On A Special S-Act Over Several Kinds of Semigroups

In this paper, we study some properties of a special S-acts D(S) = (S × S)S over several kinds of semigroups. First, we establish relationships between the diag- onal S-act D(S) of the semigroup S and the flatness properties of the diagonal act- s D(M(S; Λ; P)) of the Rees matrix semigroup M(S; Λ; P). Then, we show that for semigroups S with local units by (W F)/ of diagonal S-acts.Finally, we ascertain the direct product (S × T, *) is a semigroup when (S, ·) and (T, O) are two semigroups with operation and give flatness properties of diagonal (S × T, *)-act.