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

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 $\lambda$.