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Preprint Number 25
Luck Darnière Model completion of scaled lattices Email: Submission date: Abstract: It is known from Grzegorczyk's paper \cite{grze1951} that the lattice of real semialgebraic 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 modelcompletion, 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 modelcompletion. This leads us to a new conjecture in $p$adic semialgebraic 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 $p$. Mathematics Subject Classification: 03C10 ; 06D20 ; 06D99 Keywords and phrases: Model theory ; quantifier elimination ; scaled lattice ; Heyting algebra ; padically closed field Full text: arXiv.

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