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, ifai≤bj (i, j=1,2), then there exists ac such thatai≤c≤bj (i, j=1,2). For such groupoids possessing the descending chain condition for the positive cone and the property\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\forall a,b a \leqslant b \Rightarrow \exists u,v au = va = b,$$ \end{document} 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.