Closed preordered set

In mathematics, a closed preordered set is one whose anti-well-ordered subsets have lower bounds.

Definition

Let $\kappa$ be a cardinal. A preordered set $(P,\lesssim )$ is called $\kappa$-closed if every subset of $P$ whose opposite is well-ordered with order-type less than $\kappa$ has a lower bound.

A preordered set is $<\kappa$-closed if it is $\lambda$-closed for every $\lambda <\kappa$. A preordered set is called closed or $<\mathrm{Ord}$-closed if it is $\kappa$-closed for every $\kappa$.

A preordered set is inductive if every chain has an upper bound. Since every totally ordered set has a well-ordered cofinal subset, this is equivalent to saying that the preordered set is the opposite of a closed preordered set.

Properties

Inductive preordered sets satisfy Zorn's lemma and the Bourbaki–Witt theorem.

A $\kappa$-closed forcing preserves cofinalities less than or equal to $\kappa$, hence cardinals less than or equal to $\kappa$.