 on an S-act A endowed with a topology, θ1 ∩ θ2 is a topological S-act congruence on A if and only if the mapping ϕ : A → A/θ1 × A/θ2, defined by ϕ(a) = (aθ1, aθ2), for all a ∈ A, is ϕ-saturated continuous, where S is a topological semigroup.