Structure theorems in tame expansions of o-minimal structures by a dense set
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STRUCTURE THEOREMS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES BY A DENSE SET
BY
Pantelis E. Eleftheriou∗ Department of Mathematics and Statistics, University of Konstanz Box 216, 78457 Konstanz, Germany e-mail: [email protected]
AND
¨naydin∗∗ Ayhan Gu Department of Mathematics, Bo˘ gazi¸ci University, Bebek, Istanbul, Turkey e-mail: [email protected]
AND
Philipp Hieronymi† Department of Mathematics, University of Illinois at Urbana-Champaign 1409 West Green Street, Urbana, IL 61801, USA e-mail: [email protected]
∗ The first author was supported by an Independent Research Grant from the Ger-
man Research Foundation (DFG) and a Zukunftskolleg Research Fellowship.
∗∗ The second author was partially supported by TUBITAK Career Grant 113F119. † The third author was partially supported by NSF grant DMS-1300402.
Received April 25, 2019 and in revised form September 19, 2019
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P. E. ELEFTHERIOU ET AL.
Isr. J. Math.
ABSTRACT
We study sets and groups definable in tame expansions of o-minimal struc = M, P be an expansion of an o-minimal L-structure M tures. Let M and prove a by a dense set P . We impose three tameness conditions on M structure theorem for definable sets and functions in analogy with the cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of such M, as it achieves a decomposition of definable sets into unions of ‘cones’, instead of only boolean combinations of them. The proofs involve induction on the notion of ‘large dimension’ for definable sets, an invariant which we herewith introduce and analyze. Applications of the cone decomposition theorem include: (i) the large dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an L-definable map off a subset of the domain of smaller large dimension, and (iii) around generic elements of a definable group, the group operation is given by an L-definable map.
1. Introduction Definable groups in models of first-order theories have been at the core of model theory for at least a period of three decades (see, for example, [5, 35, 43]) and have been crucially used in important applications of model theory to other areas of mathematics (such as in [30]). An indispensable tool in their analysis has been a structure theorem for the definable sets and types: analyzability of types and the existence of a rank in the stable category, and a cell decomposition theorem and the associated topological dimension in the o-minimal setting. In this paper we establish a structure theorem for definable sets and functions in tame expansions of o-minimal structures, introduce and analyze the relevant notion of dimension and establish a local theorem for definable groups in this setting. Our structure theorem is inspired by a cone decomposition theorem known for semi-bounded o-minimal structures ([15, 17, 36]), which was also vitally used in the analysis of de
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