# ON DEFINABLE SKOLEM FUNCTIONS IN WEAKLY O-MINIMAL NONVALUATIONAL STRUCTURES

@article{Eleftheriou2017ONDS, title={ON DEFINABLE SKOLEM FUNCTIONS IN WEAKLY O-MINIMAL NONVALUATIONAL STRUCTURES}, author={Pantelis E. Eleftheriou and Assaf Hasson and Gil Keren}, journal={The Journal of Symbolic Logic}, year={2017}, volume={82}, pages={1482 - 1495} }

Abstract We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.

## 6 Citations

Tame Topology over dp-Minimal Structures

- Mathematics, Computer ScienceNotre Dame J. Formal Log.
- 2019

T tame topology is developed over dp-minimal structures equipped with definable uniformities satisfying certain assumptions to ensure that definable sets are tame.

Pillay's conjecture for groups definable in weakly o‐minimal non‐valuational structures

- MathematicsBulletin of the London Mathematical Society
- 2020

Let G be a group definable in a weakly o‐minimal non‐valuational structure M . Then G/G00 , equipped with the logic topology, is a compact Lie group, and if G has finitely satisfiable generics, then…

Strong cell decomposition property in o-minimal traces

- MathematicsArch. Math. Log.
- 2021

All o-minimal traces have strong cell decomposition property and it is shown that every expansion of o- Minimal structures by irrational nonvaluational cuts is an o-Minimal trace.

Externally definable quotients and NIP expansions of the real ordered additive group

- MathematicsTransactions of the American Mathematical Society
- 2021

Let $\mathscr{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R}, 0$ and collection $\mathcal{B}$ of bounded subsets of $\mathbb{R}^n$ such that $(\mathbb{R},<,+,\mathcal{B})$ is o-minimal.

Small sets in dense pairs

- MathematicsIsrael Journal of Mathematics
- 2019

Let $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ be an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, such that three tameness conditions hold. We prove that…

A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

- MathematicsThe Journal of Symbolic Logic
- 2019

It is proved that it is near model complete, and every definable open subset of ${\bar{M}^n}$ is already definable in $\bar{{\cal M}}$.

## References

SHOWING 1-10 OF 22 REFERENCES

Weakly o-minimal structures and real closed fields

- Mathematics
- 2000

A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced…

On the strong cell decomposition property for weakly o‐minimal structures

- MathematicsMath. Log. Q.
- 2013

It is proved that the strong cell decomposition property is preserved under elementary equivalences and fiberwise properties (of definable sets and definable functions), definable equivalence relations, and conditions implying elimination of imaginaries are investigated.

On expansions of weakly o-minimal non-valuational structures by convex predicates

- Mathematics
- 2009

We prove that if M = (M,≤,+, . . .) is a weakly o-minimal non-valuational structure expanding an ordered group (M,≤,+), then its expansion by a family of ‘non-valuational’ unary predicates remains…

Definable choice for a class of weakly o-minimal theories

- MathematicsArch. Math. Log.
- 2016

It is shown that for a properly convex subset U, the theory of the expanded structure M has definable Skolem functions precisely when M′ is valuational, and an elementary proof is obtained that any such theory does not satisfy definable choice.

Dense pairs of o-minimal structures

- Mathematics
- 1998

The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays a…

A growth dichotomy for o-minimal expansions of ordered groups

- Mathematics
- 1998

Let R be an o-minimal expansion of a divisible ordered abelian group (R, <, +, 0, 1) with a distinguished positive element 1. Then the following dichotomy holds: Either there is a 0-definable binary…

Omitting types in -minimal theories

- MathematicsJournal of Symbolic Logic
- 1986

Let L be a first order language containing a binary relation symbol <. Definition. Suppose ℳ is an L-structure and < is a total ordering of the domain of ℳ. ℳ is ordered minimal (-minimal) if and…

Non-Archimedean Tame Topology and Stably Dominated Types

- Mathematics
- 2016

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model…