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 $\lambda$.