Semiperfect countable \(C\)-finite semigroups \(S\) satisfying \(S=S+S\) (Q1306551)

For a countable abelian semigroup (without involution, possibly without zero) satisfying \(S+S=S\) and a further condition called ``\(C\)-finite'', the author proves the equivalence of the following three statements: (1) \(S\) is operator semiperfect. (2) \(S\) is semiperfect. (3) Every Archimedean component of \(S\) is isomorphic to the product of a finite group of exponent 1 or 2, and one of the semigroups \(\{0\},\mathbb{N}, \mathbb{Z}\). The result is highly non-trivial, and the proof is long and complicated.