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

Abstract

Abstract. On any semigroup S, there is an equivalence relation φ S ,called the locally equivalence relation, given by a φ S b ⇔aSa = bSb forall a,b ∈S. In Theorem 4 [4], Tiefenbach has shown that if φ S is a bandcongruence, then G a :=[a] φ S ∩(aSa) is a group. We show in this studythat G a :=[a] φ S ∩(aSa) is also a group whenever a is any idempotentelement of S. Another main result of this study is to investigate therelationships between [a] φ S and aSa in terms of semigroup theory, whereφ S may not be a band congruence. 1. IntroductionOn any semigroup S, there is an equivalence relation φ S , called the locallyequivalence relation, given byaφ S b ⇔ aSa =bSb for all a,b ∈ S.If S is a band, then φ S is a congruence, because φ S separates idempotents ofS by [3]. Also recall that φ S is not a congruence in general. In Theorem 4 [4],Tiefenbachhas shownthat if φ S isa band congruence, that is, a φ a 2 for all a ∈S, then G a :=[a] φ S ∩ (aSa)is a group and G a equals a[a] φ S a, where [a]