On the amenability of semigroups of entire maps and formal power series.

The paper’s Theorem 5 proves exactly the claim at issue and supplies a two-case proof (non-exceptional vs. exceptional rational elements). The candidate solution mirrors the same structure and key ingredients: using the equivalence for rational semigroups with a non-exceptional element (right amenable ⇔ no non-cyclic free subsemigroup), then applying Ore localization to pass from a cancellative, right-reversible semigroup to an amenable group of fractions, and handling the exceptional-only case by showing abelianness. Minor gaps in the candidate’s write-up (e.g., not explicitly justifying right reversibility and left/right orientation in amenability when invoking the group of right fractions) are addressable by citing the paper’s statements. Overall, both arguments align and are correct.