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I am trying to prove that the DLO $(\mathbb{Q}, <)$ is an $O$-minimal structure, but I am trying to prove this without using the fact that the complete theory of dense linear orderings without endpoints admits quantifier elimination.

Here is my proof thus far: Let $N = \varphi(\mathbb{Q}, \bar{a})$ where $\varphi$ is a first-order formula from the language of partial orders. We may assume that $\bar{a} = (a_0, \ldots, a_{n-1})$ and that $a_0<\cdots <a_{n-1}$. Put $a_{-1}=-\infty$ and $a_n = \infty$.

Assumption: for any $-1\leq j\leq n-1$ and $x,y \in \mathbb{Q}$ such that $a_j<x<y<a_{j+1}$ there is an order automorphism $f$ which fixes $\bar{a}$ and $f(x)=y$.

If this is true, then if $N$ contains an element of the interval $(a_j, a_{j+1})$ it must then contain the whole open interval, and from this we can infer that the structure is O-minimal.

I am having difficulty, however, proving the assumption. Any help would be greatly appreciated.

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2 Answers 2

up vote 5 down vote accepted

Edit: In the original version I very carelessly used $\frac{y}{x}$ and $\frac{x}{y}$ for the multipliers, when of course they have to be $\frac{y-a_j}{x-a_j}$ and $\frac{a_{j+1}-y}{a_{j+1}-x}$. Barring further stupidity on my part, it’s now fixed.

You want to stretch the interval $[a_j,x]$ to cover the interval $[a_j,y]$ while shrinking $[x,a_{j+1}]$ down to $[y,a_{j+1}]$. This means stretching $[a_j,x]$ by a factor of $\frac{y-a_j}{x-a_j}$. You’re going to leave $a_j$ fixed, so what really gets stretched is the distance of each point of $[a_j,x]$ from $a_j$. A function that does this is $$f_L(q) = \frac{y-a_j}{x-a_j}(q-a_j)+a_j = \frac{y-a_j}{x-a_j}q + \frac{x-y}{x-a_j}a_j;$$ since $\frac{y-a_j}{x-a_j}$ and $a_j$ are rational, $f_L$ takes rationals to rationals.

Of course this still leaves $[x,a_{j+1}]$ to be taken care of. The same basic idea works, but this time it’s the distance from $q$ to $a_{j+1}$ that matters, and it has to be multiplied by a factor of $\frac{a_{j+1}-y}{a_{j+1}-x}$. For this the function $$f_R(q) = a_{j+1}-\frac{a_{j+1}-y}{a_{j+1}-x}(a_{j+1}-q) = \frac{a_{j+1}-y}{a_{j+1}-x}q + \frac{y-x}{a_{j+1}-x}a_{j+1}$$ does the trick. Now just combine and extend these to cover $\mathbb{Q}$: $$f(q) = \begin{cases} q, &q < a_j\\ \frac{y-a_j}{x-a_j}q + \frac{x-y}{x-a_j}a_j, &a_j \le q < x\\ \frac{a_{j+1}-y}{a_{j+1}-x}q + \frac{y-x}{a_{j+1}-x}a_{j+1}, &x \le q < a_{j+1}\\ q, &q \ge a_{j+1}. \end{cases}$$

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It is sufficient to show that there is an order-preserving map taking $\mathbb{Q} \cap (0,1)$ onto itself. Look for a fractional linear transformation $$f(x) = \frac{(c+d)x}{cx+d}$$ This fixes 0 and 1. If you choose $c, d \in \mathbb{Q}$ properly, it will be increasing in $(0,1)$, take rationals onto rationals, and allow you to map any rational in $(0,1)$ onto any other.

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