比例格的模型完备化
Model-completion of scaled lattices
摘要：
从Grzegorczyk的论文\cite{grze-1951}可知，${\mathbb R}^n$的实半代数闭子集的格对每个整数$n\geq 2$都是不可判定的。更一般地说，如果$X$是实域或代数闭域$K$上的任何可定义集，则$X$闭于$X$中的所有可定义子集的格$L(X)$无论何时$\dim X\geq 2$都是不可判定的。然而，我们在本文中研究类${\rm SC\_{def}}(K,d)$的模型理论的所有这样的格$L(X)$与$\dim X\leq d$和$K$如上或一个亨塞尔值域的特征零。<p>我们表明，通用理论的${\rm SC\_{def}}(K,d)$，在一个自然的扩展定义的格语言，是相同的，每个这样的领域$K$。我们给出了它的一个有限公理化，证明了它是局部有限的，并允许一个模型完备化，它变成可判定的以及它的所有完备化。我们期望$L({\mathbb Q}\_p^d)$成为这个模型完成的一个模型（一个小变体）。这导致我们在$p$ -进半代数几何的一个新的猜想，结合本文的结果，将给予可判定性（通过一个自然的递归公理化）和消除量词的完整理论的$L({\mathbb R}\_p^d)$，一致$p$。
英文摘要：
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. <p> 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 $p$.
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