chain finite

A poset is said to be chain finite if every chain with both maximal (http://planetmath.org/MaximalElement) and minimal element is finite.

$\mathbb{Z}$ with the standard order relation is chain finite, since any infinite subset of $\mathbb{Z}$ must be unbounded (http://planetmath.org/UpperBound) above or below. $\mathbb{Q}$ with the standard order relation is not chain finite, since for example $\{x\in \mathbb{Q}\mid \mathrm{\hspace{0.17em}0}⩽x⩽1\}$ is infinite and has both a maximal element $1$ and a minimal element $0$.

Chain finiteness is often used to draw conclusions about an order from information about its covering relation (or equivalently, from its Hasse diagram).

Title	chain finite
Canonical name	ChainFinite
Date of creation	2013-03-22 16:55:07
Last modified on	2013-03-22 16:55:07
Owner	lars_h (9802)
Last modified by	lars_h (9802)
Numerical id	4
Author	lars_h (9802)
Entry type	Definition
Classification	msc 06A06
Defines	chain finite