Model completion of scaled lattices and co-Heyting algebras of p-adic semi-algebraic sets - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2018

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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Dates and versions

hal-01756160 , version 1 (31-03-2018)
hal-01756160 , version 2 (11-09-2018)
hal-01756160 , version 3 (28-10-2018)

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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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