Namespaces
Variants
Actions

Difference between revisions of "Directed order"

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 binary relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327501.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327502.png" /> with the following properties: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327504.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327505.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327506.png" />; 2) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327507.png" />, always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327508.png" />; and 3) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d0327509.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d03275010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d03275011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d03275012.png" /> (the Moore–Smith property).
+
{{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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d03275013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d03275014.png" /> together imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d03275015.png" />, as well as 1) and 2) above), and also require the underlying set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032750/d03275016.png" /> to be non-empty.
+
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.
 +
 
 +
[[Category:Order, lattices, ordered algebraic structures]]

Latest revision as of 06:37, 14 October 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
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Directed_order&oldid=33631"