On incomparability and related cardinal functions on ultraproducts of Boolean algebras

TL;DR

This paper demonstrates the consistency of certain inequalities involving cardinal characteristics of ultraproducts of Boolean algebras, addressing open problems in the field.

Contribution

It shows that for various cardinal invariants, inequalities can hold between ultraproducts and their factors, solving problems posed by Monk.

Findings

Existence of Boolean algebra sequences with specific ultraproduct properties

Inequalities between cardinal characteristics of ultraproducts and factors

Addresses open problems in Boolean algebra cardinal invariants

Abstract

Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence <B_i:i<kappa> of Boolean algebras and an ultrafilter D on kappa such that C(prod_{i<kappa}B_i/D)<|prod_{i<kappa}C(B_i)/D|. This answers a number of problems posed by Monk.