# Naturally ordered groupoid

From Encyclopedia of Mathematics

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.

#### References

- [a1] L. Fuchs, "Partially ordered algebraic systems", Pergamon (1963) Zbl 0137.02001

**How to Cite This Entry:**

Naturally ordered groupoid.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Naturally_ordered_groupoid&oldid=53557

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article