Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory

We study ultrafilters from the perspective of the algebra in the Čech–Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if $p$ is a Q-point (resp., a selective ultrafilter) and $F^p$ (resp., $G^p$) is the smallest family containing $p$ and closed under iterated sums (resp., closed under Blass–Frolík sums and Rudin–Keisler images), then $F^p$ (resp., $G^p$) contains no idempotent elements. The second of these results about a selective ultrafilter has the following interesting consequence: assuming a conjecture of Blass, in models of the form $L\left(R\right)\left[p\right]$ where $L\left(R\right)$ is a Solovay model (of $\mathrm{ZF}$ without choice) and $p$ is a selective ultrafilter, there are no idempotent elements. In particular, the theory $\mathrm{ZF}$ plus the existence of a nonprincipal ultrafilter on $\omega$ does not imply the existence of idempotent ultrafilters, which answers a question of Di Nasso and Tachtsis (Proc. Amer. Math. Soc. 146, 397–411) that was also asked by Tachtsis (J. Symb. Log. 83, 557–571). Following the line of obtaining independence results in $\mathrm{ZF}$, we finish the paper by proving that $\mathrm{ZF}$ plus ‘every additive filter can be extended to an idempotent ultrafilter’ does not imply the Ultrafilter Theorem over $R$, answering another question of Di Nasso and Tachtsis from the aforementioned paper.