I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need:
- Scalar multiplication over the reals (so you can get "2 kg" from "2 * kg")
- Addition within the same dimension (so you can have "kg + kg = 2 kg")
- Multiplication of any two elements (so you can have "J = N m = N * m")
- Inverses (so you can have "m/s = m * s^(-1)")
A tensor algebra could formalise this system -- but then you'd get all sorts of objects like "1 kg + 1 m", which make no sense.
A group would make sense -- with sub-groups like "mass measurements", "time measurements", "real numbers", "units" -- but then you can't have zero. Plus, I'd like to have some notion of units or "unit vectors"/"unit tensors".
What's a good way to formalise this?