Abstract
The relation [Rtilde] on a monoid S provides a natural generalisation of Green’s relation R. If every [Rtilde]-class of S contains an idempotentS is left semiabundant; if [Rtilde] 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 unipo-tent 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 σ on a monoid S, we construct functors is a left adjoint of F σ. 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.
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