Let p be prime number, K be a p-adically closed field, X $\subseteq$ K^m a semi-algebraic set defined over K and L(X) the lattice of semi-algebraic subsets of X which are closed in X. We prove that the complete theory of L(X) eliminates the quantifiers in a certain language LASC, the LASC-structure on L(X) being an extension by definition of the lattice structure. Moreover it is decidable, contrary to what happens over a real closed field. We classify these LASC-structures up to elementary equivalence, and get in particular that the complete theory of L(K^m) only depends on m, not on K nor even on p. As an application we obtain a classification of semi-algebraic sets over countable p-adically closed fields up to so-called "pre-algebraic" homeomorphisms.

中文翻译：

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p-adic 半代数集的标度格和 co-Heyting 代数的模型完成

设 p 是素数，K 是一个 p-adically 闭域，X $\subseteq$ K^ma 定义在 K 上的半代数集和 L(X) X 的半代数子集在 X 中封闭的格。我们证明了 L(X) 的完整理论消除了某种语言 LASC 中的量词，L(X) 上的 LASC 结构是晶格结构定义的扩展。此外，它是可判定的，这与在真实封闭场上发生的情况相反。我们将这些 LASC 结构分类到基本等价，并特别得到 L(K^m) 的完整理论仅取决于 m，而不取决于 K，甚至不取决于 p。作为一个应用，我们获得了可数 p-adically 闭域上的半代数集的分类，直到所谓的“前代数”同胚。