On biembeddable categoricity of algebraic structures
Abstract
In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to biembeddability. In this article we investigate the complexity of embeddings between biembeddable structures in two such classes, the classes of linear orders and Boolean algebras. We show that if $\mathcal L$ is a computable linear order of Hausdorff rank $n$, then for every biembeddable copy of it there is an embedding computable in $2n1$ jumps from the atomic diagrams. We furthermore show that this is the best one can do: Let $\mathcal L$ be a computable linear order of Hausdorff rank $n\geq 1$, then $\mathbf 0^{(2n2)}$ does not compute embeddings between it and all its computable biembeddable copies. We obtain that for Boolean algebras which are not superatomic, there is no hyperarithmetic degree computing embeddings between all its computable biembeddable copies. On the other hand, if a computable Boolean algebra is superatomic, then there is a least computable ordinal $\alpha$ such that $\mathbf 0^{(\alpha)}$ computes embeddings between all its computable biembeddable copies. The main technique used in this proof is a new variation of Ash and Knight's pairs of structures theorem.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2005.07829
 arXiv:
 arXiv:2005.07829
 Bibcode:
 2020arXiv200507829B
 Keywords:

 Mathematics  Logic;
 03C57
 EPrint:
 Annals of Pure and Applied Logic, vol.173 (2022), no.3, article id 103060