幂幺半群的Kleisli卷积表示
Kleisli convolution representations of power monoids
- Northwest Normal University（西北师范大学）
- Lanzhou University of Technology（兰州理工大学）
- Guizhou University（贵州大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明群的幂半群等可作为幂集单子Kleisli范畴中的卷积幺半群，统一幂半群理论相关构造，还证明真数值幺半群对应的Kleisli同态幺半群刚性，解答了Tringali-Yan猜想。
AI中文摘要：
本文证明，群的幂半群，更一般的约化有限幂幺半群，自然地作为幂集单子的Kleisli范畴中的卷积幺半群出现：（1）对于非空幂集单子，Kleisli同态空间$\text{Hom}_{\boldsymbol{\text{Kl}}(\boldsymbol{\text{P}}_+)}(1,G)$同构于幂幺半群$\boldsymbol{\text{P}}_+(G)$；（2）对于带基点集合上的约化有限幂集单子，Kleisli同态空间$\text{Hom}_{\boldsymbol{\text{Kl}}(\boldsymbol{\text{P}}_{\text{fin}})}(\boldsymbol{\text{Z}}/2\boldsymbol{\text{Z}},H)$同构于约化有限幂幺半群$\boldsymbol{\text{P}}_{\text{fin},1}(H)$。这统一了幂半群理论中的若干构造：半群的Kleisli卷积表示、满群同态下的基变换以及自同构群的刚性。作为应用，本文证明对每个真数值幺半群$S$，Kleisli同态幺半群$\text{Hom}_{\boldsymbol{\text{Kl}}(\boldsymbol{\text{P}}_{\text{fin}})}(\boldsymbol{\text{Z}}/2\boldsymbol{\text{Z}},S})$是刚性的，从而通过Kleisli范畴的语言给出Tringali-Yan猜想的肯定解答。
英文摘要：
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 $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_+)}(1,G)$ is isomorphic to the power monoid $\mathcal P_+(G)$; (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}$ $(\mathbb Z/2\mathbb Z,H)$ 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 $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}(\mathbb Z/2\mathbb Z,S)$ 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.
