Left equalizer simple semigroups

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 $\mathbb{F}[S\mathrm{/}\varrho ]$ induced by the canonical homomorphism of $S$ onto $S\mathrm{/}\varrho$. 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 $S\mathrm{/}\varrho$ is left equalizer simple.