# lo.logic – What are the definable sets in Skolem's arithmetic?

Definable subsets of $$mathbb N$$ in the language of the Presburger arithmetic, one finds exactly the possibly periodic sets and the free part of the quantizer corresponds to the integer programming with inequalities and linear variations leads to a linear programming in mixed integers, a programming in convex integers with convex constraints. What about

1. definable subsets of $$mathbb N$$ in the language of Skolem arithmetic and

2. would it be wise to look for the programming constructs to which it leads (if I'm not mistaken, the atomic formulas here could be of $$a prod_ {i = 1} ^ nx_i ^ {b_i} leq b$$ or $$a prod_ {i = 1} ^ nx_i ^ {b_i} = b$$)?

I'm not sure I'm making sense here, but if there's a reasonable way to save the job, it'll be fine.