A note on semi-pseudoorders in semigroups (Q1422279)

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).