Many classes of structures have natural functions and relations on them:
concatenation of linear orders, direct product of groups, disjoint union of
equivalence structures, and so on. Here, we study the (un)decidability of the
theory of several natural classes of structures with appropriate functions and
relations. For some of these classes of structures, the resulting theory is
decidable; for some of these classes of structures, the resulting theory is
bi-interpretable with second-order arithmetic.