Orthodox semidirect products and wreath products of monoids (Q1118701)

Let S be a monoid and T a semigroup and let \(\alpha\) : \(S\to End T\) be a given homomorphism. The semidirect product \(S\times_{\alpha}T\) is the semigroup with elements \(\{\) (s,t): \(s\in S\), \(t\in T\}\) and multiplication \((s,t)(r,u)=(sr,t^{\alpha (r)}u)\). A regular semigroup is orthodox if its idempotents form a semigroup. Conditions for \(S\times_{\alpha}T\) to be orthodox (left inverse, right inverse) are presented. These results are applied to the wreath product S \(w_ XT\), where X is a left S-act. For example, the wreath product S \(w_ XT\) is an orthodox semigroup if and only if 1) S and T are orthodox semigroups, and 2) S permutes X, or, T is a group and geX\(\subseteq eX\) for every two idempotents \(e,g\in S\).