×

Embeddings into \({\mathcal P}({\mathbb N})/\text{fin}\) and extension of automorphisms. (English) Zbl 1015.06014

Parovichenko’s theorem says that, assuming CH, every Boolean algebra \(B\) of size \(\omega_1\) embeds into \(P(\omega)\)/fin. Given such an embedding, can automorphisms of \(B\) be extended (via the embedding) to automorphisms of \(P(\omega)\)/fin? That is the general question dealt with in this paper.
It was known that embeddings in which every automorphisms can be extended exist. What about embeddings in which few or no nontrivial automorphism can be extended?
Corollaries of the results in this paper include the following: 1. Assume CH. The measure algebra has an embedding into \(P(\omega)\)/fin under which no non-trivial automorphism can be extended. 2. Assume CH. The regular open algebra on the Cantor space \(2^{\omega}\) has an embedding into \(P(\omega)\)/fin under which no non-trivial automorphism can be extended.
One of the theorems is the following: Assume CH. If \(B\) is a Boolean algebra of size \(2^{\omega}\) whose reaping number is uncountable, then \(B\) has an embedding into \(P(\omega)\)/fin under which no non-trivial automorphism can be extended.
In addition, some results under various forms of Martin’s axiom are also given.

MSC:

06E05 Structure theory of Boolean algebras
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
03E05 Other combinatorial set theory
PDFBibTeX XMLCite
Full Text: DOI arXiv