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Topic: Curious Mathematical Object: Hyperlogarithms
Replies: 1   Last Post: Aug 17, 2017 4:19 AM

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 vincent64@yahoo.com Posts: 129 Registered: 4/18/07
Curious Mathematical Object: Hyperlogarithms
Posted: Aug 16, 2017 2:47 PM

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.

Can you find all functions Hlog satisfying Hlog(XY) = Hlog(X) + Hlog(y) ?