In the same [book](http://www.cambridge.org/9780521883894) as in #17, they give an interpretation of the entropy of the meet of partitions (in a probability space), and give also an independence notion, all resembling facts about joint distributions. They prove \$$H(\sigma \wedge \tau) = H(\sigma) + H(\tau)\$$ for the Shannon entropy of the partition as above (for \$$\sigma\$$ and \$$\tau\$$ independent partitions). But while the meet of partitions is understood, then they ask, *1.4.6. (Research problem) There is no information or entropy interpretation of the join of two partitions. Find an interesting one*.