 Idempotent ultrafilter on the natural numbers

An ultrafilter $\mathcal{U}\in \beta \mathbb{N}$ is idempotent when $\mathcal{U}+\mathcal{U}=\mathcal{U}$. The Ellis–Numakura lemma guarantees such an ultrafilter and supplies the main algebraic input to the Idempotent-ultrafilter proof of Hindman's theorem.