antichain

In a preorder or poset $P$, an **antichain** is a subset $S \subseteq P$ such that no two distinct elements of $S$ are comparable.

Assuming $P$ has a bottom element $0$, a *strong antichain* is a subset $S \subseteq P$ such that for distinct $a, b \in S$, the only lower bound of $\{a, b\}$ is $0$. This definition may be extended to posets $P$ *without* a bottom element, by declaring $A \subseteq P$ to be a strong antichain if $A$ is a strong antichain in $P^+$, the poset formed by freely adjoining a bottom element to $P$.

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 December 18, 2019 at 07:43:43. See the history of this page for a list of all contributions to it.