On the number of cofinalities of cuts in ultraproducts of linear orders

TL;DR

This paper investigates the possible cofinalities of cuts in ultraproducts of linear orders, focusing on how ultrafilters influence the structure and cofinalities of these cuts.

Contribution

It characterizes the set of cofinalities of cuts in ultraproducts of linear orders with respect to non-decreasing sequences of regular cardinals.

Findings

Identifies conditions for possible cofinalities of cuts.

Provides a classification of cofinalities based on ultrafilter properties.

Enhances understanding of the structure of ultraproducts of linear orders.

Abstract

Suppose $\kappa$ is a regular cardinal and $\bar a=\langle \mu_i: i<\kappa \rangle$ is a non-decreasing sequence of regular cardinals. We study the set of possible cofinalities of cuts Pcut$(\bar a)=\{(\lambda_1, \lambda_2):$ for some ultrafilter $D$ on $\kappa$, $(\lambda_1, \lambda_2)$ is the cofinality of a cut of $\prod\limits_{i<\kappa} \mu_i / D \}$.