Dp-minimality: basic facts and examples. (English) Zbl 1258.03036

Dolich, Alfred; Goodrick, John; Lippel, David

Notre Dame J. Formal Logic 52, No. 3, 267-288 (2011).

Summary: We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of \(p\)-adic numbers is dp-minimal.

MSC:

03C45	Classification theory, stability, and related concepts in model theory
03C64	Model theory of ordered structures; o-minimality

Keywords:

independence property; dp-minimal; weakly o-minimal; \(p\)-adic field