Height of a poset

Remarks

Example: the subsets of $\set{1, 2, 3}$ ordered by inclusion have height $4$, as shown by the chain $\varnothing \subset \set{1} \subset \set{1, 2} \subset \set{1, 2, 3}$. A chain of subsets has at most one subset of each size, so no chain is longer. Non-example: the set $\set{\varnothing, \set{1}, \set{2}, \set{1, 2}}$ has $4$ elements but is not a chain, since $\set{1}$ and $\set{2}$ are incomparable. Conventions differ: some authors call the height the largest length of a chain $x_0 < x_1 < \cdots < x_n$, which is $n$, one less than the number of elements. In that second convention the height is the Krull dimension of the poset. The height of an element $x$ is often defined as the height of the down-set $\set{y \in P : y \le x}$, with the same choice of convention.