Definition 4.4.9
Let be a poset. A descending chain in is a sequence
, where all the 's are elements of . An
antichain in is a subset of with no two elements directly comparable.

For example,
is a descending chain and is an antichain in
.

Proposition 4.4.10
If is a poset with no chains of length gretaer than , then can be covered
by at most antichains.

Proof.
By induction on . Take and let be the set of all maximal elements
in . Now
has no chains of length and
is an antichain.