A counterexample to an isomorphism problem for power monoids

Let H be a (multiplicatively written) monoid. The family Pfin,1(H) of finite subsets of H containing the identity element is itself a monoid when endowed with setwise multiplication induced by H. Tringali and Yan proved that two monoids H₁ and H₂ contained in a special class of commutative, cancellative monoids are isomorphic if and only if Pfin,1(H₁) and Pfin,1(H₂) are. Moreover, they raised the question whether the same holds in the general setting of cancellative monoids. We show that if H₁ and H₂ are (commutative) valuation monoids with trivial unit groups and isomorphic quotient groups, then Pfin,1(H₁)fin,1(H₂). This provides a negative answer to Tringali and Yans question already within the class of valuation submonoids of the additive group Z².