Naturally ordered groupoid

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A partially ordered groupoid (cf. Partially ordered set; Groupoid) $H$ in which all elements are positive (that is, $a\leq ab$ and $b\leq ab$ for any $a,b\in H$) and the larger of two elements is always divisible (on both the left and the right) by the smaller, that is, $a<b$ implies that $ax=ya=b$ for some $x,y\in H$. The positive cone of any partially ordered group (cf. Ordered group) is a naturally ordered semi-group.



[a1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
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Naturally ordered groupoid. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article