Michael wrote:

> How would you translate the reflexive property \$$x \leq x\$$ in a resource theoretic way?

To answer this question, you have to 1) decide on your interpretation of \$$x \le y\$$ and then 2) see what this says in the special case where \$$x\$$ is \$$y\$$. (In other words, don't tackle \$$x \le x\$$ head on; think of it as a special case of something more fundamental.)

1) In resource theories \$$x \le y\$$ often means "if you have \$$y\$$ you can use it to make \$$x\$$". For example, if you have $50 you can use it to make$10 (just give away $40), so$40 \$$\le\$$ \$50.

2) Given this interpretation, \$$x \le x\$$ means "if you have \$$x\$$ you can use it to make \$$x\$$". And this is always true: you just do nothing.

> [...] when people assume 100% yield like when you ignore by products, this would no longer be a preorder since reflexivity is disobeyed?

That's not right if we use my interpretation of \$$x \le y\$$. I have a feeling that when you are writing \$$x \le y\$$ you are thinking \$$x \lt y\$$.