Injective hulls of S-systems over a Clifford semigroup (Q803292)

A regular semigroup with central idempotents is called Clifford semigroup. A semigroup S with adjoined 1 is denoted by \(S^*\). \(E^*\) is the set of all idempotents of \(S^*\). For a set M, a subset B of M is called semi-latticible, if \(BE^*=B\) and for each \(x\in B\), there is \(e\in E^*\) such that \(xE^*=Be\). An element a of M is called a face of a subset A of M if 1) \(A\subseteq aE^*\) and 2) \(As=At\) implies \(as=at\) for any \(s,t\in S^*\). Let G be the set of all semi-latticible subsets of M which have no face and let \(H=\{mE^*|\) \(m\in M\}\). It is proved that for an S-system M, the set \(C(M)=\{X\subseteq M|\) Xe\(\in G\cup H\) for any \(e\in E^*\}\) with the action (X,s)\(\mapsto Xs\) is an S-system and an injective hull of M.