Open core and small groups in dense pairs of topological structures☆
Introduction
Tame topological structures and expansions by a predicate have often been considered from a model-theoretical point of view. Both p-adically and real closed fields are naturally endowed with a definable topology such that the field operations are continuous. The close interaction between the topological and algebraic properties of such structures is crucial to determine their model-theoretic behavior. Several frameworks have been suggested to treat simultaneously archimedean and non-archimedean normed fields: in this paper, we will introduce geometric topological rings (Definition 1.4) following the topological approach proposed by Mathews [18], which was later on adopted by Berenstein, Dolich and Onshuus [2] to study the theory of dense pairs.
Robinson [23] showed that the theory of a real closed field M equipped with a dense proper real closed subfield is complete (we will refer to this theory as the theory of dense pairs of real closed fields). Subsequently Macintyre [16] proved the same result for dense pairs of p-adically closed fields. The model-theoretical properties of dense pairs of o-minimal expansions of ordered abelian groups were thoroughly studied by van den Dries [9], who gave an explicit description of definable unary sets and functions, up to small sets. A set is small in a pair if it is contained in the image of the -points by a semialgebraic function. Though the theory of the pair is no longer o-minimal, every definable open set in the pair is actually definable in the reduct of M as an ordered field, so the theory of the pair has o-minimal open core [9], [19], [8], [4]. A similar result on the open core of pairs of p-adically and real closed fields has been recently obtained by Point [22] and later by Cubides-Kovacsics and Point [7] (see as well work of Tressl [27]) as a by-product of the study of the theory of differentially closed topological fields. In Section 1, we will use a criterium (cf. Proposition 1.15) of geometric nature in order to provide a new proof of the following result:
Theorem A Let be a dense pair of models of a geometric theory T of topological rings, in the language with . Assume that T has elimination of imaginaries or that every -definably closed set A is special, that is, where dim is the dimension as a geometric structure. Every open -definable subset of over a special set A is -definable. Hence, every definable open subset in the pair is already definable in the reduct (so the open core of the pair is tame).
In particular, both the theory of dense pairs of o-minimal expansions of real closed fields and the theory of dense pairs of p-adically closed fields have tame open core (see Corollary 1.18).
After private communication with van den Dries, we provide in the Appendix a fix for a minor gap in the published version of [9, Corollary 3.4], and show, in the broader context of geometric topological structures, that every definable unary function in the pair agrees off a small subset with a function definable in the predicate. Note that the same gap also affects [2, Theorem 4.9].
Understanding the nature of groups definable in a specific structure is a recurrent topic in model theory. In [14], it was shown that a group definable in a p-adically or real closed field M is locally isomorphic to the connected component of the M-points of an algebraic group defined over M. In particular, the group law is locally given by an algebraic map. Furthermore, if the group is Nash affine in a real closed field, this local isomorphism extends to a global isomorphism [15], since the Nash topology on Nash affine groups is noetherian. A crucial aspect to define the local isomorphism and the corresponding algebraic group through a group configuration diagram [13] is the strong interaction between the semialgebraic dimension in M and the transcendence degree, computed in the field algebraic closure of M. This interaction will be captured in Definition 3.1 in an abstract set-up.
A first description of groups definable in dense pairs of real closed fields appears in [12]: the group law is locally semialgebraic (off a wider class of certain sets, encompassing small sets). However, if the group is small, particularly if the underlying set is a subset of some cartesian product of , the above description does not provide any relevant information. In this note, we tackle the remaining case and consider groups whose underlying set is a subset of some cartesian product of , following the proof of Hrushovski and Pillay [14]. A recent result of Eleftheriou [11] on elimination of imaginaries on with the induced structure from the pair (see Section 4) allows us to extend our description to all small groups, whenever the topological geometric field has elimination of imaginaries.
