Definable Automorphisms of P(ω)/fin

Abstract

We investigate definable automorphisms of ^(uj/fin and show that e.g. every Borel automorphism is trivial. The existence of nontrivial projective automor- phisms is consistent and independent from ZFC + CH. Throughout this paper a2(w)/fin will denote the Boolean algebra of 0>(to) modulo the ideal of finite sets. Call an automorphism F of ^(c^/fin trivial if there are two cofinite sets a,b e u and a bijection /: a -> b such that F(x) = (f(a n x)). Here (x) denotes the equivalence class of x under the equivalence relation x =+ y iff (x\y)U (y\x) is finite. Note that there are exactly 2K° trivial automorphisms. It is a well-known fact, first proved by W. Rudin (7), that the Continuum Hypothesis implies there are 22 ° automorphisms of £P(o:)/nn, hence most of them are nontrivial. It is also consistent with —,CH that there is a nontrivial automorphism of aa(w)/fin. This was proved by Baumgartner (unpublished). He noticed that if there is a family {aa: a < ux) of subsets of w such that (i) aa as is finite for every a < s < «,, (ii) {aa: a<to1}U(to)< generates a maximal ideal /, then one can build by induction a family {fa: a < tcx} of functions such that (iii) fa: aa -* aa is a permutation without fixed points,