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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2604.25892 |
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| _version_ | 1866910174828036096 |
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| author | Andrenšek, Luka |
| author_facet | Andrenšek, Luka |
| contents | We study certain dynamical and metric aspects of Kiselman's semigroup $K_n$. The level function $\mathcal{L}$ is introduced and shown to admit a simple description in terms of right multiplication by generators. We show that every sequence of partial products in $K_n$ is eventually constant. Using $\mathcal{L}$, we further study sequences of random partial products in $K_n$ and show that, in the independent and identically distributed setting where every generator is chosen with positive probability, the hitting time of the eventual constant value is distributed as a sum of $n$ independent geometric random variables. Finally, we define a natural ultrametric on $K_n$ arising from the level function and obtain some basic results on the associated metric balls and spheres. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25892 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamics, Random Products, and Ultrametric Geometry in Kiselman's Semigroup Andrenšek, Luka Group Theory Probability We study certain dynamical and metric aspects of Kiselman's semigroup $K_n$. The level function $\mathcal{L}$ is introduced and shown to admit a simple description in terms of right multiplication by generators. We show that every sequence of partial products in $K_n$ is eventually constant. Using $\mathcal{L}$, we further study sequences of random partial products in $K_n$ and show that, in the independent and identically distributed setting where every generator is chosen with positive probability, the hitting time of the eventual constant value is distributed as a sum of $n$ independent geometric random variables. Finally, we define a natural ultrametric on $K_n$ arising from the level function and obtain some basic results on the associated metric balls and spheres. |
| title | Dynamics, Random Products, and Ultrametric Geometry in Kiselman's Semigroup |
| topic | Group Theory Probability |
| url | https://arxiv.org/abs/2604.25892 |