Computational completeness of equations over sets of natural numbers

Systems of finitely many equations of the form φ(X1,…,Xn)=ψ(X1,…,Xn)φ(X1,…,Xn)=ψ(X1,…,Xn) are considered, in which the unknowns XiXi are sets of natural numbers, while the expressions φ,ψφ,ψ may contain singleton constants and the operations of union and pairwise addition S+T={m+n|m∈S,n∈T}. It is shown that the family of sets representable by unique (least, greatest) solutions of such systems is exactly the family of recursive (r.e., co-r.e., respectively) sets of numbers. Basic decision problems for these systems are located in the arithmetical hierarchy. The same results are established for equations with addition and intersection.