A counterexample to an isomorphism problem for power monoids
Abstract
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 )
Fields of Study & Subjects
semigroups and automata theory
Rings
Modules
and Algebras
Advanced Algebra and Logic
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Volume
- 154
- Issue
- 5
- Pages
- 1855–1858
- Publication date
- 13 March 2026
- Language
- EN