To wrap up the notes on o-minimal structures, let’s consider two questions that were still open at the end of the last millennium:

**Questions**

- Does every o-minimal expansion of the real field admit analytic cell decomposition?
- Is there a unique maximal o-minimal expansion of the real field?

The answer to both questions is “no”, and I want to briefly summarize here how this follows from the theorems we proved here and from some other theorems known before the turn of the millenium but not proved here.

### Analytic cell decomposition

Replacing the term “$C^p$” in the definitions of cell and cell decomposition by “$C^\infty$” or “analytic”, we obtain the definitions of **$C^\infty$-cell** and **$C^\infty$-cell decomposition** or **analytic cell** and **analytic cell decomposition**, respectively.

**Fact**

*All examples of o-minimal expansions of the real field known by the year 2000 admit analytic cell decomposition.*

These examples include the structure $\Ran$ discussed here, as well as several expansions of $\Ran$ found here and here. Moreover, if $\R$ is an o-minimal expansion of the real field that admits analytic cell decomposition, then so does its pfaffian closure $\P(\R)$ discussed here.

On the other hand, the proof of the $C^p$-cell decomposition theorem works only for finite $p$, because the concepts of being $C^\infty$ or analytic are not first-order definable, as Wilkie succinctly shows here for the $C^\infty$ case.

### Maximal o-minimal expansions

An o-minimal expansion $\R$ of the real field is **maximal**, if every proper expansion of $\R$ is not o-minimal.

By Zorn’s Lemma, every o-minimal expansion of the real field has a maximal o-minimal expansion. But is there a unique maximal o-minimal expansion of the real field, that is, are all these maximal ones identical?

This question is related to the question whether two given o-minimal expansions $\R_1$ and $\R_2$ of the real field can be **amalgamated**, that is, whether the structure generated by all sets definable in $\R_1$ or $\R_2$ is again o-minimal.

For example, the expansion of the real field by the restriction of the Euler Gamma function to $(0,\infty)$ is o-minimal, as is the expansion of the real field by the restriction of the Riemann zeta function to $(1,\infty)$ (see p. 515 of this paper); but it is not known whether the expansion of the real field by both these restrictions is o-minimal. (I conjecture that it is.)

## Back to the questions

Both questions were first answered by finding suitable examples of quasianalytic classes:

**Theorem** (see Theorem 2 here)

*There exists a quasianalytic class $\C$ satisfying Axioms (C1)–(C4) such that $\C_1$ contains a germ that is nowhere analytic.**Given any $\C^\infty$-function $f:[-1,1] \into \RR$, there exist quasianalytic classes $\C$ and $\D$, each satisfying Axioms (C1)–(C4), and there exist $g \in \C_1$ and $h \in \D_1$ such that $f = g+h$.*

Part 1 of this theorem answers Question 1, since the corresponding o-minimal structure $\RR_\C$ contains a definable function that is nowhere analytic; hence the graph of this function cannot be a finite disjoint union of analytic cells.

Part 2 of this theorem answers Question 2, since both $\RR_\C$ and $\RR_\D$ are o-minimal, while the function $f$ is definable in any expansion of the real field that expands both $\RR_\C$ and $\RR_\D$. Thus, we can take $f:[-1,1] \into \RR$ defined by $$f(x):= \begin{cases} e^{-1/x} \sin\left(\frac1x\right) &\text{if } x > 0, \\ 0 &\text{if } x \le 0 \end{cases}$$ to obtain two examples of non-amalgamable o-minimal structures.

## Final remarks

Question 1 also has “no” as an answer when “analytic” is replaced by “$C^\infty$”, as a clever genericity construction by Le Gal and Rolin shows.

A much more intriguing answer to Question 2 is given here by Rolin, Sanz and Schäfke: they show that there exists an analytic 1-distribution $d$ on $\RR^3$, definable in the real field, such that every *single* integral manifold of $d$ generates an o-minimal expansion of the real field, while the expansion of the real field by any *two* distinct integral manifolds of $d$ is not o-minimal!