Tame Expansions of ω-Stable Theories and Definable Groups

Abstract

We study groups definable in tame expansions of ω-stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and groups interpretable in these expansions. As our main example, by characterizing independence in the pair (K, G) where K is an algebraically closed field and G is a multiplicative subgroup of K × with the Mann property, we show the pair (K, G) satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in (K, G) in terms of algebraic groups in K and interpretable groups in G. Furthermore, Morley rank and U-rank are computed in (K, G) and both ranks agree.1991 Mathematics Subject Classification. 03C45.

Algebraic numbers with elements of small height

“…One way of showing the simplicity is to find a notion of independence which is symmetric and satisfying the axioms of non‐forking [, Chapter 7]. In , the author characterized the independence in a sufficiently saturated model of in the language . More precisely, let A and B be two algebraically closed sets in the pair .…”

Number fields and divisible groups via model theory

“…Then, Mann's result was generalized in [5]. For the model-theoretic approaches, the reader might consult [1,4,8]. In [9], the second author and Sertbaş proved that a number field K is either Q or an imaginary quadratic fied if and only if Equation (1.1) has only finitely many non-degenerate solutions with coordinates in O K for all a 1 , ..., a k ∈ O K .…”