In a preorder or poset , an antichain is a subset such that no two distinct elements of are comparable.
Assuming has a bottom element , a strong antichain is a subset such that for distinct , the only lower bound of is . This definition may be extended to posets without a bottom element, by declaring to be a strong antichain if is a strong antichain in , the poset formed by freely adjoining a bottom element to .
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.