A wild model of linear arithmetic and discretely ordered modules

TL;DR

This paper constructs a model of 2-linear arithmetic demonstrating that it is not model complete and shows that the associated discretely ordered module is not NIP, contrasting with simpler linear arithmetics.

Contribution

It introduces a model of 2-linear arithmetic with definable multiplication, revealing new properties about model completeness and NIP status.

Findings

LA_2 is not model complete.

The model's discretely ordered module is not NIP.

LA_1 and LA_0 are known to have quantifier elimination.

Abstract

Linear arithmetics are extensions of Presburger arithmetic (Pr) by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages. In this paper we construct a model M of the 2-linear arithmetic LA_2 (linear arithmetic with two scalars) in which an infinitely long initial segment of "Peano multiplication" on M is 0-definable. This shows, in particular, that LA_2 is not model complete in contrast to theories LA_1 and LA_0=Pr that are known to satisfy quantifier elimination up to disjunctions of primitive positive formulas. As an application, we show that M, as a discretely ordered module over the discretely ordered ring generated by the two scalars, is not NIP, answering negatively a question of Chernikov and Hils.