Spaces and Fields

A talk as part of Model Theory Day at the Istanbul Mathematical Sciences Center, Monday, April 18, 2016. Here are my notes (51 pages of size A5, 12-point type):

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The talk might be seen as a continuation of the 2013 talk called Descartes as Model Theorist, and indeed I revised my notes from that talk in preparation for this one.

Short abstract:

A field acting on an abelian group produces a vector space. A Lie ring acting on an abelian group produces a differential field. Combining the two actions produces a Lie--Rinehart pair, and we shall look at this from the model-theoretic point of view.

Long abstract:

Using areas, Euclid proved results that today we consider as algebraic. We consider them so, because Descartes justified algebra by showing how it could be considered as geometry.

Such observations can be understood as resulting from the equivalence of certain categories.

Models of a given first-order theory T are the objects of two different categories:

• Mod(T), in which the morphisms are embeddings, and
• Mod*(T), in which the morphisms are elementary embeddings.

The latter category is closed under direct limits; if the former is likewise closed, then T has universal-existential axioms (and conversely). T is called model-complete if the two categories are the same. T is called companionable if it includes a model-complete theory, called the model companion of T, in a model of which each model of T embeds.

A vector space here is a pair (K,V), where K is a field, V is an abelian group, and K acts on V. The theory of vector spaces in this sense has a model companion, which is theory of one-dimensional vector spaces.

If T is the theory of vector spaces of dimension at least two, and U is the theory of abelian groups with an appropriate notion of parallelism, then Mod(T) and Mod(U) are equivalent. If S is field theory, and T_n is the theory of n-dimensional vector spaces (where n>0) with a given basis, then Mod(S) and Mod(T_n) are equivalent.

In a vector space (K,V), V may also act on K as a Lie ring of derivations; then (K,V) becomes a Lie--Rinehart pair. Such pairs can be given universal-existential axioms, using only the signature of abelian groups for each of K and V, along with a symbol for the action of each on the other. In his 2010 dissertation, Özcan Kasal showed that the resulting theory is not companionable, although if predicates for certain definable relations are introduced, the theory becomes companionable, and the model companion is not stable. It turns out that like the theory of the integers as a group, the model companion even has the so-called tree property.