Compatibility, ccc and Knaster for posets

Let $(P,\le )$ be a poset as in Partial order and partially ordered set, with stronger conditions smaller. Conditions $p,q\in P$ are compatible if some $r\in P$ satisfies $r\le p$ and $r\le q$. Otherwise they are incompatible. A poset antichain is a set of pairwise incompatible conditions. The countable chain condition (ccc) says that every poset antichain is countable. The Knaster property says that every uncountable subset of $P$ has an uncountable subset consisting of pairwise compatible conditions. Countable includes finite, as in Finite, countably infinite, countable, uncountable.

Knaster implies ccc: if $A$ were an uncountable antichain, Knaster would give an uncountable pairwise compatible $B\subseteq A$. Choose two distinct members of $B$; they would be both compatible and incompatible. Empty and countable posets satisfy both properties, because they have no uncountable subsets. A condition is compatible with itself, using itself as lower bound; singletons are therefore pairwise compatible and also antichains under the distinct-pair convention.

Incompatibility is stronger than incomparability in a general poset. For a tree $T$, put $p{\le }_{P}q$ iff $q{\le }_{T}p$, so a lower bound is a common tree extension. By Tree predecessors and compatibility, a common extension forces the two nodes comparable. Conversely, for comparable tree nodes the higher node extends both. Thus the two antichain notions coincide for this reverse tree order. The orientation of the order is essential to that identification.