Krull dimension of a poset

Summary

The Krull dimension of a partially ordered set is the supremum of the lengths of its finite chains. Here a chain with $n + 1$ elements has length $n$. It generalizes the Krull dimension of a commutative ring, which is the Krull dimension of the set of prime ideals ordered by inclusion. For a nonempty finite poset it is one less than the height counted in elements.