Automorphisms of $\mathscr{P}(\lambda)/\mathscr{I}_\kappa$

TL;DR

This paper investigates conditions under which automorphisms of Boolean algebras formed by power sets modulo certain ideals are trivial, providing results for various cardinalities and set-theoretic assumptions.

Contribution

It establishes new criteria for automorphisms of Boolean algebras of the form P(λ)/I_κ to be trivial, including specific results under set-theoretic assumptions like MA.

Findings

Cardinality-preserving automorphisms of P(2^κ)/I_{κ^+} trivial if trivial on sets of size κ^+

Under MA_{ℵ_1}, automorphisms of P(ℝ)/Fin are trivial on a cocountable set

Conditions identified that ensure automorphisms are trivial in various set-theoretic contexts

Abstract

We study conditions on automorphisms of Boolean algebras of the form $P(\lambda)/I_\kappa$ (where $\lambda$ is an uncountable cardinal and $I_\kappa$ is the ideal of sets of cardinality less than $\kappa$) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every cardinality-preserving automorphism of $P(2^\kappa)/I_{\kappa^+}$ which is trivial on all sets of cardinality $\kappa^+$ is trivial, and that $MA_{\aleph_1}$ implies that every automorphism of $P(\mathbb{R})/Fin$ is trivial on a cocountable set.