locally finite poset

A poset $P$ is locally finite if every interval $[x,y]$ in $P$ is finite. For example, $\mathbb{Z}$ with the usual order is locally finite but not finite, while $\mathbb{Q}$ is neither.

Every locally finite poset is also chain finite, but the converse does not hold. To see this, define a partial order on $\mathbb{N}$ by the rule that $k\le \mathrm{\ell }$ if and only if $k=0$ or $\mathrm{\ell }=1$. Thus $0$ is the minimum element, $1$ is the maximum element, and the remaining elements form an infinite antichain. Every bounded chain in this poset is finite but the entire poset is an infinite interval, so the poset is chain finite but not locally finite.

Title	locally finite poset
Canonical name	LocallyFinitePoset
Date of creation	2013-03-22 14:09:15
Last modified on	2013-03-22 14:09:15
Owner	mps (409)
Last modified by	mps (409)
Numerical id	5
Author	mps (409)
Entry type	Definition
Classification	msc 06A99
Defines	locally finite