# Dilworth’s theorem

###### Theorem.

If $P$ is a poset with width $w<\infty$, 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 theorem says that if the width of $P$ is finite, then it is equal to the chain covering number of $P$. If $w$ is infinite, 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.

## References

• 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 DilworthsTheorem 2013-03-22 15:49:37 2013-03-22 15:49:37 CWoo (3771) CWoo (3771) 14 CWoo (3771) Theorem msc 06A06 msc 06A07 Dilworth chain decomposition theorem DualOfDilworthsTheorem chain covering number