Univariate Equations Over Sets of Natural Numbers

It is shown that equations of the form p(X) = ψ(X), in which the unknown X is a set of natural numbers and p, ψ use the operations of union, intersection and addition of sets S + T = {m + n |, m ∈ S, n ∈ T}, can simulate systems of equations pj(X1, …, Xn) = pj(X1, …, Xn) with 1 ≤ j ≤ e, in the sense that solutions of any such system are encoded in the solutions of the corresponding equation. This implies computational universality of least and greatest solutions of equations p(X) = ψ(X), as well as undecidability of their basic decision problems. It is sufficient to use only singleton constants in the construction. All results equally apply to language equations over a one-letter alphabet Σ = {a}.