On A Special S-Act Over Several Kinds of Semigroups

Abstract

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.