Existentially closed exponential fields

Abstract

We characterise the existentially closed models of the theory of exponential fields. They do not form an elementary class, but can be studied using positive logic. We find the amalgamation bases and characterise the types over them. We define a notion of independence and show that independent systems of higher dimension can also be amalgamated. We extend some notions from classification theory to positive logic and position the category of existentially closed exponential fields in the stability hierarchy as NSOP 1 but TP 2 .Kind 3, where Kind 1 is the usual setting of a complete first-order theory, and Kind 2, also known as Robinson theories, is like Kind 3 but where there is amalgamation over every subset of every model. (There is also Kind 4, which is now known as homogeneous abstract elementary classes.) More recently there has been interest in positive model theory, which is only slightly more general than Kind 3. This is the approach taken in this paper, and it means that we look at embeddings and existentially definable sets instead of elementary embeddings and all first-order definable sets. Following Pillay ( 2000) we call it the Category of existentially closed models rather than Kind 3. Model-theoretic backgroundWe give some background on the model theory of the category of existentially closed models of a (usually incomplete) inductive first-order theory. More details can be found in Hodges (1993), Pillay (2000 and Ben Yaacov and Poizat (2007). The only novelty in this section is the notion of a JEP-refinement of an inductive theory T (Definition 2.11), which is a useful syntactic counterpart to the choice of a monster model.

Kim-independence in positive logic

“…Remark 2.26. The idea of introducing the inconsistency witness ψ(y 1 , y 2 ) is due to Haykazyan and Kirby, [HK21]. In a first-order theory we can just take ψ(y 1 , y 2 ) to be ¬∃x(ϕ(x, y 1 ) ∧ ϕ(x, y 2 )), so we see that the definitions coincide there.…”

“…Next, we look at (the JEP refinements of) the positive theory of existentially closed exponential fields, which was shown to be NSOP 1 in [HK21] by constructing a suitable independence relation. We deduce from the known results that this theory is Hausdorff (hence thick), and then we show that Kim-independence coincides in it with the independence relation studied in [HK21]. Finally, we show that NSOP 1 is preserved under taking hyperimaginary extensions; in particular, the hyperimaginary extension of an arbitrary NSOP 1 first-order theory is a Hausdorff NSOP 1 theory.…”

“…Existentially closed exponential fields. In [HK21] the class of existentially closed exponential fields is studied. They prove that this is NSOP 1 by providing a nice enough independence relation.…”

“…In [HK21] the authors studied the theory ECEF of existentially closed exponential fields, and, working with an arbitrary JEP-refinement (which, intuitively, corresponds to a completion of an incomplete theory in the first-order setting), they have found an invariant ternary relation satisfying over models the following properties: strong finite character, existence, monotonicity, symmetry, and independence theorem. They have also proved that, for any positive theory, the existence of such a relation implies NSOP 1 , so in particular the JEP-refinements of ECEF are NSOP 1 .…”

“…Sections 6, 7 and 8 contain the proofs of the main properties of Kim-independence in thick positive theories: symmetry, independence theorem and transitivity, and Section 9 is dedicated to proving a Kim-Pillay-style characterisation of the NSOP 1 property among thick positive theories by existence of an abstract independence relation satisfying certain properties, and the characterisation of Kim-independence in NSOP 1 theories as the only relation satisfying them. In Section 10 we describe in details some examples of thick NSOP 1 theories: Poizat's example of a thick non-semi-Hausdorff theory, (JEP refinements of) the positive theory of existentially closed exponential fields studied in [HK21], and the hyperimaginary extensions of NSOP 1 theories.…”