Representation Embeddings of Cartesian Theories

TL;DR

This paper introduces a notion of representation embedding between cartesian theories, showing it preserves undecidability, and applies this to resolve a conjecture relating wild algebras and undecidable module theories.

Contribution

It defines a new concept of representation embedding for cartesian theories and proves it preserves undecidability, linking model theory and algebra representation theory.

Findings

Representation embedding preserves undecidability of theories.

Application to prove that wild algebras have undecidable module theories.

Provides a bridge between algebraic and logical properties of theories.

Abstract

A representation embedding between cartesian theories can be defined to be a functor between respective categories of models that preserves finitely-generated projective models and that preserves and reflects certain epimorphisms. This recalls standard definitions in the representation theory of associative algebras. The main result of this paper is that a representation embedding in the general sense preserves undecidability of theories. This result is applied to obtain an affirmative resolution of a reformulation in cartesian logic of a conjecture of M. Prest that every wild algebra over an algebraically closed field has an undecidable theory of modules.