Dilworth’s theorem


If P is a poset with width w<, then w is also the smallest integer such that P can be written as the union of w chains.

Remark. The smallest cardinal c such that P can be written as the union of c chains is called the chain covering number of P. So Dilworth’s theoremMathworldPlanetmath says that if the width of P is finite, then it is equal to the chain covering number of P. If w is infiniteMathworldPlanetmath, then statement is not true. The proof of Dilworth’s theorem and its counterexample in the infinite case can be found in the reference below.


  • 1 J.B. Nation, “Lattice Theory”, http://www.math.hawaii.edu/ jb/lat1-6.pdfhttp://www.math.hawaii.edu/ jb/lat1-6.pdf
Title Dilworth’s theorem
Canonical name DilworthsTheorem
Date of creation 2013-03-22 15:49:37
Last modified on 2013-03-22 15:49:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Theorem
Classification msc 06A06
Classification msc 06A07
Synonym Dilworth chain decomposition theorem
Related topic DualOfDilworthsTheorem
Defines chain covering number