Niovi Kehayopulu
Michael Tsingelis

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.
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