Enlargements, semiabundancy and unipotent monoids

The relation (R) over tilde on a monoid S provides a natural generalisation of Green's relation R. If every (R) over tilde-class of S contains an idem potent, S is left semiabundant; if (R) over tilde is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant mono ids is much larger, as we illustrate with a number of examples. This is the first of three related papers exploring the relationship betwee n unipotent monoids and left semiabundancy. We consider the situations wher e the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with (CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence sigma on a monoid S, we construct functors (<(circle)over tilde>)(SR) : U --> F and F-sigma : F --> U such t hat (<(circle)over tilde>)(SR) is a left adjoint of F-sigma. Here U is the category of unipotent monoids and F is a category of left semiabundant mono ids with properties echoing those of F-inverse monoids.