Graph expansions of semigroups

Abstract

We construct a graph expansion from a semigroup with a given generating set, thereby generalizing the graph expansion for groups introduced by Margolis and Meakin. We then describe structural properties of this expansion. The semigroup graph expansion is itself a semigroup and there is a map onto the original semigroup. This construction preserves many features of the original semigroup including the presence of idempotent/periodic elements, maximal group images (if the initial semigroup is E-dense), finiteness, and finite subgroup structure. We provide necessary and sufficient graphical criteria to determine if elements are idempotent, regular, periodic, or related by Green’s relations. We also examine the relationship between the semigroup graph expansion and other expansions, namely the Birget and Rhodes right prefix expansion and the monoid graph expansion. If S is a -generated semigroup, its graph expansion is generally not -generated. For this reason, we introduce a second construction, the path expansion of a semigroup. We show that it is a -generated subsemigroup of the semigroup graph expansion. The semigroup path expansion possesses most of the properties of the semigroup graph expansion. Additionally, we show that the path expansion construction plays an analogous role with respect to the right prefix expansion of semigroups that the group graph expansion plays with respect to the right prefix expansion of groups.