I know the Cartesian product for an algebraic structure: A x B = {(a,b): a ∈ A, b ∈ B}
Which naturally gives A^{n} = {(a_{1}, a_{2}, ... , a_{n}): a_{i} ∈ A ∀ i}
Some of the time, at least we can also have a non integer n.
For example [A x A x A]^{2/3} = A x A.
Is there any way of continuing the exponent n into non integer numbers for any exponent? It might involve creating an algebraic structure very unlike the original A, but which is still isomorphic when raised to an appropriate power, with respect to the Cartesian product.
I have another question as well:
Is there a way to define a group, which has as elements all groups, or all algebraic structures of a certain kind? Could this be made into a ring? If it could, perhaps we could create a structure exponential using the two operations. A metric might also be required on the set of all structures to make sure the series converges.
Anyway, I would be grateful if anyone could tell me if any of this exists already, is complete nonsense, or which area of mathematics it would belong to.
