Thorn independence in the field of real numbers with a small multiplicative group

Abstract

We characterize þ-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.

NIP for some pair-like theories

“…We apply our result to theories of the real field with a predicate for a subgroup of the unit circle which were studied by Belegradek and Zilber in [1]. Similar theories were studied by van den Dries and Günaydın in [5] and by Berenstein, Ealy and Günaydın in [3] and shown to have NIP in [6].…”

Structures having o-minimal open core

“…As the conditions of (2) are first order, it suffices to consider the case that R is saturated. We now show that (2) is equivalent to 6.7 (2).…”

“…In Sections 5 and 6, we show that these examples extend to our more general setting and we investigate them in more detail, with particular attention paid to EP, definable Skolem functions, elimination of imaginaries, atomic models, and NIP (the "non-independence property"). During the preparation of this paper, some other interesting examples related to dense pairs came to light via Belegradek and Zilber [1,41] and Berenstein, Ealy, and Günaydın [2]; we discuss this briefly at the end of Section 5.…”

“…Some interesting theories related to dense pairs have recently come to light via work of others [1,2,41]. We shall not attempt to describe them fully, but as examples, if E is either a noncyclic, finitely generated subgroup of (R >0 , · ), or the set of all complex roots of unity (regarded as a subset of R 2 ), then the theory of real closed fields is an open core of Th(R, +, ·, E).…”

“…We must show that f −1 (G J ) = ∅ with all data as in (2). Let S be the parameters used in defining f .…”