I recently learned about the concept limits in categories. From R we can construct C the H etc... by iterating the Cayley-Dickson construction.
My question is: Can we construct a (non-associative)ring that is the union(?) of all the hypercomplex unital algebras regarded as rings? (If not, why?)
What are the interesting properties that it may have?
Below, I use "rings" loosely for "nonassociative rings."
Each member of the Cayley-Dickson tower embeds into the member above it as a subring (sharing identity) via the ring homomorphism $x\mapsto (x,0)$. As far as I know, there is nothing wrong with considering the directed union $\bigcup_{i\in \mathbb N}C_i$ where $C_i$ is the $i$'th Cayley algebra. (Someone earlier asked a related question here)
It may be hard to find many properties shared by the whole thing, but certain ones will work out. For example, all the rings satisfy the flexible identity and so will the union. Based on things I'm reading it looks like all nonzero elements have inverses, too.
This should work for any property which is defined by specifying a condition on finitely many elements, because you will always be able to find one of the algebras that contains them all (and hence says the condition is true for that set of elements.)