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 idempotent, 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 monoids is much larger, as we illustrate with a number of examples. This is the first of three related papers exploring the relationship between unipotent monoids and left semiabundancy. We consider the situations where 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 that (<(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 monoids with properties echoing those of F-inverse monoids.