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Monoids with Idempotent Cross-Sections

In this paper we consider monoids M with an idempotent cross-section Λ of regular -classes such that for all idempotents e, f of M with e≥f, there exists x in the unit group such that x-1ex, x-1fx∈Λ. Such monoids arise frequently in connection with group representations. We show that Λ is then necessarily a semilattice and that the idempotent structure of M can for the most part, be determined within the unit group. We identify a particular class of "canonical monoids" inspired by some projective compactifications of algebraic groups. We show that these canonical monoids are 0-semilattice unions of completely 0-simple semigroups.