@Jonathan #1, Puzzle **JMC7**: Great question! For general resource theories, I haven't seen a precise definition of when a monoidal monotone "merely forgets things". But in general, there are three main types of monoidal monotones:

* Those which add capabilities: those are the monoidal monotones that are given by the identity map \\(x\mapsto x\\) between monoidal preorders with the same underlying set and monoid structure, \\((X,\leq,+) \to (X,\leq',+)\\). In other words, we only add further comparison relations while not doing anything else. For example, the discovery of a new technology is often like this: the set of resources and how they combine does not change, but suddenly you can do stuff that you could not do before!

* Those which only change the underlying set: those are the monoidal monotones of the form \\(f:(X,\leq,+)\to (Y,\leq,+)\\), where \\(x\leq x'\\) holds if and only if \\(f(x)\leq f(x')\\), and for every \\(y\in Y\\) there is \\(x\in X\\) with \\(y\leq f(x)\leq y\\). For example, introducing a new currency may be an example of this: when you include a new currency in your resource theory, then you've expanded the theory, so that the original one is included in the new one. But it's still pretty much the same theory, since the new currency is perfectly interconvertible with your original one. (I'm ignoring conversion fees for simplicity.) So monoidal monotones of this type don't really do anything.

* Those which discover new kinds of resources: these are inclusion maps of some symmetric monoidal preordered *subset* of some other symmetric monoidal preorder \\(X\\), where both the monoid structure and the order are simply given by taking the given structures on \\(X\\) and restricting them. For example, when your company hires a new employee, then the time of that employee becomes a new type of resource. But the things that you can do without using the time of that employee are still exactly the same.

The neat thing is that you can write every monoidal monotone as a combination of one of these: first, add capabilities; then, change the underlying set; then, discover new resources!

I'll leave it open to find more examples of monoidal monotones that do not "merely forget things", but this comment may have given some hints.

* Those which add capabilities: those are the monoidal monotones that are given by the identity map \\(x\mapsto x\\) between monoidal preorders with the same underlying set and monoid structure, \\((X,\leq,+) \to (X,\leq',+)\\). In other words, we only add further comparison relations while not doing anything else. For example, the discovery of a new technology is often like this: the set of resources and how they combine does not change, but suddenly you can do stuff that you could not do before!

* Those which only change the underlying set: those are the monoidal monotones of the form \\(f:(X,\leq,+)\to (Y,\leq,+)\\), where \\(x\leq x'\\) holds if and only if \\(f(x)\leq f(x')\\), and for every \\(y\in Y\\) there is \\(x\in X\\) with \\(y\leq f(x)\leq y\\). For example, introducing a new currency may be an example of this: when you include a new currency in your resource theory, then you've expanded the theory, so that the original one is included in the new one. But it's still pretty much the same theory, since the new currency is perfectly interconvertible with your original one. (I'm ignoring conversion fees for simplicity.) So monoidal monotones of this type don't really do anything.

* Those which discover new kinds of resources: these are inclusion maps of some symmetric monoidal preordered *subset* of some other symmetric monoidal preorder \\(X\\), where both the monoid structure and the order are simply given by taking the given structures on \\(X\\) and restricting them. For example, when your company hires a new employee, then the time of that employee becomes a new type of resource. But the things that you can do without using the time of that employee are still exactly the same.

The neat thing is that you can write every monoidal monotone as a combination of one of these: first, add capabilities; then, change the underlying set; then, discover new resources!

I'll leave it open to find more examples of monoidal monotones that do not "merely forget things", but this comment may have given some hints.