Coset Monoids and Embeddings

Speaker: James East, University of Sydney

Abstract: There is a well-established theory of embeddings of semigroups in different kinds of transformation semigroups. In this talk we will look at embeddings of semigroups in coset monoids. Coset monoids are factorisable inverse monoids which contain all the information encoded in a group and the cosets of its subgroups. Schein introduced coset monoids in 1966, and in 1980 McAlister showed that every inverse semigroup embeds in the coset monoid of some group (indeed some symmetric group). It is then natural to ask how "small" this group may be. The class of semigroups which embed as a "co-full" submonoid of the coset monoid of the "smallest possible" group is very easy to describe. We will give some examples of semigroups which belong to this class, as well as some which "almost" do.