# Chains of of Theories

A talk in the Istanbul Model Theory Seminar, February 28, 2013.

## Abstract

I shall speak about properties of theories that are preserved under taking unions of increasing chains of theories. I talked about these things last fall, on November 29, but I shall try both:

- not to assume that you were at that talk or remember it,
- to say things that were not in that talk.

The main examples are the following.

For each natural number *m*, the theory, called *m*-DF, of
fields with *m* commuting derivations has a model companion (which
is the theory of the existentially closed models of *m*-DF, this
class being elementary). Call this model-companion *m*-DCF. The
theories *m*-DCF are mutually inconsistent, and therefore the union
of the *m*-DF, which is consistent, has no model companion.

However, the theory of fields of characteristic 0 with omega-many
commuting derivations does have a model-companion, which preserves the
quantifier-elimination, completeness, and stability of the theories
*m*-DCF_{0}, but not their omega-stability.

The theory of vector-spaces with the scalar-field as an additional
sort is highly dependent on the chosen signature. If there are 2-ary,
3-ary, …, *m*-ary predicates for linear dependence of
vectors, let the theory be *m*-VS. Then the existentially closed
models of *m*-VS have dimension *m*, and the scalar field is
algebraically closed. Thus the union of the model-companions of the
*m*-VS is inconsistent. But the union of the *m*-VS has a
model-companion (as I said last time), whose completions, obtained by
specifying a characteristic for the field, are omega-stable. If
*m* is at least 2, the *m*-ary predicate for linear dependence
can be used to interpret the scalar field in the vector space (if this
has dimension at least *m*) in such a way that, in the foregoing
observations, *m*-VS can be replaced with the the theory of
(one-sorted) abelian groups with the appropriate *m*-ary predicate.

## Notes

8 pp., A5 paper: