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EMBEDDINGS IN COSET MONOIDS

Part of: Semigroups

Published online by Cambridge University Press:  01 August 2008

JAMES EAST
JAMES EAST*
Affiliation:
School of Mathematics and Statistics, Carslaw Building F07, University of Sydney, New South Wales 2006, Australia (email: [email protected])
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Abstract

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A submonoid S of a monoid M is said to be cofull if it contains the group of units of M. We extract from the work of Easdown, East and FitzGerald (2002) a sufficient condition for a monoid to embed as a cofull submonoid of the coset monoid of its group of units, and show further that this condition is necessary. This yields a simple description of the class of finite monoids which embed in the coset monoids of their group of units. We apply our results to give a simple proof of the result of McAlister [D. B. McAlister, ‘Embedding inverse semigroups in coset semigroups’, Semigroup Forum20 (1980), 255–267] which states that the symmetric inverse semigroup on a finite set X does not embed in the coset monoid of the symmetric group on X. We also explore examples, which are necessarily infinite, of embeddings whose images are not cofull.

Keywords

factorizable inverse monoidcoset monoidsymmetric inverse semigroup

MSC classification

Secondary: 20M18: Inverse semigroups 20M30: Representation of semigroups; actions of semigroups on sets
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 1 , August 2008 , pp. 75 - 80
Copyright
Copyright © 2008 Australian Mathematical Society

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