Extension of congruences and homomorphisms to translational hulls

Abstract

L. M. Gluskin has shown that if $\alpha$ is an isomorphism of a weakly reductive semigroup $S$ onto a semigroup $T$, if $V$ is a dense extension of $S$ and $T$ is densely embedded in $W$ then $\alpha$ extends uniquely to an isomorphism of $V$ into W. P. Grillet and M. Petrich have shown that this result can be interpreted in terms of extending $\alpha$ to certain subsemigroups of the translational hull $\Omega(S)$ of $S$. Here the problem of extending homomorphisms between inverse semigroups is considered. As a preliminary to the main results the problem of extending congruences from $S$ to $\Omega(S)$ is considered and various classes of congruences are shown to be extendable. The main result shows that any homomorphism $\theta$ of an inverse semigroup $S$ into an inverse semigroup $T$ such that the ideal, in the semilattice $E$ of idempotents of $T$, generated by the image of the idempotents of $S$ intersects any principal ideal of $E_{T}$ in a principal ideal extends naturally to a homomorphism of $\Omega(S)$ into $\Omega(T)$. The extension described is unique with respect to certain natural restrictions.