Computational completeness of equations over sets of natural numbers

Systems of finitely many equations of the form φ( X₁,..., X_n)=ψ(X₁,...,X_n) are considered, in which the unknowns ^Xi are sets of natural numbers, while the expressions φ,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.