Coding in the automorphism group of a computably categorical structure

TL;DR

This paper introduces new methods to construct structures with specific categoricity and spectral properties, including a structure with degree of categoricity but infinite spectral dimension, and others with unique computability characteristics.

Contribution

It develops novel techniques for controlling the categoricity spectrum, enabling the construction of structures with previously unattainable combinations of properties.

Findings

Constructed a structure with degree of categoricity and infinite spectral dimension.

Built a computably categorical structure with non-computable Scott rank.

Created a structure of computable dimension 2 with no hyperarithmetic isomorphism between copies.

Abstract

Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimulin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank, and a structure of computable dimension 2 such that there is no hyperarithmetic isomorphism between the two copies.