An analog of the fundamental theorem of arithmetic in ordered groupoids

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.