The first obstacle we encountered is the lack of a sensible notion of dimension in the pairs. In an arbitrary pair of topological structures, there is a rudimentary notion of dimension, given by the small closure: the union of all small definable sets. Since we are only interested in small sets, the small closure is not useful for our purposes. In Section 2, we attach a dimension to definable subsets of in terms of their honest definition, as introduced in [6], when the topological geometric structure does not have the independence property. Using this dimension, we produce in Section 3 a suitable group configuration diagram and show the following:
Theorem B Consider a dense pair of models of a geometric NIP theory T of topological rings, in the language with . Assume that T is stably controlled with respect to a strongly minimal -theory (see Definition 3.1). If is an -definable group over a special set such that for some k, then the group law is locally -definable. Furthermore, if T has elimination of imaginaries, then the same holds for all small groups .
In case of the reals or the p-adics, we conclude that the group law is locally an algebraic group law. We do not know whether the group law is globally semialgebraic for small groups in dense pairs of p-adically or real closed fields.
We thank the anonymous referee of a previous version for the suggestions which have improved the presentation of this article.
Section snippets
Dense pairs of topological fields
In this section, we will recall the basic results on topological geometric structures [9], [2].
Definition 1.1 A structure M in the language is a topological structure if there is a formula , where x is a single variable, such that the collection forms a basis of a proper topology, that is, a topology with no isolated points.
Remark 1.2 If M is a topological structure, then so is every model of the theory of M. Since the formula has the order property [21, Proposition 1.2], the theory of
Honesty and dimension
By Fact 1.8.(3), any -definable subset Y of is externally -definable: there is some -definable subset Z in , possibly with parameters from M, such that . In a stable theory, externally definable subsets of any predicate are actually definable using parameters from the predicate, since φ-types are definable. This is not longer true for NIP theories, which is the context of interest in the sequel.
Fact 2.1 [2, Corollary 4.10] If the geometric theory T of topological rings has NIP, then so
Small groups
Every definable group G in a geometric structure M becomes naturally a topological group with respect to a definable topology on G (which need not coincide with the induced topology) [20, Theorem 7.1.10]. The study of groups in [14] relies on the close relation between a real or p-adically closed field and its algebraic closure. For a similar analysis of groups for geometric topological fields, we introduce the following notion.
Definition 3.1 A geometric theory T is stably controlled if there is a strongly
Digression: Elimination of imaginaries for the predicate
Eleftheriou showed [11] that, given any expansion of an o-minimal ordered group by a new predicate, the induced structure on the predicate has elimination of imaginaries whenever it satisfies three natural (yet technical) conditions, which hold for all canonical examples of pairs of an o-minimal expansion of an ordered group by a predicate. His result allows to characterize small sets and it will be crucial in order to describe arbitrary small groups. For the sake of the presentation, we will
References (29)
- et al.
Thorn independence in the field of real numbers with a small multiplicative group
Ann. Pure Appl. Log.
(2007) Ensembles définissables dans les corps ordonnés différentiellement clos
C. R. Acad. Sci. Paris
(2011)- et al.
Paires de structures o-minimales
J. Symb. Log.
(1998) - et al.
The independence property in generalized dense pairs of structures
J. Symb. Log.
(2011) - et al.
Expansions which introduce no new open sets
J. Symb. Log.
(2012) - et al.
Stable theories with a new predicate
J. Symb. Log.
(2001) - et al.
Externally definable sets and dependent pairs
Isr. J. Math.
(2013) - et al.
Topological fields with a generic derivation
- et al.
Structures having o-minimal open core
Trans. Am. Math. Soc.
(2010) Dense pairs of o-minimal structures
Fundam. Math.
(1998)
The fields of real and complex numbers with a small multiplicative group
Proc. Lond. Math. Soc.
Small sets in dense pairs
Isr. J. Math.
Structure theorems in tame expansions of o-minimal structures by a dense set
Isr. J. Math.
Contributions to stable model theory
Cited by (3)
Definable groups in dense pairs of geometric structures
2022, Archive for Mathematical LogicPillay's conjecture for groups definable in weakly o-minimal non-valuational structures
2021, Bulletin of the London Mathematical Society
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The first author is partially supported by MTM2014-59178-P, MTM2017-82105-P (Ministerio de Economía y Competitividad, Spain) and Grupos UCM 910444. The second author is partially supported by the project Lógica Matemática (grant number MTM2017-86777-P) as well as by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project number 2100310201, within the ANR-DFG program GeoMod.