Systems of equations of the form φ_j(X₁,...,X _n)=ψ_j(X₁,...,X_n) with 1 ≤ j ≤ m are considered, in which the unknowns X_i are sets of natural numbers, while the expressions φ_j,ψ_j may contain singleton constants and the operations of union (possibly replaced by intersection) 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.