= Idempotent ultrafilter
{title2=$\mathcal U+\mathcal U=\mathcal U$}
An additive idempotent <ultrafilter> satisfies $\mathcal U+\mathcal U=\mathcal U$. On positive integers it is necessarily nonprincipal, since a principal ultrafilter at $n$ adds to itself to give the principal ultrafilter at $2n$. The <idempotent-ultrafilter star-set lemma> converts this algebraic property into a recursive construction of finite sums. If zero is included, the trivial principal idempotent at zero must be excluded for that increasing-sequence application.
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