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Isomorphism Spectra and Computably Composite Structures

Joey Lakerdas-Gayle

TL;DR

The paper addresses how degrees of categoricity interact with isomorphism spectra and investigates whether some degrees are never strong. It develops computably composite structures and an infinite base structure $\mathcal{H}$ to encode unions of spectra via component isomorphisms, proving that $IsoSpec$ is closed under computable unions and constructing an example with a non-finitely generated spectrum using Thomason’s framework. It also connects these ideas to automorphism spectra and categoricity spectra, and introduces the notion of uniform $\mathbf{d}$-computable categoricity within collections, highlighting open questions about uniformity and strong degrees. These methods provide a robust framework for building complex isomorphism spectra and clarifying how spectral properties propagate through composite constructions.

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.

Isomorphism Spectra and Computably Composite Structures

TL;DR

The paper addresses how degrees of categoricity interact with isomorphism spectra and investigates whether some degrees are never strong. It develops computably composite structures and an infinite base structure to encode unions of spectra via component isomorphisms, proving that is closed under computable unions and constructing an example with a non-finitely generated spectrum using Thomason’s framework. It also connects these ideas to automorphism spectra and categoricity spectra, and introduces the notion of uniform -computable categoricity within collections, highlighting open questions about uniformity and strong degrees. These methods provide a robust framework for building complex isomorphism spectra and clarifying how spectral properties propagate through composite constructions.

Abstract

Adapting a result of Bazhenov, Kalimullin, and Yamaleev, we show that if a Turing degree is the degree of categoricity of a computable structure and is not the strong degree of categoricity of any computable structure, then 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.

Paper Structure

This paper contains 10 sections, 22 theorems, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Computably composite structures
  3. Isomorphisms between computably composite structures
  4. Degrees of categoricity that are never strong
  5. An infinite base structure
  6. Automorphisms of the base structure
  7. Computable unions of isomorphism spectra
  8. An isomorphism spectrum that is not finitely generated
  9. Automorphism spectra of computably composite structures
  10. Categoricity spectra of some computably composite structures

Key Result

Proposition 10

Suppose $\mathcal{M}$ is a computable copy of $\mathcal{S}[\textbf{A}]$ via isomorphism $\rho:\mathcal{S}[\textbf{A}]\cong\mathcal{M}$. Let $\mathcal{G}$ be the substructure of $\mathcal{M}$ generated by $\rho(S)$. For each $g\in G$ let $\mathcal{B}_g$ be the substructure of $\mathcal{M}$ generated

Figures (4)

  • Figure 2.1: The black arrows represent the directed edge relation $E$ and the red arrows represent the directed edge relation $\mu$. The dotted boxes indicate the subsets that generate substructures $\mathcal{A}_0$, $\mathcal{A}_1$, and $\mathcal{A}_2$. The $\mu$-self-loops on $0,1,2$ are omitted.
  • Figure 3.1: Subsets of $\{0,1,2\}$ and edge relations $E_0$ (red), $E_1$ (blue), and $E_2$ (green).
  • Figure 3.2: Subsets of $\{0,1,2\}$, edge relations $E_0$ (red), $E_1$ (blue), and $E_2$ (green), hyperfaces $\{0,1,2\}\times\{0,1\}$, and edge relations $D_0$ (red arrows), $D_1$ (blue arrows), and $D_2$ (green arrows).
  • Figure 3.3: The first three dimensions of $\mathcal{M}=\mathcal{H}[\textbf{M}]$ (left) and $\mathcal{N}=\mathcal{H}[\textbf{N}]$ (right).

Theorems & Definitions (62)

  • Example 1
  • Remark 2
  • Definition 3
  • Remark 4
  • Definition 5: Computably composite structures
  • Remark 6
  • Remark 7
  • Remark 8
  • Example 9
  • Proposition 10
  • ...and 52 more