Abstract

Linear arithmetics are extensions of Presburger arithmetic () 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 of the 2‐linear arithmetic (linear arithmetic with two scalars) in which an infinitely long initial segment of “Peano multiplication” on is ‐definable. This shows, in particular, that is not model complete in contrast to theories and that are known to satisfy quantifier elimination up to disjunctions of primitive positive formulas. As an application, we show that, as a discretely ordered module over the discretely ordered ring generated by the two scalars, does not have the NIP, answering negatively a question of Chernikov and Hils.