Ramsey property for Boolean algebras with ideals and $\mathcal{P}(\omega_1)/\rm{fin}$

TL;DR

This paper establishes the Ramsey property for classes of finite Boolean algebras with ideals using the Dual Ramsey Theorem, enabling the computation of universal minimal flows of certain automorphism groups.

Contribution

It applies the Dual Ramsey Theorem to Boolean algebras with ideals and computes universal minimal flows of automorphism groups of specific quotient algebras.

Findings

Proves Ramsey property for Boolean algebras with ideals

Computes universal minimal flows of automorphism groups

Analyzes automorphism groups of quotient algebras

Abstract

We apply the Dual Ramsey Theorem of Graham and Rothschild to prove the Ramsey property for classes of finite Boolean algebras with distinguished ideals. This allows us to compute the universal minimal flow of the group of automorphisms of $\mathcal{P}(\omega_1)/\rm{fin},$ should it be isomorphic to $\mathcal{P}(\omega)/\rm{fin}$ or not, and of other quotients of power set algebras. Taking Fra\"iss\'e limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.