Model-completion of scaled lattices

by L. Darnière.

History

This preprint was first submitted to APAL on November 04, 2004, revised and re-submitted on April 05, 2005, and finally rejected on March 24, 2006. In spite of this it was the foundation of all my later papers, both on co-Heyting algebras and on p-adically closed fields, and thus a decisive work for me.

In March 2018, a major revision appeared with better proofs and more results:
Model-completion of scaled lattices and co-Heyting algebras of p-adic semi-algebraic sets.

Abstract

This preprint introduces and studies "scaled lattices", which are expansions by definition of distributive lattices intended to mimic the behavior of the lattice L(X) of Zariski closed subsets of X, an affine variety, or the lattice of closed definable subsets of a semi-algebraic set over a real closed or a p-adically closed field K. Their lattice structure is expanded by operations A-B and Cⁱ(A) which in the geometric cases are respectively the closure of A\B and the i-pure component of A. We give a recursive axiomatisation of the universal theories of these classes of geometric lattices, and prove that it admits a model-completion. We conjecture that when K is p-adically closed, L(X) is a model of (a variant) of this model-completion, or equivalently that L(X) satisfies a certain "Splitting Property" which had never been supected until now.

Mathematics Subject Classification (2010)

03C10 Quantifier elimination, model completeness and related topics
06D20 Heyting algebras [See also 03G25]
12L99 Field theory and polynomials

Electronic version of the paper

Version June 2006 (27 pages) dvi, pdf

Version "poster" July 2005 (5 pages): dvi, pdf.
Version April 2005 (26 pages): dvi, Postscript.

Version July 2004 (44 pages): dvi, Postscript.