# Difference between revisions of "Naturally ordered groupoid"

From Encyclopedia of Mathematics

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− | A partially ordered groupoid (cf. [[Partially ordered set|Partially ordered set]]; [[Groupoid|Groupoid]]) | + | {{TEX|done}} |

+ | 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. | ||

## Revision as of 11:31, 9 November 2014

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.

#### Comments

#### References

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

**How to Cite This Entry:**

Naturally ordered groupoid.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Naturally_ordered_groupoid&oldid=19093

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