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Let $\mathcal{M}=(M,<,\ldots)$ be an o-minimal structure, namely a linearly ordered (by $<$) first order structure such that every definable set in $M$ is finite union of points and intervals $(a,b)$ where $a\in M\cup\{-\infty\}$ and $b=M\cup \{+\infty\}$.

Is there an expansion of $\mathcal{M}$ to an o-minimal structure that has definable choice (i.e. definable Skolem functions)?

Extra questions:

What about an expansion to an o-minimal structure that expands an ordered group $(M,0,+,<)$, or an ordered field $(M,0,1,+,\cdot, <)$?

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Anguepa
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    Why the close vote? This is a very good question. – Alex Kruckman Sep 17 '18 at 20:03
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    About your extra questions: every o-minimal group is an ordered divisible abelian group (ODAG), and every ODAG is densely ordered. So If you start with an o-minimal structure which is not densely ordered, it can't be expanded to an expansion of an ordered group or an ordered field. I don't know the answer to your main question, or to your extra questions if you assume $\mathcal{M}$ is densely ordered by $<$. – Alex Kruckman Sep 17 '18 at 20:08
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    In the previous comment, of course I meant to say "If you start with an o-minimal structure which is not densely ordered, it can't be expanded to an o-minimal expansion of an ordered group or an ordered field." – Alex Kruckman Sep 19 '18 at 20:39
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    Observation: it seems like an o-minimal structure should be decomposable into finitely many intervals, each of which is either dense without endpoints or discrete (possibly finite). You could simply add a (divisible ordered abelian) group structure on each of the dense intervals and it seems like this might preserve o-minimality. But I'm not sure what you should do with the discrete parts. Is it clear that $({\mathbf Z},<)$ has an o-minimal expansion with Skolem functions? – tomasz May 23 '19 at 14:30
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    @tomasz seems to me that any disctete o-minimal structure with a constant must have definable skolem functions. Just take the minimum or the maximum within the left-most interval or the constant. For densely ordered structures one might have to go to the the Trichotmy Theorem of o-minimality for answers. – Anguepa Aug 31 '19 at 09:11

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