  In this paper 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
$({\mathbb F}[\varrho]:_r{\mathbb F}[S])=\{ a\in {\mathbb F}[S]:\ {\mathbb
F}[S]a\subseteq {\mathbb F}[\varrho ]\}$ in general, and in that special case
when $\varrho$ has the property that the factor semigroup $S/\varrho$ is left
equalizer simple.

Group Theory