I was hoping this algebra could have physical significance. For example, you could make a line segment by combining two nodes. You can then combine three such line segments into a triangle. And then you can combine four triangles into a tetrahedron. And you can use techniques from differential geometry. E.g., you could use a "∂" operator to get the boundary of a given structure. And you can transform the graph into an adjacency matrix, and solve differential equations.
I'm the author of the blog post -- thank you for your comment! I was wondering whether anyone would be interested in a different version of the algebra that modells hypergraphs with k-edges. For example, you can change the decomposition axiom to: a.b.c.d = a.b.c + a.b.d + a.c.d + b.c.d, which allows you to have an algebraic structure with nodes, 2-edges (pairs of nodes), and 3-edges (triples of nodes). I can write about this in the following blog post if you are interested. (And I'd love to hear further thoughts from you on the connection to the boundary operator -- do get in touch by email!)
EDIT: Used dots (.) for the connect operator not to mess up with formatting.
These are all well-known things in simplicial geometry [1] which is used everywhere in computational topology and discrete differential geometry. It turns out all of these operations can be done efficiently by considering a matrix over F_2 and considering its several forms. These manipulations lead to a natural (computable!) discretization of differential/topological methods because they're fully constructive and have beautiful theorems. I'd highly recommend looking into it if you're interested.
There may even be a nice way of taking elements from the algebra and extending these definitions to fit this idea of '∂' or closure or interior, etc; but I have little intuition for what the algebra might look like and whether it would admit nice expressions of this (personally, I was more motivated by the possible connection to computability theory and languages). Once I have some time, I may sit down and toy around with the idea.
I would love to read even a short post on such applications. One thing I realized while learning abstract math is that abstraction is pretty useless unless you have things to abstract. You don't need to apply any abstractions to what you're doing. Leave that to other people.
