Some results on the locally equivalence on a non-regular semigroup
Commun. Korean Math. Soc. 2013 Vol. 28, No. 1, 63-69
https://doi.org/10.4134/CKMS.2013.28.1.63
Printed January 31, 2013
Sevgi Atlihan
Gazi University
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
MSC numbers : 20M10
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