Isomorphism Spectra and Computably Composite Structures

TL;DR

This paper explores the relationship between isomorphism spectra of computable structures and their degrees of categoricity, introducing computably composite structures to demonstrate spectra that are not finitely generated.

Contribution

It introduces computably composite structures and shows how their isomorphism spectra relate to unions of spectra, expanding understanding of spectra complexity.

Findings

Existence of isomorphism spectra that are not finitely generated

Characterization of isomorphisms in computably composite structures

Any computable union of isomorphism spectra is itself an isomorphism spectrum

Abstract

Adapting a result of Bazhenov, Kalimullin, and Yamaleev, we show that if a Turing degree $\textbf{d}$ is the degree of categoricity of a computable structure $\mathcal{M}$ and is not the strong degree of categoricity of any computable structure, then $\mathcal{M}$ has a pair of computable copies whose isomorphism spectrum is not finitely generated. Motivated by this result, we introduce a class of computable structures called computably composite structures with the property that the isomorphisms between arbitrary computable copies of these structures are exactly the unions of isomorphisms between the computable copies of their components. We use this to show that any computable union of isomorphism spectra is also an isomorphism spectrum. In particular, this gives examples of isomorphism spectra that are not finitely generated.