guest post by Tobias Fritz
Hi! I am Tobias Fritz, a mathematician at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. I like to work on all sorts of mathematical structures which pop up in probability theory, information theory, and other sorts of applied math. Today I would like to tell you about my latest paper:
It should be of interest to Azimuth readers as it forms part of what John likes to call ‘green mathematics’. So let’s get started!
Resources and their management are an essential part of our everyday life. We deal with the management of time or money pretty much every day. We also consume natural resources in order to afford food and amenities for (some of) the 7 billion people on our planet. Many of the objects that we deal with in science and engineering can be considered as resources. For example, a communication channel is a resource for sending information from one party to another. But for now, let’s stick with a toy example: timber and nails constitute a resource for making a table. In mathematical notation, this looks like so:
We interpret this inequality as saying that “given timber and nails, we can make a table”. I like to write it as an inequality like this, which I think of as stating that having timber and nails is at least as good as having a table, because the timber and nails can always be turned into a table whenever one needs a table.
To be more precise, we should also take into account that making the table requires some tools. These tools do not get consumed in the process, so we also get them back out:
Notice that this kind of equation is analogous to a chemical reaction equation like this:
So given a hydrogen molecules and an oxygen molecule, we can let them react such as to form a molecule of water. In chemistry, this kind of equation would usually be written with an arrow ‘’ instead of an ordering symbol ‘’ , but here we interpret the equation slightly differently. As with the timber and the nails and nails above, the inequality says that if we have two hydrogen atoms and an oxygen atom, then we can let them react to a molecule of water, but we don’t have to. In this sense, having two hydrogen atoms and an oxygen atom is at least as good as having a molecule of water.
So what’s going on here, mathematically? In all of the above equations, we have a bunch of stuff on each side and an inequality ‘’ in between. The stuff on each side consists of a bunch of objects tacked together via ‘’ . With respect to these two pieces of structure, the collection of all our resource objects forms an ordered commutative monoid:
Definition: An ordered commutative monoid is a set equipped with a binary relation a binary operation and a distinguished element such that the following hold:
• and equip with the structure of a commutative monoid;
• equips with the structure of a partially ordered set;
• addition is monotone: if then also
Here, the third axiom is the most important, since it tells us how the additive structure interacts with the ordering structure.
Ordered commutative monoids are the mathematical formalization of resource convertibility and combinability as follows. The elements are the resource objects, corresponding to the ‘collections of stuff’ in our earlier examples, such as or Then the addition operation simply joins up collections of stuff into bigger collections of stuff. The ordering relation is what formalizes resource convertibility, as in the examples above. The third axiom states that if we can convert into then we can also convert together with into together with for any for example by doing nothing to
A mathematically minded reader might object that requiring to form a partially ordered set under is too strong a requirement, since it requires two resource objects to be equal as soon as they are mutually interconvertible: and implies However, I think that this is not an essential restriction, because we can regard this implication as the definition of equality: ‘’ is just a shorthand notation for ‘ and ’ which formalizes the perfect interconvertibility of resource objects.
We could now go back to the original examples and try to model carpentry and chemistry in terms of ordered commutative monoids. But as a mathematician, I needed to start out with something mathematically precise and rigorous as a testing ground for the formalism. This helps ensure that the mathematics is sensible and useful before diving into real-world applications. So, the main example in my paper is the ordered commutative monoid of graphs, which has a resource-theoretic interpretation in terms of zero-error information theory. As graph theory is a difficult and traditional subject, this application constitutes the perfect training camp for the mathematics of ordered commutative monoids. I will get to this in Part 3.
In Part 2, I will say something about what one can do with ordered commutative monoids. In the meantime, I’d be curious to know what you think about what I’ve said so far!