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

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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/}
}
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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/