Dynamics, Random Products, and Ultrametric Geometry in Kiselman's Semigroup

TL;DR

This paper explores the dynamical, probabilistic, and geometric properties of Kiselman's semigroup, including the behavior of random products, a level function, and an ultrametric structure.

Contribution

It introduces a level function, analyzes random partial products, and defines an ultrametric on Kiselman's semigroup, revealing new structural insights.

Findings

Sequences of partial products are eventually constant.

Hitting times of the constant value follow a sum of independent geometric distributions.

An ultrametric structure is established with basic properties of metric balls and spheres.

Abstract

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.