Decidability of extended theories of addition of the natural numbers and the integers (Q1071748)

The aim of this note is to prove the decidability of the elementary theories of the models \({\mathfrak N}=<{\mathbb{N}};+,Pw_ p(x),|_{\Pr}>\) and \({\mathfrak M}=<{\mathbb{Z}};+,<,Pw_ p(x),|_{\Pr}>\), where \({\mathbb{N}}\) is the set of natural numbers, \({\mathbb{Z}}\) is the set of integers, \(+\) is the operation of addition, \(<\) is the natural order on \({\mathbb{Z}}\), \(Pw_ p(x)\) is the predicate that distinguishes the powers of a fixed prime p, and \(x|_{\Pr}y\) means that x is a power of p and x divides the natural number y. From the decidability of the theory of the model \({\mathfrak M}\), using the ideas of \textit{Yu. G. Penzin} [Algebra (Irkutsk) 1974, No.3, 120-148 (1974), p. 138], we can prove the decidability of the elementary theory of the two-base model \(<{\mathbb{Z}};M;+,<,\in >\), where M is the chain of subgroups \({\mathbb{Z}}\), p\({\mathbb{Z}}\), \(p^ 2{\mathbb{Z}},...\), and \(\in\) is the relation of belonging to a subgroup.