> **Puzzle 194.** From [Lecture 11](https://forum.azimuthproject.org/discussion/1991/lecture-11-chapter-1-the-poset-of-partitions/p1) we know that for any set \$$X\$$ the set of partitions of \$$X\$$, \$$\mathcal{E}(X)\$$, becomes a poset with \$$P \le Q\$$ meaning that \$$P\$$ is finer than \$$Q\$$. It's a monoidal poset with product given by the meet \$$P \wedge Q\$$. Is this monoidal poset closed? How about if we use the join \$$P \vee Q\$$?

If \$$P\multimap Q\$$ needs to be the coarsest partition that distinguishes everything distinguished in Q that is not distinguished in P, I think that there is no such thing in general, which means this monoidal poset is not closed. For instance, if \$$P=((1,2),(3,4))\$$ and \$$Q=((1),(2),(3),(4))\$$, then \$$P\wedge((1,4),(2,3))=P\wedge((1,3),(2,4)) = Q\$$, but \$$((1,4),(2,3))\vee((1,3),(2,4))=I\$$.