Let’s start by laying out some common components of all these conversions. All of them involve transforming a set of resources right into a set of products. Every resource has some products that it can make and others that it cannot make. These transformations must be composable, that is that if we are able to turn A into B and B into C, then we should always give you the option to show A into C through multiple steps. All of those ideas will be modelled by a symmetric monoidal category. That’s a sophisticated expression, let’s see what that’s and provides an example.

We define a symmetric monoidal category as (S, >, I, *). That is numerous structures, I’ll undergo each in turn.

S is solely all of the set of all objects that we’re enthusiastic about. If we’re applying this structure to a chemical problem, it might be all of the chemicals now we have access to, in addition to those we want to create. That is similar to the usual mathematical set.

> defines an order on S. I could simply list the properties of the order, but I feel it’s more intuitive to provide an example.

How can we interpret this diagram by way of >? By taking a look at Figure 1, we see that there’s an arrow from A to B, so A > B. We also can compose arrows, so A > C and A > D. I didn’t include them in Figure 1, but every point also has an arrow going to itself, so A > A, B > B, etc. Additionally it is possible for A > B and B > A if I had drawn in an arrow going from B to A.

What’s the interpretation of this? It’s pretty easy, if A > B, then we are able to turn A into B by a process. Notice that C and D can’t be become anything (besides themselves), they’re stuck of their current state. Since A > A, we are able to turn A into A by a trivial process. For the reason that arrows will be combined, we all know that A > B and B > C means A > C. This is smart after we take into consideration composition. To summarize, the objects contained in > tell us what objects in S will be become other objects in S.

Now let’s turn to I and *. These parts tell us concerning the actual act of performing a process to convert elements into one other. * is a binary operation that acts as A * B = C. This operation represents actually turning A and B into C. I is just is the “neutral resource” where A * I = I * A = A for all elements. Again, there are a bunch of properties needed for this operation, but there’s one which is significantly more essential and serves to attach * and >.

This property is named monotonicity, and is defined as A1 > B1, A2 > B2, implies that A1 * A2 > B1 * B2. We will consider this property as “if we are able to turn A1 into B1 and A2 into B2, then we are able to turn the mix of A1 and A2 into the mix of B1 into B2.” Pondering this fashion is intuitive for Resource Theory, but must be formalized in the maths.