Kleisli convolution representations of power monoids

Abstract

We show that power semigroups of groups, and more generally reduced finitary power monoids, arise naturally as convolution monoids in Kleisli categories of powerset monads: (1) for the non-empty powerset monad, the Kleisli Hom-space ${\mathrm{Hom}}_{\mathbf{Kl}({\mathcal{P}}_{+})}(1,G)$ is isomorphic to the power monoid ${\mathcal{P}}_{+}(G)$; (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space ${\mathrm{Hom}}_{\mathbf{Kl}({\mathcal{P}}_{\mathrm{fin}})}$ $(\mathbb{Z}/2\mathbb{Z},H)$ is isomorphic to the reduced finitary power monoid ${\mathcal{P}}_{\mathrm{fin},1}(H)$. This unifies several constructions in power semigroup theory: Kleisli convolution representations of semigroups, base change along surjective group homomorphisms, and rigidity of automorphism groups. As an application, we prove that for every proper numerical monoid $S$, the Kleisli Hom-monoid ${\mathrm{Hom}}_{\mathbf{Kl}({\mathcal{P}}_{\mathrm{fin}})}(\mathbb{Z}/2\mathbb{Z},S)$ is rigid. This gives a proof of the Tringali--Yan conjecture in the language of Kleisli categories. It should be mentioned that this conjecture was already proved by Bhowmik and Tringali in a preprint posted on arXiv on July 25, 2026.

arXiv comment29 pages; we updated the latest work of Bhowmik and Tringali on the Tringali-Yan conjecture