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Does $\mathcal P(ω) / \mathrm{fin}$ know its right hand from its left?

Will Brian

TL;DR

The paper resolves a longstanding question about the shift automorphism $\sigma$ on $\mathcal{P}(\omega)/\mathrm{fin}$: under CH, $\sigma$ and its inverse $\sigma^{-1}$ are conjugate in the automorphism group of the Boolean algebra, while earlier work showed consistency of non-conjugacy relative to ZFC. The author develops a detailed algebraic-dynamical framework, reducing the problem to a generalized lifting problem for countable/incompressible dynamical systems and exploiting a transfinite back-and-forth construction guided by a key Lifting Lemma. A polarized, finitary-combinatorial core (via digraphs and a diagonal argument) ensures sufficient solvability of “nice” lifting instances to carry the back-and-forth through ω1 steps under CH, yielding a global conjugacy and elementary-equivalence results. The work also derives corollaries concerning elementary equivalence of the two dynamical structures and the existence of nontrivial automorphisms commuting with the shift, highlighting CH-induced rigidity vs. independence phenomena in the automorphism group of $\mathcal{P}(\omega)/\mathrm{fin}$. Overall, the paper advances our understanding of how CH shapes the algebraic-dynamical landscape of Stone spaces and their automorphisms, with implications for the dynamics of $\omega^*$ and the structure of autohomeomorphisms under various forcing axioms.

Abstract

Let $σ$ denote the shift automorphism on $\mathcal{P}(ω) / \mathrm{fin}$, defined by setting $σ([A]) = [A+1]$ for all $A \subseteq ω$. We show that the Continuum Hypothesis implies the shift automorphism $σ$ and its inverse $σ^{-1}$ are conjugate in the automorphism group of $\mathcal{P}(ω) / \mathrm{fin}$. Due to work of van Douwen and Shelah, it has been known since the 1980's that it is consistent with $\mathsf{ZFC}$ that $σ$ and $σ^{-1}$ are not conjugate. Our result shows that the question of whether $σ$ and $σ^{-1}$ are conjugate is independent of $\mathsf{ZFC}$. As a corollary to the main theorem, we deduce that the structures $\langle \mathcal{P}(ω) / \mathrm{fin},σ\rangle$ and $\langle \mathcal{P}(ω) / \mathrm{fin},σ^{-1} \rangle$ are elementarily equivalent in the language of algebraic dynamical systems (Boolean algebras together with an automorphism). This corollary does not depend on the Continuum Hypothesis.

Does $\mathcal P(ω) / \mathrm{fin}$ know its right hand from its left?

TL;DR

The paper resolves a longstanding question about the shift automorphism on : under CH, and its inverse are conjugate in the automorphism group of the Boolean algebra, while earlier work showed consistency of non-conjugacy relative to ZFC. The author develops a detailed algebraic-dynamical framework, reducing the problem to a generalized lifting problem for countable/incompressible dynamical systems and exploiting a transfinite back-and-forth construction guided by a key Lifting Lemma. A polarized, finitary-combinatorial core (via digraphs and a diagonal argument) ensures sufficient solvability of “nice” lifting instances to carry the back-and-forth through ω1 steps under CH, yielding a global conjugacy and elementary-equivalence results. The work also derives corollaries concerning elementary equivalence of the two dynamical structures and the existence of nontrivial automorphisms commuting with the shift, highlighting CH-induced rigidity vs. independence phenomena in the automorphism group of . Overall, the paper advances our understanding of how CH shapes the algebraic-dynamical landscape of Stone spaces and their automorphisms, with implications for the dynamics of and the structure of autohomeomorphisms under various forcing axioms.

Abstract

Let denote the shift automorphism on , defined by setting for all . We show that the Continuum Hypothesis implies the shift automorphism and its inverse are conjugate in the automorphism group of . Due to work of van Douwen and Shelah, it has been known since the 1980's that it is consistent with that and are not conjugate. Our result shows that the question of whether and are conjugate is independent of . As a corollary to the main theorem, we deduce that the structures and are elementarily equivalent in the language of algebraic dynamical systems (Boolean algebras together with an automorphism). This corollary does not depend on the Continuum Hypothesis.
Paper Structure (12 sections, 59 theorems, 117 equations)

This paper contains 12 sections, 59 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. A blueprint
  3. Incompressibility
  4. Embeddings of Boolean algebras and dynamical systems
  5. A false hope cruelly dashed
  6. The Lifting Lemma, part 1: a back-and-forth argument
  7. The Lifting Lemma, part 2: directed graphs
  8. The Lifting Lemma, part 3: a diagonal argument
  9. The Lifting Lemma, part 4: polarization
  10. The Lifting Lemma, part 5: the heart of the lemma
  11. The Lifting Lemma, part 6: completing the proof
  12. A few corollaries

Key Result

Theorem 1.1

It is consistent with $\mathsf{ZFC}$ that $\langle\omega^*,\sigma\rangle$ and $\langle\omega^*,\sigma^{-1}\rangle$ are not conjugate.

Theorems & Definitions (129)

  • Theorem 1.1: van Douwen and Shelah
  • Definition 2.1
  • Theorem 2.2: Parovičenko, 1963 Parovicenko
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • proof
  • Definition 3.1
  • ...and 119 more