## A note on semi-pseudoorders in semigroups.(English)Zbl 1041.06006

In a previous paper [ibid. 11, 19–21 (2002; Zbl 1015.06016)] the authors defined a semi-pseudoorder of a semigroup $$S$$ as a reflexive and transitive relation $$\sigma$$ on $$S$$ which is compatible with multiplication on both sides. The relation $$\overline\sigma= \sigma\cap\sigma^{-1}$$ is a congruence on $$S$$ such that $$S/\overline\sigma$$ is a partially ordered semigroup with respect to: $$a\overline\sigma\sqsubseteq b\overline\sigma$$ iff $$a\,\sigma b$$. In the paper under review the converse is shown: If $$\rho$$ is any congruence on a semigroup $$S$$ such that with respect to some partial order $$\preceq$$, $$(S/\rho,*,\preceq)$$ is a partially ordered semigroup, then there exists a semi-pseudoorder $$\sigma$$ on $$S$$ with $$\rho= \overline\sigma$$ and $$\preceq= \sqsubseteq$$. Hence, in this way, for any semigroup $$S$$ the homomorphic images which are partially ordered semigroups are obtained. It should be noted that the partial order on the factor semigroup of $$S$$ does not depend on a partial order given on $$S$$. (Caution: there are several misprints).
Reviewer: H. Mitsch (Wien)

### MSC:

 06F05 Ordered semigroups and monoids 20M10 General structure theory for semigroups

Zbl 1015.06016
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