An (strong) antichain A of \mathbb{P} is a subset such that if p,q \in A , then p and q are incompatible (written p \perp q ), meaning there is no r in \mathbb{P} such that r \leq p and r \leq q . In the example on Borel sets, incompatibility means that p \cap q has zero measure. In the example on finite partial functions, incompatibility means that p \cup q is not a function, in other words, p and q assign different values to some domain input.
