On universal \(\mathcal P\)-regular semigroups. (Q874851)

An inverse system for a regular semigroup \(S\) is a family \(\{Q(x)\mid x\in S\}\) of subsets of \(S\) such that for \(Q(x)\subseteq V(x)\) (\(V(x)\) the set of inverses of \(x\)) and \(x\in Q(y)\Rightarrow y\in Q(x)\) for all \(x,y\in S\). An inverse system is called product-inverse if \(Q(x)Q(y)\subseteq V(yx)\) for all \(x,y\in S\). It is shown that a regular semigroup \(S\) has a product-inverse system iff there is a unary operation \(^*\) on \(S\) with the properties \(xx^*x=x\), \((x^*x)^*=x^*x\), \((xx^*)^*=xx^*\), \(xyy^*x^*xy=xy\).