Automorphisms of $\mathcal P(\lambda )/\mathcal I_\kappa $

Paul Larson¹ ; Paul McKenney¹

¹ Department of Mathematics Miami University Oxford, OH 45056, U.S.A.

Fundamenta Mathematicae, Tome 233 (2016) no. 3, pp. 271-291

Résumé

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.

Détail

DOI : 10.4064/fm129-12-2015

Keywords: study conditions automorphisms boolean algebras form mathcal lambda mathcal kappa where lambda uncountable cardinal mathcal kappa ideal sets cardinality kappa which allow conclude given automorphism trivial among other things every automorphism mathcal kappa mathcal kappa which trivial sets cardinality kappa trivial aleph implies every automorphism mathcal mathbb fin trivial cocountable set every automorphism mathcal mathbb ctble trivial