nLab antichain




In a preorder or poset PP, an antichain is a subset SPS \subseteq P such that no two distinct elements of SS are comparable.

Assuming PP has a bottom element 00, a strong antichain is a subset SPS \subseteq P such that for distinct a,bSa, b \in S, the only lower bound of {a,b}\{a, b\} is 00. This definition may be extended to posets PP without a bottom element, by declaring APA \subseteq P to be a strong antichain if AA is a strong antichain in P +P^+, the poset formed by freely adjoining a bottom element to PP.

In the context of set theory, for example in discussions of forcing and countable chain conditions, “strong antichain” is often abbreviated to just “antichain”.

Last revised on May 28, 2024 at 06:29:05. See the history of this page for a list of all contributions to it.