 Addition on the Stone-Čech compactification of the natural numbers

For ultrafilters $\mathcal{U},\mathcal{V}\in \beta \mathbb{N}$, define $\mathcal{U}+\mathcal{V}$ byThis operation is associative, and each right translation $\mathcal{U}\mapsto \mathcal{U}+\mathcal{V}$ is continuous, so $\beta \mathbb{N}$ is a compact Hausdorff left-topological semigroup.