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PARTIAL ACTIONS OF INVERSE AND WEAKLY LEFT E-AMPLE SEMIGROUPS

Part of: Semigroups

Published online by Cambridge University Press:  01 June 2009

VICTORIA GOULD  and
CHRISTOPHER HOLLINGS
VICTORIA GOULD
Affiliation:
Department of Mathematics, University of York, Heslington York YO10 5DD, UK (email: varg1@york.ac.uk)
CHRISTOPHER HOLLINGS*
Affiliation:
Centro de Álgebra da Universidade de Lisboa, Avenida Professor Gama Pinto, 2, 1649-003 Lisboa, Portugal (email: cdh500@cii.fc.ul.pt)
*
✉For correspondence; e-mail: cdh500@cii.fc.ul.pt
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Abstract

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We introduce partial actions of weakly left E-ample semigroups, thus extending both the notion of partial actions of inverse semigroups and that of partial actions of monoids. Weakly left E-ample semigroups arise very naturally as subsemigroups of partial transformation semigroups which are closed under the unary operation α↦α+, where α+ is the identity map on the domain of α. We investigate the construction of ‘actions’ from such partial actions, making a connection with the FA-morphisms of Gomes. We observe that if the methods introduced in the monoid case by Megrelishvili and Schröder, and by the second author, are to be extended appropriately to the case of weakly left E-ample semigroups, then we must construct not global actions, but so-called incomplete actions. In particular, we show that a partial action of a weakly left E-ample semigroup is the restriction of an incomplete action. We specialize our approach to obtain corresponding results for inverse semigroups.

Keywords

augmented actionexpansionincomplete actioninverse semigrouppartial actionpremorphismweakly left E-ample semigroup

MSC classification

Secondary: 20M30: Representation of semigroups; actions of semigroups on sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 3 , June 2009 , pp. 355 - 377
Copyright
Copyright © Australian Mathematical Society 2009

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