Representation Embeddings of Cartesian 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.