Curious Mathematical Object: Hyperlogarithms

Logarithms turn a product of numbers into a sum of numbers: log(xy) log(x) log(y). Hyperlogarithms generalize the concept as follows: Hlog(XY) Hlog(X) Hlog(y), where X and Y are any kind of objects, and the product and sum are replaced by operators in some arbitrary space. Here we focus exclusively on operations on sets: XY becomes the intersection of the sets X and Y, and X Y the union of X and Y. The question is: which functions satisfy Hlog(XY) Hlog(X) Hlog(y). We assume here that the argument for Hlog is a set X, and the returned value Hlog(X) Y is another set Y from the same set of sets. Let E {X, Y, ... } be the sets of all potential arguments for Hlog.