On embeddability of idempotent separating extensions of inverse semigroups (Q1576295)

Let \(K\) be a semigroup and \(T\) an inverse semigroup. If \(S\) is a semigroup and \(\rho\) a congruence on \(S\) such that \(S/\rho\) is isomorphic to \(T\) and the kernel of \(\rho\) is isomorphic to \(K\) then \(S/\rho\) is called an extension of \(K\) by \(T\). If \(A\) and \(T\) are inverse semigroups then a \(\lambda\)-semidirect product, denoted by \(A*_\lambda T\) is an inverse semigroup defined on a certain subset of \(A\times T\) through a left action of \(T\) on \(A\) by endomorphisms. The extension \((A*_\lambda T,\ker\pi_2)\), where \(\pi_2\) is the projection onto \(T\) is referred to as a \(\lambda\)-semidirect product extension of \(A\) by \(T\). The main theorem, which has interesting corollaries not listed here, involves an idempotent separating congruence \(\rho\) on an inverse semigroup \(S\). It is proved that the extension \(S/\rho\) is embeddable into a \(\lambda\)-semidirect product extension of a group \(F\) by \(S/\rho\), where \(F\) belongs to the variety of groups generated by the idempotent classes of \(\rho\).