locally finite poset

A poset P is locally finitePlanetmathPlanetmathPlanetmath if every interval [x,y] in P is finite. For example, with the usual order is locally finite but not finite, while is neither.

Every locally finite poset is also chain finite, but the converseMathworldPlanetmath does not hold. To see this, define a partial orderMathworldPlanetmath on by the rule that k if and only if k=0 or =1. Thus 0 is the minimum element, 1 is the maximum element, and the remaining elements form an infiniteMathworldPlanetmath antichainMathworldPlanetmath. Every boundedPlanetmathPlanetmathPlanetmathPlanetmath 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