On automorphisms of $\mathcal P(\lambda)/[\lambda]^{<\lambda}$

TL;DR

This paper explores the conditions under which automorphisms of certain Boolean algebras are trivial, showing implications of Martin's Axiom, and constructing models where non-trivial automorphisms exist for large cardinals.

Contribution

It establishes new results linking set-theoretic axioms and large cardinal properties to the triviality of automorphisms of Boolean algebras.

Findings

Martin's Axiom implies trivial automorphisms for regular uncountable λ<2^{ℵ₀}

Non-trivial automorphisms exist for measurable λ when 2^λ=λ^+

Models with densely trivial automorphisms for inaccessible λ can be forced with 2^λ=λ^{++}

Abstract

We investigate the statement ``all automorphisms of $\mathcal P(\lambda)/[\lambda]^{<\lambda}$ are trivial''. We show that MA implies the statement for regular uncountable $\lambda<2^{\aleph_0}$; that the statement is false for measurable $\lambda$ if $2^\lambda=\lambda^+$; and that for ``densely trivial'' it can be forced (together with $2^\lambda=\lambda^{++}$) for inaccessible $\lambda$.