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

by L. Darnière.

Mathematical Logic Quarterly 65 (2019), no. 3, 305-331

Submitted on March 31, 2018.

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 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.

Mathematics Subject Classification (2010)

03C10 Quantifier elimination, model completeness and related topics
06D20 Heyting algebras [See also 03G25]
06D99 None of the above, but in this section

Electronic version of the paper

Version October 2018 (30 pages) pdf

This paper is a major revision of my old preprint (2004):
Model-completion of scaled lattices.