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Model completion of scaled lattices and co-Heyting algebras of p-adic semi-algebraic sets

Abstract : Let p be prime number, K be a p-adically closed field, X ⊆ 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.
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https://hal.archives-ouvertes.fr/hal-01756160
Contributor : Luck Darnière Connect in order to contact the contributor
Submitted on : Sunday, October 28, 2018 - 12:21:01 AM
Last modification on : Wednesday, November 3, 2021 - 9:18:32 AM

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  • HAL Id : hal-01756160, version 3
  • ARXIV : 1804.01421

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Luck Darnière. Model completion of scaled lattices and co-Heyting algebras of p-adic semi-algebraic sets. 2018. ⟨hal-01756160v3⟩

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