Computable Bi-Embeddable Categoricity

Abstract

We study the algorithmic complexity of isomorphic embeddings between computable structures. Suppose that L is a language. We say that L-structures A and B are bi-embeddable (denoted A ≈ B) if there are isomorphic embeddings f : A → B and g : B → A. The systematic investigation of the bi-embeddability relation in computable structure theory was initiated by Montalbán [1, 2]: he proved that any hyperarithmetical linear order is bi-embeddable with a computable one. In [3], similar results were obtained for Abelian p-groups, Boolean algebras, and compact metric spaces. The paper [4] studies degree spectra with respect to bi-embeddability.Definition 1. Let d be a Turing degree. We say that a computable structure S is d-computably bi-embeddably categorical if, for any computable structure A ≈ S, there are d-computable isomorphic embeddings f : A → S and g : S → A. The bi-embeddable categoricity spectrum of S is the set CatSpec ≈ (S) = {d : S is d-computably bi-embeddably categorical}.

Bi‐embeddability spectra and bases of spectra

“…They studied the complexity of embeddings structures. To facilitate this study, they introduced computable bi‐embeddable categoricity and classified the degrees of computable bi‐embeddable categoricity for equivalence structures .…”

Degrees of bi-embeddable categoricity of equivalence structures

“…In this article we focus on general results, especially, on the question which Turing degrees can and can not be degrees of categoricity. Some of the results of the paper were announced in [Baz+18a].…”