Embeddings in Coset Monoids

Abstract

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 Forum 20 (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.2000 Mathematics subject classification: primary 20M18, secondary 20M30.

On a Class of Factorizable Inverse Monoids Associated with Braid Groups

“…If M is an (inverse) monoid which embeds in some coset monoid G , then the cardinality of G is bounded below by the cardinality of G M . The class of (inverse) monoids which may be embedded in the coset monoid of their group of units was investigated in East (2006bEast ( , 2007.…”

“…Although it follows from Theorem 12 (and Theorem 3.5 of Easdown et al, 2002) that n embeds in B for n = 1 2, we are unable to use Theorem 12 to conclude that n does not embed in B for n ≥ 3. By results of East (2007)…”