Generic expansions by a reduct

Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present a setup in which this expansion admits a model companion [Formula: see text]. We show that some of the nice features of the theory [Formula: see text] transfer to [Formula: see text]. In particular, we study conditions for which this expansion preserves the [Formula: see text]-ness, the simplicity or the stability of the starting theory [Formula: see text]. We give concrete examples of new [Formula: see text] not simple theories obtained by this process, among them the expansion of a perfect [Formula: see text]-free PAC field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.

Model theory of R-trees

We show the theory of pointed $\R$-trees with radius at most $r$ is axiomatizable in a suitable continuous signature. We identify the model companion $\rbRT_r$ of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of $\rbRT_r$ are $\R$-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of $\rbRT_r$; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of $\rbRT_r$. Among other things, we show that the space of $2$-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that $\rbRT_r$ has no atomic model.

Interpolative fusions

We define the interpolative fusion [Formula: see text] of a family [Formula: see text] of first-order theories over a common reduct [Formula: see text], a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each [Formula: see text] is model-complete, [Formula: see text] coincides with the model companion of [Formula: see text]. By obtaining sufficient conditions for the existence of [Formula: see text], we develop new tools to show that theories of interest have model companions.

ABOUT EXISTENCE MODEL COMPANION

Found syntactic conditions sufficient for the existence model companion inductive theory.

DEFINABLE GROUPS IN DCFA

E. Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion. We denote it DCFA. In this paper we study definable abelian groups in a model of DCFA. First we prove that such a group is embeddable on an algebraic group. Then, we study one-basedeness, stability and stable embeddability of abelian definable groups.

Differential-Henselian Fields with Many Constants

This chapter considers differential-henselian fields with many constants. Here d-henselian includes having small derivation, so d-henselian valued differential fields with many constants are monotone. The goal here is to derive Scanlon's extension in [382, 383] of the Ax-Kochen-Eršov theorems to d-henselian valued differential fields with many constants. Among the results to be established is Theorem 8.0.1: Suppose K and L are d-henselian valued differential fields with many constants. Then K = L as valued differential fields if and only if res K = res L as differential fields and Γ‎ subscript K = Γ‎ subscript L as ordered abelian groups. The chapter also discusses an angular component map on K, equivalence over substructures, relative quantifier elimination, and a model companion.

THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

A model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.