# A note on semi-pseudoorders in semigroups

## (*Lobachevskii Journal of Mathematics, Volume XIII*)

An important problem for studying the structure of an ordered semigroup *S* is to know conditions under which for a given congruence *ρ* on *S* the set *S/ρ* 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 *σ* on *S* such that *S/σ* 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 *σ* on *S* such that *S/σ* is an ordered semigroup. In this note we prove that the converse of the last statement also holds. That is each congruence *σ* on a semigroup *(S, .)* such that *S/σ* is an ordered semigroup induces a semi-pseudoorder on *S*.