Abstract : On any semigroup $S$, there is an equivalence relation $\phi ^{S}$, called the {\it locally equivalence relation}, given by $a\ \phi ^{S}b\Leftrightarrow aSa=bSb$ for all $a, b\in S$. In Theorem 4 \cite{3}, 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.
Keywords : $\phi ^{S}$-class, idempotent, finite order, group
