Width of a poset

Remarks

Example: the set $\set{1, \ldots, 10}$ ordered by divisibility has width $5$. The set $\set{6, 7, 8, 9, 10}$ is an antichain, and the five chains $\set{1, 2, 4, 8}$, $\set{3, 6}$, $\set{5, 10}$, $\set{7}$, $\set{9}$ cover the set, so no antichain has more than $5$ elements. Non-example: the set $\set{2, 3, 4}$ has $3$ elements but is not an antichain there, since $2 \mid 4$, so it does not show that the width is at least $3$. A nonempty poset has width $1$ exactly when it is a total order. The width of an infinite poset can be infinite even when each antichain is finite. Dilworth's theorem says that a finite poset of width $w$ is a union of $w$ chains and of no fewer.