|dc.description.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.||en_US