Dp-minimality: basic facts and examples

We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. 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.Comment: 19 pages; simplified proof for the p-adic

Representing Scott sets in algebraic settings

We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic

Discrete sets definable in strong expansions of ordered Abelian groups

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive elements. In particular, if the burden of the structure is at most n, then the result of applying the operation mapping D to D' n times must be a finite set (Theorem 2.13). In the case when the structure is densely ordered and has burden 2, we show that any definable unary discrete set must be definable in some elementary extension of the structure (R; <, +, Z) (Theorem 3.1).Comment: 41 pages. This newly revised version corrects some errors from the original version (pointed out by the anonymous referee) and some arguments have been significantly revised for clarit