Difference between revisions of "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. | 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==== | ====References==== | ||
| − | + | * {{Ref|a1}} L. Fuchs, "Partially ordered algebraic systems", Pergamon (1963) {{ZBL|0137.02001}} | |
[[Category:Order, lattices, ordered algebraic structures]] | [[Category:Order, lattices, ordered algebraic structures]] | ||
Latest revision as of 09:15, 2 April 2023
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=34412
Naturally ordered groupoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Naturally_ordered_groupoid&oldid=34412
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article