Statement:etale-monoid-generation

Is there a finite generating set?

The page argues that no finite set of maps, with the automorphisms, generates the étale monoid of ℂ³. Degrees multiply under composition, so each prime degree $p\ge3$ can only come from a generator of degree $p$. Gallagher's family realizes every such prime, so infinitely many generators are needed. Two things are worth checking before this is trusted. First, the multiplicativity of generic fiber degree for compositions of étale maps (field degrees in a tower). Second, that Gallagher's members of prime degree are genuinely non-automorphic, which is his uniform proof (unrefereed); the members up to 100 are certified on jacobianfun.org. If both hold, the page's revised question has a one-line negative answer too, and the right question is about generators up to a weaker equivalence. — atlas import, 2026-09-30 05:09 UTC