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.