# 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. |

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+ | [[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