Cofull Embeddings in Coset Monoids

James East

Abstract

Easdown, East and FitzGerald (2004) gave a sufficient condition for a (factorizable inverse) monoid to embed as a cofull submonoid of the coset monoid of its group of units. We show that this condition is also \emph{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 short proof of the result of McAlister (1980) 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 present examples which show that the word "cofull" may not be removed.

Keywords: Factorizable inverse monoid, coset monoid, symmetric inverse semigroup.

: Primary 20M18; Secondary 20M30.

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 Thursday, July 14, 2005