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On Special Semigroups Derived From an Arbitrary Semigroup

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