Least and greatest solutions of equations over sets of integers

Systems of equations with sets of integers as unknowns are considered, with the operations of union, intersection and addition of sets, S+T={m+n|m∈S,n∈T}. These equations were recently studied by the authors (“Representing hyper-arithmetical sets by equations over sets of integers”, Theory of Computing Systems  , 51 (2012), 196–228), and it was shown that the class of sets representable by their unique solutions is exactly the class of hyper-arithmetical sets. In this paper it is demonstrated that greatest solutions of such equations represent exactly the Σ11-sets in the analytical hierarchy, and all those sets can already be represented by systems in the resolved form  Xi=φi(X1,…,Xn)Xi=φi(X1,…,Xn). Least solutions of such resolved systems represent exactly the recursively enumerable sets.