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]( 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".