Primitive recursive categoricity spectra

TL;DR

This paper explores the primitive recursive analogue of computable categoricity spectra across various classes of structures, revealing where these notions align or differ.

Contribution

It establishes the equivalence of primitive recursive and computable categoricity spectra for specific classes like equivalence structures, linear orders, Boolean algebras, and trees.

Findings

Equivalence structures and linear orders have coinciding spectra for relatively Δ₂⁰-categorical cases.

Boolean algebras show spectrum coincidence at the relatively Δ₃⁰ level.

Trees as partial orders are computably categorical, aligning with primitive recursive notions.

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 $\Delta_{2}^{0}$-categorical equivalence structures and linear orders, relatively $\Delta_{3}^{0}$-categorical Boolean algebras, and computably categorical tree as partial orders.