Equicontinuity, uniform almost periodicity, and regionally proximal relation for topological semiflows

Let π:T×X→X, where T is any discrete monoid, be a topological semiflow on a compact Hausdorff space X such that each of its transition maps πt is a surjection of X. We prove that this semiflow (T,X,π) is equicontinuous if and only if it is uniformly almost periodic if and only if its regionally proximal relation is equal to the diagonal Δ of X×X.