Difference between revisions of "Directed order"
From Encyclopedia of Mathematics
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| − | A binary relation | + | {{TEX|done}} |
| + | A binary relation $\leq$ on a set $A$ with the following properties: 1) if $x\leq y$, $y\leq z$, then $x\leq z$, for any $x,y,z\in A$; 2) for any $x\in A$, always $x\leq x$; and 3) for any $x,y\in A$ there exists a $z\in A$ such that $x\leq z$, $y\leq z$ (the Moore–Smith property). | ||
====Comments==== | ====Comments==== | ||
| − | Many authors require a directed order to be a [[Partial order|partial order]] (i.e. to satisfy the condition that | + | Many authors require a directed order to be a [[Partial order|partial order]] (i.e. to satisfy the condition that $x\leq y$ and $y\leq x$ together imply $x=y$, as well as 1) and 2) above), and also require the underlying set $A$ to be non-empty. |
Revision as of 18:37, 14 August 2014
A binary relation $\leq$ on a set $A$ with the following properties: 1) if $x\leq y$, $y\leq z$, then $x\leq z$, for any $x,y,z\in A$; 2) for any $x\in A$, always $x\leq x$; and 3) for any $x,y\in A$ there exists a $z\in A$ such that $x\leq z$, $y\leq z$ (the Moore–Smith property).
Comments
Many authors require a directed order to be a partial order (i.e. to satisfy the condition that $x\leq y$ and $y\leq x$ together imply $x=y$, as well as 1) and 2) above), and also require the underlying set $A$ to be non-empty.
How to Cite This Entry:
Directed order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Directed_order&oldid=14517
Directed order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Directed_order&oldid=14517
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article