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]]) 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
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