L. M. Gluskin has shown that if
α 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 α 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 α to certain subsemigroups of the
translational hull Ω(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 Ω(S) is considered and
various classes of congruences are shown to be extendable. The main result
shows that any homomorphism 𝜃 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
Er in a principal ideal extends naturally to a homomorphism of Ω(S) into
Ω(T). The extension described is unique with respect to certain natural
restrictions.
