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REGULAR CONGRUENCES ON AN IDEMPOTENT-REGULAR-SURJECTIVE SEMIGROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 2 / October 2013
- Published online by Cambridge University Press:
- 30 July 2013, pp. 190-196
- Print publication:
- October 2013
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- Article
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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.
