Antichain

Definition

Let $(P, \le)$ be a partially ordered set. A subset $A$ of $P$ is an antichain when, for all $a, b \in A$ with $a \neq b$, neither $a \le b$ nor $b \le a$.

Remarks

In the power set of $\{1, 2, 3\}$, the set $\{\{1\}, \{2\}, \{3\}\}$ is an antichain. A chain and an antichain have at most one common element. The largest size of an antichain is the width of the poset.

Explanations

L1 human

An antichain is the opposite of a chain: its elements are side by side. Under divisibility, the primes form an antichain, because no prime divides another prime.