On Special Semigroups Derived From an Arbitrary Semigroup

TL;DR

This paper introduces a new semigroup construction based on an arbitrary semigroup and explores its structural properties, including connections between related quotient semigroups and an embedding theorem for specific classes of semigroups.

Contribution

It presents a novel semigroup construction using a set and a mapping, establishing links between various quotient semigroups and providing an embedding theorem for left equalizer simple semigroups.

Findings

Established a connection between $S$, $S/ heta$, and $S/ heta^*$ semigroups.

Proved an embedding theorem for $(S, S/ heta, ullet_P)$ when $S$ is left equalizer simple without idempotents.

Defined a new semigroup construction $(S, ext{set}, ullet_P)$ with specific algebraic properties.

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 $S/\theta ^*=(S/\theta)/(\theta ^*/\theta)$, 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.