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

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

Una caracterización de los grupos abelianos fuertemente dependientes

We characterize all ordered Abelian groups whose first-order theory in the language {+, &lt;, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].Damos una caracterización completa de los grupos abelianos ordenados cuyas teorías completas en el lenguaje {+, &lt;, 0} son fuertamente dependientes. El resultado principal de este artículo fue obtenido de manera independiente por Halevi y Hasson [7] y Farré [5]

STRUCTURES HAVING O-MINIMAL OPEN CORE

The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: Let R be an expansion of a densely ordered group (R, &lt;, ∗) that is definably complete and satisfies the uniform finiteness property. Then the open core of R is o-minimal. Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core