On Special Semigroups Derived From an Arbitrary Semigroup

Abstract

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 $S/{\theta }^{\ast }=(S/\theta )/({\theta }^{\ast }/\theta )$, where $\theta$ and ${\theta }^{\ast }/\theta$ 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.