Naturally ordered groupoid
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)|
Naturally ordered groupoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Naturally_ordered_groupoid&oldid=19093