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

Title & Authors
SOME RESULTS ON THE LOCALLY EQUIVALENCE ON A NON-REGULAR SEMIGROUP
Atlihan, Sevgi;

Abstract
On any semigroup S, there is an equivalence relation $\small{{\phi}^S}$, called the locally equivalence relation, given by a $\small{{\phi}^Sb{\Leftrightarrow}aSa=bSb}$ for all $\small{a}$, $\small{b{\in}S}$. In Theorem 4 [4], Tiefenbach has shown that if $\small{{\phi}^S}$ is a band congruence, then $\small{G_a}$ := $\small{[a]_{{\phi}^S}{\cap}(aSa)}$ is a group. We show in this study that $\small{G_a}$ := $\small{[a]_{{\phi}^S}{\cap}(aSa)}$ is also a group whenever a is any idempotent element of S. Another main result of this study is to investigate the relationships between $\small{[a]_{{\phi}^S}}$ and $\small{aSa}$ in terms of semigroup theory, where $\small{{\phi}^S}$ may not be a band congruence.
Keywords
$\small{{\phi}^S}$-class;idempotent;finite order;group;
Language
English
Cited by
References
1.
J. M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, 1995.

2.
F. Pastijn, Regular locally testable semigroup as semigroups of quasi-ideals, Acta Math. Acad. Sci. Hungar. 36 (1980), no. 1-2, 161-166.

3.
A. Tiefenbach, Locale Unterhalbgruppen, Ph. D. Thesis, University of Vienna, 1995.

4.
A. Tiefenbach, On certain varieties of semigroups, Turkish J. Math. 22 (1998), no. 2, 145-152.