It is well known that the sum $\alpha+\beta$ of two ordinals $\alpha,\beta$ can be defined "geometrically" as the order type of the sum $(\{0\}\times \alpha)\cup(\{1\}\times\beta)$ endowed with the lexicographic order.

Also the product $\alpha\cdot\beta$ of ordinals $\alpha,\beta$ is the order type of the Cartesian product $\beta\times\alpha$ endowed with the lexicographic order.

What about the exponentiation of ordinals?

Does $\alpha^\beta$ have some nice "geometric'' or combinatorial model?

Maybe as some set of (partial) functions endowed with a suitable well-order?

ANDHow do you define ordinal exponentiation without induction? (the comments might be of interest). $\endgroup$