Geodesic
A~note on semi-pseudoorders in semigroups
Lobachevskii journal of mathematics, Tome 13 (2003), pp. 51-55

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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 $S$.
Keywords: pseudocongruence
Mots-clés : Pseudoorder, semi-pseudoorder. 
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     author = {N. Kehayopulu and M. Tsingelis},
     title = {A~note on semi-pseudoorders in semigroups},
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N. Kehayopulu; M. Tsingelis. A~note on semi-pseudoorders in semigroups. Lobachevskii journal of mathematics, Tome 13 (2003), pp. 51-55. http://geodesic.mathdoc.fr/item/LJM_2003_13_a5/