J Austral Math Soc Ser A 47 pp399--417, 1989.
(Received 6 January 1988)
It is shown that a so-called shortly connected combinatorial semigroup is strongly lattice-determined "modulo semilattices". One of the conseqences of this theorem is the known fact that a simple inverse semigroup with modular lattice of full inverse subsemigroups is strongly lattice-determined [7]. The partial automorphism semigroup of an inverse semigroup S consists of all isomorphisms between inverse subsemigroups of S. It is proved that if S is a shortly connected combinatorial inverse semigroup, T an inverse semigroup and the partial automorphism semigroups of S and T are isomorphic, then either S and T are isomorphic or they are dually isomorphic chains (with respect to the natural partial order); moreover, any isomorphism between the partial automorphism semigroups of S and T is induced either by an isomorphism or, if S and T are dually isomorphic chains, by a dual isomorphism between S and T. Counter-examples are constructed to demonstrate that the assumptions about S being shortly connected and combinatorial are essential.
1980 AMS Subject Classification (1985 Revision): 20M10, 20M18, 20M20
Last Modified: Wed Feb 19 10:27:50 2003