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DEFINABLE SKOLEM FUNCTIONS IN WEAKLY O-MINIMAL NON-VALUATIONAL EXPANSIONS OF ORDERED GROUPS

Part of: Model theory

Published online by Cambridge University Press:  26 August 2025

SOMAYYEH TARI  and
MOHSEN KHANI
SOMAYYEH TARI*
Affiliation:
DEPARTMENT OF MATHEMATICS https://ror.org/05pg2cw06 AZARBAIJAN SHAHID MADANI UNIVERSITY TABRIZ 5375171379, IRAN
MOHSEN KHANI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES https://ror.org/00af3sa43 ISFAHAN UNIVERSITY OF TECHNOLOGY ISFAHAN 84156-83111, IRAN E-mail: mohsen.khani@iut.ac.ir
*
E-mail: s_tari@azaruniv.ac.ir
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Abstract

Let $\mathcal {M}=(M,<,+, \dots )$ be a weakly o-minimal non-valuational structure expanding an ordered group. We show that the full first-order theory $\operatorname {\mathrm {Th}}(\mathcal {M})$ has definable Skolem functions if and only if isolated types in $S_{n}^{\mathcal M}(A)$ are dense for each $ A\subseteq M $ and $ n\in \mathbb {N} $. Using this, we prove that no strictly weakly o-minimal non-valuational expansion of an ordered group has definable Skolem functions, thereby answering Conjecture 1.7 of Eleftheriou et al. (On definable Skolem functions in weakly o-minimal non-valuational structures. J. Symb. Logic, vol. 82 (2017), no. 4).

Keywords

weakly o-minimal structuredefinable Skolem functionnon-valuational cuts

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 11
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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