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.



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> What about the exponentiation of ordinals?
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> Does $\alpha^\beta$ have some nice "geometric'' or combinatorial model?
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> Maybe as some set of (partial) functions endowed with a suitable well-order?
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> 



Has the exponentiation of ordinals a nice geometric model?
Ordinal exponentiation is a special case of *linear order* exponentiation. For any linear order $L$, element $a\in L$, and ordinal $\beta$ we can define the **$\beta$th power of $L$ at $a$**, which I'll call "$L\_a^\beta$," as the set of functions $f:\beta\rightarrow L$ such that all but finitely many $\alpha\in\beta$ have $f(\alpha)=a$. The ordering on this set is given by looking at the last point of difference: $$f\trianglelefteq g\iff f=g\mbox{ or } f(\max\{x:f(x)\not=g(x)\})<g(\max\{x:f(x)\not=g(x)\}).$$


For ordinals $\alpha,\beta$ we have $\alpha^\beta=\alpha\_0^\beta$. [Rosenstein's book](https://books.google.com/books?id=y3YpdW-sbFsC&pg=PA131&lpg=PA131&dq=linear+order+exponentiation&source=bl&ots=eSIBSAQXDQ&sig=ACfU3U2HK4B4tjBMVyn_McrGCo5lU-kaRA&hl=en&sa=X&ved=2ahUKEwiO1YTC7oHpAhUIH80KHa8WBtAQ6AEwFnoECAkQAQ#v=onepage&q=linear%20order%20exponentiation&f=false) treats this in some detail (and is generally an awesome book all-around - it's a huge tragedy that it's so hard to find).


What I like about this definition is that it very nicely complements the definition of cardinal exponentiation: they start with the same basic idea of counting functions between sets, but ordinal exponentiation is the "finite support" version.


