# Model-theoretic research

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The current focus of my research is definable Skolem functions in weakly o-minimal structures, specifically constructive
definitions of Skolem functions. I have found an explicit construction of Skolem functions for a subclass of valuational weakly
o-minimal theories called *T*-immune, where it was known that there were definable Skolem functions, but for which there was
no construction. I have also analyzed a certain subclass of nonvaluational weakly o-minimal structures, regarded colloquially to be
as close as possible to being o-minimal, and found that such theories in fact do not have definable Skolem functions.

I currently have a paper in preparation which narrates the results outlined above and some extensions to more general cases.

## Background

For the purpose of this introduction, a *model* **M** is a set (called the *universe* of the
model) together with a
specified algebraic structure of *definable sets*, which may be subsets of the universe **M** itself, or of
**M**^{n} for a finite integer *n*. A canonical example of a model is the field of real numbers,
(**R**,+,·,0,1,<), in which case the definable subsets are precisely the semialgebraic subsets of
**R**^{n}.

An *o-minimal structure* is a model **M** whose definable sets include a dense linear order on the universe, and for
which any definable set of **M** is a finite union of points and intervals (whose infema and suprema are elements of the
universe).
The real field is the archetype for this study. During the past several decades, a rash of work has led to a powerful structure
theory for general o-minimal structures (*cf.* [6] [8] [11] [12]). Among
the properties enjoyed by this class, every o-minimal structure has a strong *cellular decomposition* property which
guarantees all definable subsets **M**^{n} can be written as a finite union of simple definable subsets, called
*cells*.

Independently of this, *Skolem functions* were developed initially in order to prove what is now known as the
Löwenheim-Skolem theorem (*cf.* [2]). Given a model **M** and
definable set *D* ⊆ **M**^{n}, a Skolem function is a function *f* such that for every *a* ∈ **M**^{n-1}, if there is some *y* ∈ **M**
so that (*a*,*y*) ∈ *D*, then
(*a*, f(*a*)) ∈ *D*.

Informally, one says that a Skolem function finds a witness for *D*, if there is one. Skolem functions are useful in their
own right, both in providing conditions for model completeness, and as a tool used in automated theorem-proving. Any o-minimal
model with a group operation (an *o-minimal group*) can also be shown to have definable Skolem functions. The algorithm for
determining Skolem functions expands upon the following simple case: if **D**(x,y) ⊆ **M**^{2} defines,
for every fixed value of *x*, an interval, then value of the Skolem function for each *a* is the midpoint of the interval
defined by the set *D*(*a,y*). O-minimal structures also satisfy the related property of having *uniform elimination of
imaginaries*.

*Weakly o-minimal structures* generalize o-minimal structures by allowing each definable subset of the model **M** to be
a finite union of *convex sets* which are not necessarily intervals. Consider the ordered group of rational numbers,
(**Q**,+,<). This structure is o-minimal. If we add to the structure the definable set *P*={*x* ∈ **Q**:
x<π}, then the resulting expansion (**Q**, +, <, *P*) is not o-minimal: the supremum of the set named by *P* is not
a rational number, thus *P* cannot represent an interval in **Q**. But this set is *convex* in **Q**, and it can be
shown that the expanded structure is weakly o-minimal. In fact, it is shown by Baizhanov in [1] that any o-minimal theory, if new convex subsets are introduced, yields a weakly o-minimal theory. A
more complex structure of this type is the real-closed valued field (**R**, +, ·, 0, 1, <, *V*), in which **R**
is a real-closed field with value ring *V*. This theory (called *RCVF*) is also weakly o-minimal, and the model theory is
explored at length in [3], [4], and [10].

In view of these facts, there is a large program of study concerned with determining which of the properties of o-minimal groups
also hold true in the weakly o-minimal case. The authors of [9] distinguish between a
*valuational* weakly o-minimal group, in which there is a proper definable subgroup, and a *nonvaluational* weakly
o-minimal group. They showed that while weakly o-minimal groups in general need not have cellular decomposition, the class of
nonvaluational weakly o-minimal groups does have an analogue of this property.

## My research projects

*Let T be a weakly o-minimal theory with uniform elimination of imaginaries and definable Skolem
functions,
and M⊧ T. Then M is nonvaluational.
*

This proposition actually arises as a simple corollary of a deeper lemma, which may yet have some broader consequences.

*Let M be a model of a weakly o-minimal theory T which has definable Skolem functions and uniform
elimination of imaginaries. Then there is no equivalence relation E definable on M with infinitely many convex equivalence classes
of nonzero length.
*

It is shown in [10] that under certain limitations, real-closed valued fields have elimination of imaginaries. A possible consequence of this along with the proposition is that such structures will fail to have definable Skolem functions; in future research I hope to be able to determine whether this is the case.

Hoping to understand the implications of the above results, we began studying the class of properly nonvaluational weakly o-minimal models, in order to see whether there in fact are any such models which satisfy the conditions of the proposition. A natural class of these is the nonvaluational weakly o-minimal theories obtained by adding a predicate for a new nonvaluational convex subset to an o-minimal structure. However, this work turned up the surprising result that in fact such structures cannot have definable Skolem functions at all.

