Right, now that I think about it, it's only coincidental that the objects in question represent the subsets in a partition of the reals. It seems what we have

Right, now that I think about it, it's only coincidental that the objects in question represent the subsets in a partition of the reals. It seems what we have here is a finite set A (in arithmetic, A would be a finite collection of sets which constitute a partition of the reals) and two operations: f, g: P(A) × P(A) -> P(A), taking a pair of subsets of A to another subset of A, with the property that f(W ∪ X, Y) = f(W, Y) ∪ f(X, Y), and so on. This can all be represented as a bitmask table of size |A|^3, or half if commutativity is enforced. We also have the "inverse" which takes X and Z, and returns the set {a ∈ A : f(X, {a}) ∩ Z ≠ Ø}. If I'm not mistaken, the mechanisms for doing all of this in hardware would come for free, and it represents quite a variety of problem domains beyond arithmetic. Perhaps some arithmetic-specific optimization would be ruled out (e.g. the bit twiddling to get additive or multiplicative inverses).