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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{LJM_2003_13_a5,
     author = {N. Kehayopulu and M. Tsingelis},
     title = {A~note on semi-pseudoorders in semigroups},
     journal = {Lobachevskii journal of mathematics},
     pages = {51--55},
     publisher = {mathdoc},
     volume = {13},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2003_13_a5/}
}
TY  - JOUR
AU  - N. Kehayopulu
AU  - M. Tsingelis
TI  - A~note on semi-pseudoorders in semigroups
JO  - Lobachevskii journal of mathematics
PY  - 2003
SP  - 51
EP  - 55
VL  - 13
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/LJM_2003_13_a5/
LA  - en
ID  - LJM_2003_13_a5
ER  - 
%0 Journal Article
%A N. Kehayopulu
%A M. Tsingelis
%T A~note on semi-pseudoorders in semigroups
%J Lobachevskii journal of mathematics
%D 2003
%P 51-55
%V 13
%I mathdoc
%U http://geodesic.mathdoc.fr/item/LJM_2003_13_a5/
%G en
%F LJM_2003_13_a5
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/

[1] N. Kehayopulu, M. Tsingelis, “On subdirectly irreducible ordered semigroups”, Semigroup Forum, 50 (1995), 161–177 | DOI | MR | Zbl

[2] N. Kehayopulu, M. Tsingelis, “Pseudoorder in ordered semigroups”, Semigroup Forum, 50 (1995), 389–392 | DOI | MR | Zbl

[3] N. Kehayopulu, M. Tsingelis, “A note on pseudocongruences in semigroups”, Lobachevskii J. Math., 11 (2002), 19–21 | MR | Zbl