SOME RESULTS ON THE LOCALLY EQUIVALENCE ON A NON-REGULAR SEMIGROUP

On any semigroup S, there is an equivalence relation ${\phi}^S$, called the locally equivalence relation, given by a ${\phi}^Sb{\Leftrightarrow}aSa=bSb$ for all $a$, $b{\in}S$. In Theorem 4 [4], Tiefenbach has shown that if ${\phi}^S$ is a band congruence, then $G_a$ := $[a]_{{\phi}^S}{\cap}(aSa)$ is a group. We show in this study that $G_a$ := $[a]_{{\phi}^S}{\cap}(aSa)$ is also a group whenever a is any idempotent element of S. Another main result of this study is to investigate the relationships between $[a]_{{\phi}^S}$ and $aSa$ in terms of semigroup theory, where ${\phi}^S$ may not be a band congruence.