Equations over Sets of Natural Numbers with Addition Only

Systems of equations of the form X = Y Z and X = C are considered, in which the unknowns are sets of natural numbers, “+” denotes pairwise sum of sets S+T = {m + n | m ∈ S, n ∈ T}, and C is an ultimately periodic constant. It is shown that such systems are computationally universal, in the sense that for every recursive (r.e., co-r.e.) set S ⊆ N there exists a system with a unique (least, greatest) solution containing a component T with S = {n | 16n+13 ∈ T}. This implies undecidability of basic properties of these equations. All results also apply to language equations over a one-letter alphabet with concatenation and regular constants.