On Special Semigroups Derived From an Arbitrary Semigroup

Let $S$ be a semigroup, $\Lambda$ a non-empty set and $P$ a mapping of $\Lambda$ into $S$. The set $S\times \Lambda$ together with the operation $\circ _P$ defined by $(s, \lambda)\circ _P(t, \mu )=(sP(\lambda)t, \mu )$ form a semigroup which is denoted by $(S, \Lambda , \circ _P)$. Using this construction, we prove a common connection between the semigroups $S$, $S/\theta$ and $S/\theta ^*=(S/\theta)/(\theta ^*/\theta)$, where $\theta$ and $\theta ^*/\theta$ are the kernels of the right regular representations of $S$ and $S/\theta$, respectively. We also prove an embedding theorem for the semigroup $(S, S/\theta , \circ _p)$, where $S$ is a left equalizer simple semigroup without idempotents, and $P$ maps every $\theta$-class of $S$ into itself.