View source for Naturally ordered groupoid

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A partially ordered groupoid (cf. [[Partially ordered set|Partially ordered set]]; [[Groupoid|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|Ordered group]]) is a naturally ordered semi-group.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>

[[Category:Order, lattices, ordered algebraic structures]]