Theorem 1   If is a poset with width , then is also the smallest integer such that can be written as the union of chains.

Remark. The smallest cardinal such that can be written as the union of chains is called the chain covering number of . So Dilworth's theorem says that if the width of is finite, then it is equal to the chain covering number of . If 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.

Bibliography

1

J.B. Nation, ``Lattice Theory", http://www.math.hawaii.edu/˜jb/lat1-6.pdf