On page 3 of *Introduction to Lattices and Order*, Davey and Priestley define an antichain in a poset $\langle P,\leq\rangle$ as a set of pairwise **incomparable** elements:

The ordered set P is an antichain if $x\leq y$ in P only if $x=y$

Gratzer's definition is equivalent, but stated in a manner which is difficult to excerpt.

On page 53 of *Set Theory, an Introduction to Independence Proofs*, Kunen defines an antichain in $\langle P,\leq\rangle$ as a set of pairwise **incompatible** elements, saying that two elements $p$ and $q$:

are

incompatible($p\bot q$) iff $\neg\exists r\in P(r\leq p\wedge r\leq q)$. Anantichainin $P$ is a subset $A\subset P$ such that $\forall p,q\in A(p\neq q\rightarrow p\bot q)$.

So, given a three-element partially ordered set $\{0,a,b\}$ with $0\leq a$, $0\leq b$ the only (non-reflexive) related pairs in the partial order, it appears that $\{a,b\}$ is an antichain in the lattice sense but not in the forcing sense.

**Question**: is it in fact true that "antichain in a poset" means something different to set theorists than to lattice theorists?