Vol. 54, No. 1, 1974

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Extension of congruences and homomorphisms to translational hulls

Norman R. Reilly

Vol. 54 (1974), No. 1, 209–228

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.

Mathematical Subject Classification 2000
Primary: 20M10
Received: 28 March 1973
Revised: 19 September 1973
Published: 1 September 1974
Norman R. Reilly