On cuts in ultraproducts of linear orders II

TL;DR

This paper investigates the structure of cuts in ultraproducts of linear orders, providing a combinatorial characterization of possible cofinalities and conditions under which these cofinalities are equal.

Contribution

It offers a new combinatorial description of the class of cofinalities of cuts in ultraproducts and establishes conditions for when these cofinalities are equal.

Findings

Characterization of the class (D) of cofinalities of cuts.

Conditions under which (D) contains only equal cofinalities.

Results for ultrafilters that are -complete or when cofinalities exceed 2^.

Abstract

We continue our study of the class $\mathscr{C}(D)$, where $D$ is a uniform ultrafilter on a cardinal $\kappa$ and $\mathscr{C}(D)$ is the class of all pairs $(\theta_1, \theta_2),$ where $(\theta_1, \theta_2)$ is the cofinality of a cut in $J^\kappa /D$ and $J$ is some $(\theta_1+\theta_2)^+$-saturated dense linear order. We give a combinatorial characterization of the class $\mathscr{C}(D)$. We also show that if $(\theta_1, \theta_2) \in \mathscr{C}(D)$ and $D$ is $\aleph_1$-complete or $\theta_1 + \theta_2 > 2^\kappa,$ then $\theta_1=\theta_2.$