Tobias wrote:

> Background: in a resource theory, one may want to call a resource \$$x\$$ *quality-like* if it satisfies \$$x \leq x\otimes x\$$ and \$$x\otimes x \leq x\$$. This means that no matter how many copies of \$$x\$$ you have, you can turn them into any other number of copies, so quantity doesn't matter at all. For example, knowledge is often of this type: the cost of reproducing knowledge is often negligible. So the puzzle really asks: if all resources are quality-like, what does this imply about the mathematical structure of our resource theory?

This is very nice. I've begun discussing this at the end of [Lecture 23](https://forum.azimuthproject.org/discussion/2086/lecture-23-chapter-2-commutative-monoidal-posets/p1). However, I start from the other end: I assume we have poset with finite meets, and pose as a puzzle to show that every resource (=element in the poset) is what you're calling "quality-like".