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.^[1]^{: 214, Definition VII.6.12}

A preordered set is called 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 opposite of the preordered set is $\kappa$ -closed for all $\kappa$ .

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 $\kappa$ .^[1]^{: 215, Corollary 2.6.15}

References

^ ^a ^b Kunen, Kenneth (1980). Set theory: an introduction to independence proofs. Studies in Logic and the Foundations of Mathematics. Vol. 102. North-Holland. ISBN 978-0-444-86839-8. MR 0597342. Zbl 0534.03026. Archived from the original on 2016-09-11. Retrieved 2016-08-14.

[Kunen-1] Kunen, Kenneth (1980). Set theory: an introduction to independence proofs. Studies in Logic and the Foundations of Mathematics. Vol. 102. North-Holland. ISBN 978-0-444-86839-8. MR 0597342. Zbl 0534.03026. Archived from the original on 2016-09-11. Retrieved 2016-08-14.

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