Left Equalizer Simple Semigroups

Abstract

We characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence ${\varrho}$ on a semigroup S, let ${{\mathbb F}[\varrho]}$ denote the ideal of the semigroup algebra ${{\mathbb F}[S]}$ which determines the kernel of the extended homomorphism of ${{\mathbb F}[S]}$ onto ${{\mathbb F}[S/\varrho]}$ induced by the canonical homomorphism of S onto ${S/\varrho}$. We examine the right colons (${{\mathbb F}[\varrho] :_{r} {\mathbb F}[S]) = {a \epsilon {\mathbb F}[S] : {\mathbb F}[S]a \subseteqq {\mathbb F}[\varrho]}}$ in general, and in that special case when ${\varrho}$ has the property that the factor semigroup ${S/\varrho}$ is left equalizer simple.