Automorphisms of $\mathcal P(\lambda )/\mathcal I_\kappa $
Volume 233 / 2016
Fundamenta Mathematicae 233 (2016), 271-291
MSC: Primary 03E35; Secondary 06E05.
DOI: 10.4064/fm129-12-2015
Published online: 2 December 2015
Abstract
We study conditions on automorphisms of Boolean algebras of the form $\mathcal P(\lambda )/\mathcal I_{\kappa }$ (where $\lambda $ is an uncountable cardinal and $\mathcal I_{\kappa }$ 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 automorphism of $\mathcal P(2^{\kappa })/\mathcal I_{\kappa ^{+}}$ which is trivial on all sets of cardinality $\kappa ^{+}$ is trivial, and that MA$_{\aleph _{1}}$ implies both that every automorphism of $\mathcal {P}(\mathbb {R})/\tt{Fin} $ is trivial on a cocountable set and that every automorphism of $\mathcal P(\mathbb R)/\tt {Ctble}$ is trivial.
