Products of derived structures on topological spaces

We consider topological spaces X   equipped with an algebra AA of subsets of X   and an ideal II of AA. Motivated by the example of the Jordan measurable subsets of RR, we consider the derived structure   obtained by replacing AA by the algebra ∂A={E∈A:∂E∈I}∂A={E∈A:∂E∈I} of sets with negligible boundaries, and II by ∂I=I∩∂A∂I=I∩∂A. In a previous paper by M.R. Burke et al. (2012) [7], the authors classified these derived structures (under some assumptions) and computed densities for them. In the present paper, we extend that work in the context of products of derived structures. We study in greater detail the box cross product γ⊠δγ⊠δ of two set maps γ∈P(X)Aγ∈P(X)A, δ∈P(Y)Bδ∈P(Y)B introduced in joint work of the authors with K. Musiał (2009) [6], examining when it preserves densities and other types of liftings. For preservation of monotonicity, we introduce a variation on the localization property of ideals which is well-known for the meager ideal. An examination of skew products provides a class of structures to which our results apply.