The Prefix Expansion of a Monoid

I will begin by defining the concept of an expansion, in the sense of Birget & Rhodes (1984), before moving swiftly to a detailed discussion of a specific expansion: the prefix expansion. In the study of non-regular semigroups, the prefix expansion has been largely over-shadowed by the related Szendrei expansion (Szendrei 1989). I will remedy this situation by demonstrating necessary and sufficient conditions for the prefix expansion of a monoid to be weakly left ample. (These conditions are analogous to those already obtained for the Szendrei expansion by Fountain, Gomes & Gould (1999).) I will next present a number of (surprisingly natural) examples of monoids satisfying these conditions. Finally, I will obtain, as a corollary to the "weakly left ample" results, conditions for the prefix expansion to be inverse.