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Model completion of scaled lattices and co‐Heyting algebras of p ‐adic semi‐algebraic sets
Author(s) -
Darnière Luck
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201800021
Subject(s) - mathematics , algebraic extension , countable set , decidability , algebraic number , equivalence (formal languages) , lattice (music) , pure mathematics , closed set , discrete mathematics , mathematical analysis , differential algebraic equation , ordinary differential equation , physics , acoustics , differential equation
Abstract Let p be prime number, K be a p ‐adically closed field, X ⊆ K ma 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 quantifiers in a certain language L ASC , the L ASC ‐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 for m > 1 . We classify these L ASC ‐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.