Tringali–Yan conjecture

The Tringali–Yan isomorphism problem asks whether reduced finitary power monoids determine their underlying monoids. Tringali and Yan established the positive Puiseux- and numerical-monoid cases in earlier work, while broader versions remain more complicated.

Known results

• Tringali and Yan (October 26, 2023): for Puiseux monoids, $\mathcal{P}_{\mathrm{fin},0}(S_1)\cong\mathcal{P}_{\mathrm{fin},0}(S_2)$ exactly when $S_1\cong S_2$.
• Tringali and Yan: the analogous implication holds for cancellative monoids when one monoid is torsion; the arbitrary-groups case was reported open.
• Rago (September 28, 2025): nonisomorphic cancellative valuation monoids can have isomorphic reduced finitary power monoids, disproving the broader general statement.
• A separate preprint claims triviality of $\operatorname{Aut}(\mathcal{P}_{\mathrm{fin},0}(H))$ for proper numerical monoids.

August 27, 2026 Kleisli-convolution claim

A preprint announcement claims a Kleisli-category proof resolving the Tringali–Yan conjecture for all proper numerical monoids. This is an asserted complete resolution, but it is unrefereed and has no independent confirmation; a survey separately says an associated automorphism conjecture remains open.

Current status (as of August 2026): The numerical-monoid cases have substantial positive results, but the claimed Kleisli-convolution resolution is unverified, broader cancellative analogues are false, and a distinct automorphism conjecture remains open.