A note on semi-pseudoorders in semigroups

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 $S/\rho$ 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 $S/\rho$ 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 $S/\rho$ is an ordered semigroup induces a semi-pseudoorder on $S$.