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Primitive recursive categoricity spectra

Nikolay Bazhenov, Heer Tern Koh, Keng Meng Ng

Abstract

We study the primitive recursive analogue of computable categoricity spectra for various natural classes of structures. We show that these notions coincide for all relatively $Δ_{2}^{0}$-categorical equivalence structures and linear orders, relatively $Δ_{3}^{0}$-categorical Boolean algebras, and computably categorical tree as partial orders.

Primitive recursive categoricity spectra

Abstract

We study the primitive recursive analogue of computable categoricity spectra for various natural classes of structures. We show that these notions coincide for all relatively -categorical equivalence structures and linear orders, relatively -categorical Boolean algebras, and computably categorical tree as partial orders.
Paper Structure (6 sections, 21 theorems, 2 equations, 3 figures)

This paper contains 6 sections, 21 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence Structures
  4. Linear Orders
  5. Boolean Algebras
  6. Trees as Partial Orders

Key Result

Theorem 1.1

In each of the following classes, every computable structure has a punctual presentation: equivalence structures, linear orders, Boolean algebras, torsion-free Abelian groups, and Abelian $p$-groups.

Figures (3)

  • Figure 1: Active intervals if $P(0,s)$ and $P(1,t)$ fires.
  • Figure 2: The tree of generators for $A\cong B\cong\mathop{\mathrm{\mathtt{Int}}}\nolimits(\omega*\eta)$.
  • Figure 3: $A$ and $B$ if $P(0,s)$ fires infinitely often.

Theorems & Definitions (40)

  • Theorem 1.1: kmn17
  • Theorem 1.2: kmn17
  • Definition 1.3: km21
  • Definition 1.4: bk21
  • Definition 1.5: bk21
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • ...and 30 more