Independence in computable algebra

TL;DR

This paper establishes a new sufficient condition for algebraic structures to have computable presentations with or without computable bases, applying it to various fields and groups, and introducing a novel technique of safe extensions.

Contribution

It introduces a new technique of safe extensions and provides a unifying condition applicable to multiple algebraic structures for their computable presentations.

Findings

Derived new corollaries on the number of computable presentations.

Applied the condition to different classes of fields and groups.

Introduced a novel technique of safe extensions.

Abstract

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.