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Structures having o-minimal open core

Authors: Alfred Dolich, Chris Miller and Charles Steinhorn
Journal: Trans. Amer. Math. Soc. 362 (2010), 1371-1411
MSC (2000): Primary 03C64; Secondary 06F20
Published electronically: October 8, 2009
MathSciNet review: 2563733
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Abstract: 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 $ \mathfrak{R}$ be an expansion of a densely ordered group $ (R,<,*)$ that is definably complete and satisfies the uniform finiteness property. Then the open core of  $ \mathfrak{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.

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Additional Information

Alfred Dolich
Affiliation: Department of Mathematics and Computer Science, Chicago State University, Chicago, Illinois 60628
Address at time of publication: Department of Mathematics, East Stroudsberg University, East Stroudsberg, Pennsylvania 18301

Chris Miller
Affiliation: Department of Mathematics, 231 West 18th Avenue, The Ohio State University, Columbus, Ohio 43210

Charles Steinhorn
Affiliation: Department of Mathematics, Vassar College, Poughkeepsie, New York 12604

Keywords: o-minimal, open core, ordered group, ordered field, uniform finiteness, definably complete, exchange property, definable Skolem functions, elimination of imaginaries, independence property, atomic model, dense pair, generic predicate, tame pair
Received by editor(s): January 7, 2008
Published electronically: October 8, 2009
Additional Notes: The second author was partially supported by NSF Grant DMS-9988855. The third author was partially supported by NSF Grant DMS-0070743.
Article copyright: © Copyright 2009 American Mathematical Society

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