Let H H be a (multiplicatively written) monoid. The family P fin, 1 (H) P ₅₈₍, 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 H₁ and H 2 H₂ contained in a special class of commutative, cancellative monoids are isomorphic if and only if P fin, 1 (H 1) P ₅₈₍, 1 (H₁) and P fin, 1 (H 2) P ₅₈₍, 1 (H₂) are. Moreover, they raised the question whether the same holds in the general setting of cancellative monoids. We show that if H 1 H₁ and H 2 H₂ are (commutative) valuation monoids with trivial unit groups and isomorphic quotient groups, then P fin, 1 (H 1)
Balint Rago (Fri,) studied this question.