We consider topological spaces X equipped with an algebra A of subsets of X and an ideal I of A. Motivated by the example of the Jordan measurable subsets of R, we consider the "derived structure" obtained by replacing A by the algebra of elements of A whose boundaries belong to I, and I by its intersection with that algebra.

In previous joint work with N. D. Macheras and W. Strauss, these derived structures were classified (under some assumptions) and densities were computed for them. In more recent work with the same co-authors, we extend that work in the context of products of derived structures. We examine when product operations preserve densities and other types of liftings. We will discuss some topological and set-theoretic issues that arise in the context of this work.