Difference between revisions of "Naturally ordered groupoid"
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A partially ordered groupoid (cf. [[Partially ordered set|Partially ordered set]]; [[Groupoid|Groupoid]]) 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