2024/12/18 by David J. Fernández-Bretón, Fernández-Bretón, David, Jareb Navarro‐Castillo +3 · 1 citation
Computer Science · Mathematics · #20M10 #22A15 #54D35 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Optimization and Variational Analysis #Primary 03E25 #Secondary 03E30
paper · pdf · doi:10.48550/arxiv.2412.13499
openalex publication_date 2024/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 \mathscr Fp (resp. \mathscr Gp) is the smallest family containing p and closed under iterated sums (resp. closed under Blass--Frol'ık sums and Rudin--Keisler images), then \mathscr Fp (resp. \mathscr Gp) 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(\mathbb R)[p] where L(\mathbb R) is a Solovay model (of ZF without choice) and p is a selective ultrafilter, there are no idempotent elements. In particular, the theory ZF plus the existence of a nonprincipal ultrafilter on ω does not imply the existence of idempotent ultrafilters, which answers a question of DiNasso and Tachtsis (Proc. Amer. Math. Soc. 146, 397-411). Following the line of obtaining independence results in ZF, we finish the paper by proving that ZF plus "every additive filter can be extended to an idempotent ultrafilter" does not imply the Ultrafilter Theorem over \mathbb R, answering another question of DiNasso and Tachtsis from the same paper.