Closed preordered set

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

Let ${\displaystyle \kappa }$ be a cardinal. A preordered set ${\displaystyle (P,\lesssim )}$ is called ${\displaystyle \kappa }$-closed if every subset of ${\displaystyle P}$ whose opposite is well-ordered with order-type less than ${\displaystyle \kappa }$ has a lower bound.^{[1]: 214, Definition VII.6.12 [2]: Definition 15.7 [3]: §2}

A preordered set is ${\displaystyle {<}\kappa }$-closed if it is ${\displaystyle \lambda }$-closed for every ${\displaystyle \lambda <\kappa }$. A preordered set is called closed or ${\displaystyle {<}{\operatorname {Ord} }}$-closed if it is ${\displaystyle \kappa }$-closed for every ${\displaystyle \kappa }$.^{[4]: Lemma 4.0.10}

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.

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

A ${\displaystyle \kappa }$-closed forcing preserves cofinalities less than or equal to ${\displaystyle \kappa }$, hence cardinals less than or equal to ${\displaystyle \kappa }$.^{[1]: 215, Corollary 2.6.15}