Description of commutative monoids over which all polygons are \(\omega\)-stable (Q916648)

A monoid \(S\) is called \(\omega\)-stabilizer [stabilizer, superstabilizer] if every left polygon over \(S\) has \(\omega\)-stable [stable, superstable] theory. Stabilizers and superstabilizers were characterized by \textit{T. G. Mustafin} [Transl., Ser. 2, Am. Math. Soc. 195, 205--223 (1999); translation from Tr. Inst. Mat. 8, 92--108 (1988; Zbl 0725.03019)]. In this paper the following theorem is proved. A countable commutative monoid \(S\) is an \(\omega\)-stabilizer if and only if \(S\) is an abelian group with finite or countable number of subgroups, or \(S\) is a finite monoid with a unique proper ideal.