Definability and decidability issues in extensions of the integers with the divisibility predicate

Let M be a first-order structure; we denote by DEF(M) the set of all first-order definable relations and functions within Rm: Let pi be any one-to-one function from N into the set of prime integers. Let \ and . be respectively the divisibility relation and multiplication as function. We show that the sets DEF(N, pi, \) and DEF(N, pi, .) are equal. However there exists function pi such that the set DEF(N, pi, \), or, equivalently, DEF(N, pi, .) is not equal to DEF(N, +, .). Nevertheless, in all cases there is an {pi, .}- definable and hence also {pi, \}-definable structure over N which is isomorphic to <N, +, .>. Hence theories TH(N, pi, \) and TH(N, pi, .) are undecidable. The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice. The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables.