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REGULAR CONGRUENCES ON AN IDEMPOTENT-REGULAR-SURJECTIVE SEMIGROUP

Part of: Lattices Semigroups

Published online by Cambridge University Press:  30 July 2013

ROMAN S. GIGOŃ
ROMAN S. GIGOŃ*
Affiliation:
Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland email romekgigon@tlen.pl
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Abstract

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A semigroup $S$ is called idempotent-surjective (respectively, regular-surjective) if whenever $\rho $ is a congruence on $S$ and $a\rho $ is idempotent (respectively, regular) in $S/ \rho $, then there is $e\in {E}_{S} \cap a\rho $ (respectively, $r\in \mathrm{Reg} (S)\cap a\rho $), where ${E}_{S} $ (respectively, $\mathrm{Reg} (S)$) denotes the set of all idempotents (respectively, regular elements) of $S$. Moreover, a semigroup $S$ is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.

Keywords

semigroupregular congruenceregular-surjective semigroupidempotent-surjective semigroupstructurally regular semigroupkerneltrace

MSC classification

Secondary: 20M99: None of the above, but in this section 06B10: Ideals, congruence relations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 190 - 196
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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