On cuts in ultraproducts of linear orders I

TL;DR

This paper investigates the existence of specific cuts in ultraproducts of dense linear orders, focusing on the relationship between saturation levels, ultrafilter properties, and cofinalities, especially when certain regular cardinals exceed the power set of .

Contribution

It characterizes conditions under which ultraproducts of linear orders have cuts of given cofinalities based on ultrafilter and saturation properties.

Findings

Identifies pairs of regular cardinals for which cuts exist in ultraproducts.

Analyzes the impact of ultrafilter saturation on the structure of ultraproducts.

Focuses on cases where _1, _2 > 2^.

Abstract

For an ultrafilter $D$ on a cardinal $\kappa,$ we wonder for which pair $(\theta_1, \theta_2)$ of regular cardinals, we have: for any $(\theta_1+\theta_2)^+-$saturated dense linear order $J, J^{\kappa}/ D$ has a cut of cofinality $(\theta_1, \theta_2).$ We deal mainly with the case $\theta_1, \theta_2 > 2^\kappa.$