*Let M be an o-minimal expansion of an ordered group in the language L. Let U be a new unary predicate
symbol, L'=L &cup {U}, and M'=(M,U), where U^{M'} is a downward-closed convex set which defines a
properly convex nonvaluational cut. Then M' does not have definable Skolem functions in L'.*

The proof of this theorem is based in part on work by L. van Den Dries on dense pairs of o-minimal structures (*cf.* [5]), and relied on my lemma below, which establishes the connection between weakly
o-minimal structures and dense pairs of o-minimal structures.

*
Let M be o-minimal with language L; let L' = L ∪ {U}, and M' = (M, U) with U^{M'} a
downward-closed nonvaluational convex set, and N = pr(M ∪ {b}), where b realizes tp_{C}(sup U / M). Then
for any X ⊆ M definable in M', there is an L-formula φ_{X}
(x,y) such that X=φ_{X}(N^{n},b) ∩
M^{n}.
*

**M** be an o-minimal expansion of an ordered group, and *V* ⊆ **M** be a convex set. We say that the pair
(**M**, *V*) is *T-immune* if *V* ⊆ **M** and for any 0-definable function *F*: **M** →
**M** and
any open convex set *I* ⊆ *V*^{M}, if *F* restricted to *I* is continuous, then
*F*(*V*) ⊆ *V*.

As an example of *T*-immunity, consider a nonstandard model of the real group, **M**=(**R***, +, <, 0), where
**R** ⊆ **R***, and consider *V* interpreted by **R**. Then (**M**, *V*) is valuational and has a weakly
o-minimal theory, and in particular is *T*-immune.

*
Let ( M, +, <, 0, ε, ...) be an o-minimal expansion of a group with named positive element ε in a language
L which admits elimination of quantifiers, and V⊆ M such that (M,V) is T-immune. (Note that since ε
∈ L, then ε^{M} ∈ V.) Let c be a new constant symbol and c^{M}>0 an element of
M\V. Then (M, V, c) has definable Skolem functions in the language L ∪ {V,c}.
*

L. van den Dries has studied theories with a property known as *T*-convexity, which generalizes the notion of
*T*-immunity. The authors of [7] showed that *T*-convex theories also have
definable Skolem functions. I am currently working on generalizing the algorithm for calculating Skolem functions in a
*T*-immune theory to the *T*-convex case in order to give an explicit construction.

A natural extension of these results would be a precise set of conditions for definable Skolem functions in any weakly o-minimal
theory. For technical reasons, there are many theories which fail to be *T*-convex, but may be made so by augmenting the
language in a simple way. Modulo a reasonable concept of "almost *T*-convexity," I am investigating now whether it is true
that no weakly o-minimal theory obtained by the Baizhanov technique which fails to be "almost *T*-convex" has definable Skolem
functions. For now, the chief method for doing this is to generalize the concept of dense pairs to theories which may be
valuational.

*last updated 2010*

## References

[1] B. Baizhanov, "Expansion of a model of a weakly o-minimal structure by a family of convex predicates," J. Symb. Log.**66**No. 3, 1382-1414 (2001).

[2] C. Chang, H. J. Keisler,

*Model Theory*, Studies in Logic and the Foundations of Mathematics, vol 73 (Elsevier, Amsterdam 1973).

[3] G. Cherlin, M. A. Dickmann, "Real closed rings I: Residue rings of rings of continuous functions," Fund. Math.

**126**147-183 (1986).

[4] G. Cherlin, M. A. Dickmann, "Real closed rings II: Model theory," Ann. Pure App. Logic

**25**213-231 (1983).

[5] L. van den Dries, "Dense pairs of o-minimal structures," Fund. Math.

**157**, 61-78 (1998).

[6] L. van den Dries,

*Tame Topology and O-minimal Structures*, London Mathematical Society Lecture Note Series, vol. 248 (Cambridge University Press, Cambridge 1998).

[7] L. van den Dries, A. H. Lewenberg, "

*T*-Convexity and tame extensions," J. Symb. Log.

**60**No. 1, 74-102 (1995).

[8] J. Knight, A. Pillay, C.Steinhorn, "Definable sets and ordered structures II," Trans. AMS

**295**, 593-605 (1986).

[9] D. MacPherson, D. Marker, C. Steinhorn, "Weakly o-minimal structures and real closed fields," Trans. Amer. Math. Soc. \textbf{352} No. 12, 5435-5483 (2000).

[10] T. Mellor, "Imaginaries in real closed valued fields," Ann. Pure App. Logic

**139**, 230-279 (2006).

[11] A. Pillay, C. Steinhorn, "Definable sets in ordered structures I," Trans. AMS

**295**, 565-592 (1986).

[12] A. Pillay, C. Steinhorn, "Definable sets in ordered structures III," Trans. AMS

**309**, 469-476 (1988).