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Inverse semigroups with isomorphic partial automorphism semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Simon M. Goberstein
Simon M. Goberstein
Affiliation:
Department of Mathematics, California State University Chico, California 95929, U.S.A.
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Abstract

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It is shown that a so-called shortly connected combinatorial inverse semigroup is strongly lattice-determined “modulo semilattices”. One of the consequences 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.

MSC classification

Secondary: 20M10: General structure theory 20M18: Inverse semigroups 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 47 , Issue 3 , December 1989 , pp. 399 - 417
Copyright
Copyright © Australian Mathematical Society 1989

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