A counterexample to an isomorphism problem for power monoids

Let H H be a (multiplicatively written) monoid. The family P fin , 1 ( H ) \mathcal {P}_{\text {fin},1}(H) of finite subsets of H H containing the identity element is itself a monoid when endowed with setwise multiplication induced by H H . Tringali and Yan proved that two monoids H 1 H_1 and H 2 H_2 contained in a special class of commutative, cancellative monoids are isomorphic if and only if P fin , 1 ( H 1 ) \mathcal {P}_{\text {fin},1}(H_1) and P fin , 1 ( H 2 ) \mathcal {P}_{\text {fin},1}(H_2) are. Moreover, they raised the question whether the same holds in the general setting of cancellative monoids. We show that if H 1 H_1 and H 2 H_2 are (commutative) valuation monoids with trivial unit groups and isomorphic quotient groups, then P fin , 1 ( H 1 )