Proper restriction semigroups -- semidirect products and \(W\)-products. (Q2250838)

Left restriction semigroups are unary semigroups isomorphic to unary subsemigroups of partial transformation semigroups, where the unary operation is \(x\to x^+=I_{\mathrm{dom}\;x}\). The semilattice of projections of a left restriction semigroup \(S\) is \(E=\{x^+,\;x\in S\}\); \(\sigma_S\) is the least congruence identifying all the elements of \(E\). A left restriction semigroup is proper if \(a^+=b^+\), \(a\sigma_Sb\) yields \(a=b\), and left ample if \(xz=yz\) yields \(xz^+=yz^+\). It is known that any proper left ample semigroup embeds into a so-called \(W\)-product, which is a subsemigroup of a reverse semidirect product of a semilattice \(\mathcal Y\) by a monoid \(T\), where the action of \(T\) on \(\mathcal Y\) is injective with images of the action being order ideals of \(\mathcal Y\). Here necessary and sufficient conditions are given for a proper left restriction semigroup to be embeddable into a \(W\)-product.