(Lobachevskii Journal of Mathematics, Vol.13, pp.51-56)

An important problem for studying the structure of an
ordered semigroup $S$ is
to know conditions under which for a
given congruence $\rho$ on $S$ the set $S/\rho$
is an ordered
semigroup. In [1] we introduced the concept of pseudoorder in
ordered
semigroups and we proved that each pseudoorder on an
ordered semigroup $S$ induces
a congruence $\sigma$ on $S$
such that $S/ \sigma$ is an ordered semigroup.
In [3] we
introduced the concept of semi-pseudoorder (also called
pseudocongruence)
in semigroups and we proved that each
semi-pseudoorder on a semigroup $S$ induces
a congruence
$\sigma$ on $S$ such that $S/ \sigma$ is an ordered semigroup.
In
this note we prove that the converse of the last
statement also holds. That is each
congruence $\sigma$ on
a semigroup $(S, .)$ such that $S/ \sigma$ is an ordered
semigroup
induces a semi-pseudoorder on $S$.