Model-completion of scaled lattices

Details Model-completion of scaled lattices Model-completion of scaled lattices

2006-06-30

arxiv.org/abs/math/0606792v1

Mathematics

Logic

27 pages

Scientific paper

It is known from Grzegorczyk's paper \cite{grze-1951} that the lattice of real semi-algebraic closed subsets of ${\mathbb R}^n$ is undecidable for every integer $n\geq 2$. More generally, if $X$ is any definable set over a real or algebraically closed field $K$, then the lattice $L(X)$ of all definable subsets of $X$ closed in $X$ is undecidable whenever $\dim X\geq 2$. Nevertheless, we investigate in this paper the model theory of the class ${\rm SC\_{def}}(K,d)$ of all such lattices $L(X)$ with $\dim X\leq d$ and $K$ as above or a henselian valued field of characteristic zero.

We show that the universal theory of ${\rm SC\_{def}}(K,d)$, in a natural expansion by definition of the lattice language, is the same for every such field $K$. We give a finite axiomatization of it and prove that it is locally finite and admits a model-completion, which turns to be decidable as well as all its completions. We expect $L({\mathbb Q}\_p^d)$ to be a model of (a little variant of) this model-completion. This leads us to a new conjecture in $p$-adic semi-algebraic geometry which, combined with the results of this paper, would give decidability (via a natural recursive axiomatization) and elimination of quantifiers for the complete theory of $L({\mathbb R}\_p^d)$, uniformly in $p$.

Affiliated with

Also associated with

No associations

Rating

Model-completion of scaled lattices does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Model-completion of scaled lattices, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Model-completion of scaled lattices will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-117520