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”.

Revised on June 11, 2017 14:49:29
by Todd Trimble
(24.146.226.222)