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

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.