Chains of Theories and Companionability

TL;DR

This paper investigates the companionability of theories involving fields with derivations and vector spaces, revealing conditions under which unions of chains of theories are or are not companionable, with implications for model theory.

Contribution

It demonstrates how certain chains of theories can have union theories that are either companionable or not, depending on additional axioms and structures.

Findings

Characteristic zero fields with derivations are companionable when axioms are added.

Vector space theories with predicates for linear dependence are companionable, but their chains' model-companions are inconsistent.

Union of non-companionable theories can sometimes be companionable.

Abstract

The theory of fields that are equipped with a countably infinite family of commuting derivations is not companionable; but if the axiom is added whereby the characteristic of the fields is zero, then the resulting theory is companionable. Each of these two theories is the union of a chain of companionable theories. In the case of characteristic zero, the model-companions of the theories in the chain form another chain, whose union is therefore the model-companion of the union of the original chain. However, in a signature with predicates, in all finite numbers of arguments, for linear dependence of vectors, the two-sorted theory of vector-spaces with their scalar-fields is companionable, and it is the union of a chain of companionable theories, but the model-companions of the theories in the chain are mutually inconsistent. Finally, the union of a chain of non-companionable theories may be companionable.