Namespaces
Variants
Actions

Difference between revisions of "Naturally ordered groupoid"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Category:Order, lattices, ordered algebraic structures)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
A partially ordered groupoid (cf. [[Partially ordered set|Partially ordered set]]; [[Groupoid|Groupoid]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066120/n0661201.png" /> in which all elements are positive (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066120/n0661202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066120/n0661203.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066120/n0661204.png" />) and the larger of two elements is always divisible (on both the left and the right) by the smaller, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066120/n0661205.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066120/n0661206.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066120/n0661207.png" />. The positive cone of any partially ordered group (cf. [[Ordered group|Ordered group]]) is a naturally ordered semi-group.
+
{{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.
  
  
Line 8: Line 9:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 +
 +
[[Category:Order, lattices, ordered algebraic structures]]

Latest revision as of 12:22, 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