On bi-embeddable categoricity of algebraic structures

TL;DR

This paper explores the complexity of embeddings between bi-embeddable structures, specifically linear orders and Boolean algebras, revealing bounds related to computability and the structure's Hausdorff rank.

Contribution

It establishes precise computability bounds for embeddings in bi-embeddable structures and introduces a new variation of Ash and Knight's pairs of structures theorem.

Findings

Embeddings in linear orders are computable within 2n-1 jumps for Hausdorff rank n.

The bound of 2n-1 jumps is optimal for linear orders.

Superatomic Boolean algebras have a least computable ordinal for embeddings, unlike non-superatomic ones.

Abstract

In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable 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 bi-embeddable copy of it there is an embedding computable in $2n-1$ 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 $\mathbf 0^{(2n-2)}$ does not compute embeddings between it and all its computable bi-embeddable copies. We obtain that for Boolean algebras which are not superatomic, there is no hyperarithmetic degree computing embeddings between all its computable bi-embeddable 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 bi-embeddable copies. The main technique used in this proof is a new variation of Ash and Knight's pairs of structures theorem.