It is well-known that the operation of addition of two ultrafilters on the set $\mathbb{N}$ of natural numbers which extends the natural addition on $\mathbb{N}$ to $\beta\mathbb{N}$, the Cech-Stone compactification of $\mathbb{N}$, is not continuous (it is only right-continuous).


I am thus looking for examples of continuous binary operations on $\beta\mathbb{N}$ i.e. such functions $\beta\mathbb{N}\times\beta\mathbb{N}\to\beta\mathbb{N}$ which are (jointly) continuous (not only separately continuous). I am only interested in operations that engage both variables (so e.g. projections and their compositions with mappings $\beta\mathbb{N}\to\beta\mathbb{N}$ drop out), but they need not have any additional properties like associativity etc.



Continuous binary operations on $\beta\mathbb{N}$
In this paper, [Dimension phenomena associated with $\beta\mathbb{N}$-spaces](https://doi.org/10.1016/S0166-8641(01)00281-4), Ilijas Farah proved that continuous maps from $\beta\mathbb{N}^2$ (and other powers) to $\beta\mathbb{N}$ are quite simple: there is a finite disjoint cover such that the map depends on one coordinate on each member of the cover. 


