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

Atlihan, Sevgi

  • Received : 2012.03.02
  • Published : 2013.01.31

Abstract

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.

Keywords

${\phi}^S$-class;idempotent;finite order;group

References

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  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.