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\\).

> 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\\).