Presburger arithmetic and Visser's conjecture
Presburger Arithmetic the true theory of natural numbers with addition. We show that the interpretations of Presburger Arithmetic in itself are definably isomorphic to the trivial one, confirming the conjecture of A. Visser. To prove that, we develop a characterization of linear orderings interpretable in (N, +). We show that all interpretable linear orderings can be expressed as a restriction of the lexicographical ordering on Z_k for some k to some Presburger-definable set. This generalizes the results of Zapryagaev, A., Pakhomov F.: Interpretations of Presburger arithmetic in itself. International Symposium on Logical Foundations of Computer Science. Springer, Cham (2018) where the one-dimensional result was proven.