Height and width of a nonempty finite poset

Let $P$ be a nonempty finite poset. Its height is

$h(P):=\max\{\,|C|:C\subseteq P\text{ is a chain}\,\},$

and its width is

$w(P):=\max\{\,|A|:A\subseteq P\text{ is an antichain}\,\}.$

Here $|P|$ and the cardinalities below are finite cardinalities (The cardinality $\lvert A\rvert$ of a finite set).

The maxima exist. A chain is a subset of $P$ whose elements are pairwise comparable and an antichain one whose distinct elements are pairwise incomparable (Chain in a poset, Antichains, chain covers, and antichain covers of a poset); each is a subset of $P$, hence finite with cardinality at most $|P|$ by A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$. The possible cardinalities therefore form nonempty subsets of the finite set $\{0,1,\ldots,|P|\}$ — the empty subset is vacuously both a chain and an antichain, so $0$ occurs, and every singleton is both, so some cardinality $1$ occurs. A nonempty finite set of natural numbers has a greatest member, as follows from The well-ordering principle by applying leastness to the corresponding differences from $|P|$. Thus $h(P)$ and $w(P)$ are natural numbers with $1\le h(P),w(P)\le |P|$.

The empty poset is excluded so that $h(P)$ and $w(P)$ are at least $1$: on the empty poset the only chain and the only antichain are empty, so both maxima would be $0$ and every statement below with a nonzero lower bound would need a separate convention.