The Stone–Čech compactification $\beta \mathbb N$ of the positive integers has extensive applications in combinatorial number theory.

A feature of $\beta \mathbb N$ that makes these applications possible is that it inherits some (but not all) of the algebraic structure from $\mathbb N$. The defining property of the Stone–Čech compactification of a space $X$ is that any continuous function from $X$ to a compact space $Z$ can be uniquely extended to a continuous function $\beta f \colon \beta X \to Z$. Hence, we can equip $\beta \mathbb N$ with addition operation given by $$p + q = \lim_{n \to p} \lim_{m \to q} (n+m),$$ where the limits are taken in $\beta \mathbb{N}$ and make sense thanks to the extension property. Accordingly, $\beta \mathbb N$ has the multiplication operation, given by $$p \cdot q = \lim_{n \to p} \lim_{m \to q} (n \cdot m).$$

Unlike the addition and multiplication in $\mathbb N$, the corresponding operations in $\beta \mathbb N$ are neither continuous or commutative, nor do we have the familiar distributivity properties. We do have continuity in one argument, as well as all the usual identities if one of the terms is in $\mathbb N$, such as $n+p = p+n$ ($p \in \beta \mathbb N$, $n \in \mathbb N$). The payoff is that $\beta \mathbb N$ is compact, which in turn leads to existence of (non-zero) additive and multiplicative idempotents, minimal ideals, and so on.

The lack of commutativity means in particular the order in which the limits above are taken matters: $$\lim_{n \to p} \lim_{m \to q} (n+m) \neq \lim_{m \to q} \lim_{n \to p} (n+m).$$ However, this distinction is not significant: changing between the two amounts to reordering the summands (this is very similar to how taking the opposite group changes left actions to right actions).

Somewhat to my surprise, I recently run into an application for the exponential map $$\beta \mathbb N \ni p \mapsto 2^p := \lim_{n \to p} 2^n \in \beta \mathbb N.$$ This lead me to wonder (purely out of curiosity, I have no application in mind):

*What is the "right" notion of the exponential map $\beta \mathbb N \times \beta \mathbb N \to \beta \mathbb N$, if any?*

Two obvious possibilities come to mind: $$ p^q = \lim_{n \to p} \lim_{m \to q} n^m \qquad \text{ or } \qquad p^q = \lim_{m \to q} \lim_{n \to p} n^m. $$ It seems that in this case, changing the order of the limits leads to an essentially different operation. I'm curious if any of these operations have ever been seriously considered and found applications to any "natural" problems.