Geodesic
An analog of the fundamental theorem of arithmetic in ordered groupoids
Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 910-915.

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

In the note we consider ordered groupoids with the Riesz interpolation property, that is, if $a_i\le b_j$ ($i,j=1,2$), then there exists a $c$ such that $a_i\le c\le b_j$ ($i,j=1,2$). For such groupoids possessing the descending chain condition for the positive cone and the property $$ \forall a,b \quad a\le b \implies\exists u,v \quad au=va=b, $$ a theorem analogous to the fundamental theorem of arithmetic is proved. The result is a generalization of known results for lattice-ordered monoids, loops, and quasigroups. 
@article{MZM_1997_62_6_a11,
     author = {V. A. Testov},
     title = {An analog of the fundamental theorem of arithmetic in ordered groupoids},
     journal = {Matemati\v{c}eskie zametki},
     pages = {910--915},
     publisher = {mathdoc},
     volume = {62},
     number = {6},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a11/}
}
TY  - JOUR
AU  - V. A. Testov
TI  - An analog of the fundamental theorem of arithmetic in ordered groupoids
JO  - Matematičeskie zametki
PY  - 1997
SP  - 910
EP  - 915
VL  - 62
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a11/
LA  - ru
ID  - MZM_1997_62_6_a11
ER  - 
%0 Journal Article
%A V. A. Testov
%T An analog of the fundamental theorem of arithmetic in ordered groupoids
%J Matematičeskie zametki
%D 1997
%P 910-915
%V 62
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a11/
%G ru
%F MZM_1997_62_6_a11
V. A. Testov. An analog of the fundamental theorem of arithmetic in ordered groupoids. Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 910-915. http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a11/

[1] Birkgof G., Teoriya reshetok, Nauka, M., 1984

[2] Fuks L., Chastichno uporyadochennye algebraicheskie sistemy, Mir, M., 1965

[3] Evans T., “Lattice-ordered loops and quasigroups”, J. Algebra, 16:2 (1970), 218–226 | DOI | MR | Zbl

[4] Sklifos P. I., “Ob uporyadochennykh lupakh”, Algebra i logika, 9:6 (1970), 714–730 | MR | Zbl

[5] Testov V. A., “K teorii reshetochno uporyadochennykh kvazigrupp”, Tkani i kvazigruppy, Kalinin, 1981, 153–157 | MR | Zbl

[6] Bosbach B., “Lattice ordered binary systems”, Acta Sci. Math., 52 (1988), 257–289 | MR | Zbl

[7] Testov V. A., Reshetochno uporyadochennye gruppoidy, Dep. VINITI, No 119-93, 1993