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PreprintarXiv licencearXiv

Kleisli convolution representations of power monoids

  • Haicun Wen
  • Jian He
  • Yu-Zhe Liu
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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 HomKl(𝒫+)(1,G) is isomorphic to the power monoid 𝒫+(G); (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space HomKl(𝒫fin) (β„€/2β„€,H) is isomorphic to the reduced finitary power monoid 𝒫fin,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 HomKl(𝒫fin)(β„€/2β„€,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.

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Cite

BibTeXRIS
BibTeX
@misc{wen2026kleisli,
  title = {{Kleisli convolution representations of power monoids}},
  author = {Haicun Wen and Jian He and Yu-Zhe Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.27292},
  eprint = {2608.27292},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27292},
}
RIS
TY  - UNPB
TI  - Kleisli convolution representations of power monoids
AU  - Haicun Wen
AU  - Jian He
AU  - Yu-Zhe Liu
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.27292
UR  - https://arxiv.org/abs/2608.27292
AB  - 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 Hom_Kl(𝒫_+)(1,G) is isomorphic to the power monoid 𝒫_+(G); (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space Hom_Kl(𝒫_fin) (β„€/2β„€,H) is isomorphic to the reduced finitary power monoid 𝒫_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 Hom_Kl(𝒫_fin)(β„€/2β„€,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.
ER  -