Antichains, chain covers, and antichain covers of a poset

Let $(P,\le)$ be a poset (Partial order and partially ordered set).

An antichain is a subset $A\subseteq P$ whose distinct elements are incomparable. Thus $x,y\in A$ and $x\ne y$ imply that neither $x\le y$ nor $y\le x$. An antichain is maximal if it is contained in no larger antichain, and maximum if its cardinality is at least that of every antichain in $P$. These notions are different: maximal refers to inclusion, whereas maximum refers to cardinality.

A chain cover of $P$ is a family $\mathcal C$ of chains (Chain in a poset) with $P=\bigcup_{C\in\mathcal C}C$. An antichain cover is a family $\mathcal A$ of antichains with $P=\bigcup_{A\in\mathcal A}A$. A cover may have overlapping members. For a finite cover, order its members, assign each point to the first member that contains it, and delete it from the others. This produces a partition into no more chains, or no more antichains, so minimum cover numbers are unchanged if partitions are required.