Stable groups and expansions of
Tom 242 / 2018
Fundamenta Mathematicae 242 (2018), 267-279
MSC: Primary 03C45; Secondary 03C64.
DOI: 10.4064/fm403-8-2017
Opublikowany online: 26 February 2018
Streszczenie
We prove that if G is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then G is superstable of finite U-rank. Combined with recent work of Palacín and Sklinos, this shows that (\mathbb Z,+,0) has no proper stable expansions of finite weight. A corollary is that if P\subseteq \mathbb Z^n is definable in a finite dp-rank expansion of (\mathbb Z,+,0), and (\mathbb Z,+,0,P) is stable, then P is definable in (\mathbb Z,+,0). In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.
