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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 $\neg\exists r\in P(r\leq p\wedge r\leq q)$. An antichain in $P$ 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?

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up vote 6 down vote accepted

Adam:

Yes, the notions are different, but I believe the ambiguity is older than forcing; doesn't Halmos use "antichain" for the forcing notion in his book on Boolean algebras?

Typically, when the need arises of distinguishing both notions, I've seen used (and used myself) "$A$ is a weak antichain" for "the elements of $A$ are pairwise incomparable", while "$A$ is a strong antichain" is reserved for the forcing version, "the elements of $A$ are pairwise incompatible."

Usually context suffices to know which version is used. In combinatorial contexts I would think using "antichain" for the "weak" version is more common. Certainly whenever forcing is used, it is the "strong", Boolean- (or forcing-)theoretic version that is used. In any paper where ambiguity could be an issue, I've seen at least a remark.

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