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Inverse semigroups determined by their partial automorphism monoids

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Simon M. Goberstein
Simon M. Goberstein
Affiliation:
Department of Mathematics and Statistics, California State University, Chico, CA 95929, USA, e-mail: SGoberstein@csuchico.edu
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Abstract

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The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup S with no isolated nontrivial subgroups is lattice determined ‘modulo semilattices’ and if T is an inverse semigroup whose partial automorphism monoid is isomorphic to that of S, then either S and T are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if T is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of S and T, respectively, are isomorphic. Moreover, for these results to hold, the conditions that S be tightly connected and have no isolated nontrivial subgroups are essential.

MSC classification

Secondary: 20M10: General structure theory 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 185 - 198
Copyright
Copyright © Australian Mathematical Society 2006

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