An antichain of a partially ordered set $X$ is a subset of $X$ such that no two distinct elements are comparable. A chain of a partially ordered set $X$ is a subset of $X$ such that every pair of its elements are comparable. Then, is the emptyset of some partially ordered set $X$ both a chain and an antichain? I reasoned the way Halmos showed in his own works about vacuous conditions: since the empty set has no elements, it can't fail to satisfy the definition, therefore it is a chain, and also an antichain? Thank you in advance.
Is the Empty Both a Chain and an Antichain
order-theory
Best Answer
Yes that is correct. The empty set, $\varnothing$, and singleton sets are the only subsets of a poset that are both chains and antichains. Subsets with cardinality greater than or equal to two, can not be both chains and antichains, they can be either one or neither.