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

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.

• The paper introduces the level function in Kiselman’s semigroup, which measures the distance from any element to the zero element via generator eliminations.
• It demonstrates that random products stabilize with convergence characterized by a sum of independent geometric random variables yielding an expected time of n².
• The study defines an ultrametric on the semigroup, linking combinatorial geometry with algebraic and probabilistic dynamics for applications in noncommutative systems.

Dynamics and Ultrametric Geometry of Kiselman's Semigroup

Introduction and Context

This work investigates the algebraic, probabilistic, and metric structure of Kiselman's semigroups $K_n$. Kiselman's semigroup, defined by the generators $a_1, ..., a_n$ and relations $a_i^2 = a_i$ and $a_i a_j a_i = a_j a_i a_j = a_i a_j$ for $1 \leq j < i \leq n$, arises in convex analysis and combinatorics. The semigroup possesses a canonical zero element $f = e_{[n]}$ and exhibits rich combinatorial behavior with a number of elements growing double-exponentially with $n$.

The Level Function and its Algebraic Role

A central concept introduced is the level function $\mathcal{L}: K_n \rightarrow \{0, 1, ..., n\}$, defined as the minimum integer $i$ such that the deletion endomorphism $$\overline{\partial}_{[i]}(x) = e_{[n] \setminus [i]}$$. This function serves as a stratification over $a_1, ..., a_n$0, measuring the "distance" from $a_1, ..., a_n$1 to the zero element in terms of generator eliminations. The level function's dynamics under right multiplication by generators $a_1, ..., a_n$2 are summarized by the explicit recurrence:

$a_1, ..., a_n$3

This formula exposes a decremental property tied to generator indices, providing an effective means to analyze products and their stabilization.

Partial and Random Products: Dynamics and Stochasticity

The study extends to infinite sequences of partial products. It is proven that every such sequence stabilizes: for any sequence $a_1, ..., a_n$4 with $a_1, ..., a_n$5, the sequence of products $a_1, ..., a_n$6 is eventually constant. Under mild conditions on the sequence (specifically, when every generator appearing does so infinitely often), the limiting value is the idempotent associated to the set of indices that occur in the sequence. If all $a_1, ..., a_n$7 occur infinitely often, the limit is the zero element $a_1, ..., a_n$8.

The analysis becomes particularly rich in the probabilistic setting. When $a_1, ..., a_n$9 is an i.i.d. sequence over the generators, the process $a_i^2 = a_i$0 forms a Markov chain under $a_i^2 = a_i$1. The transition matrix is lower-bidiagonal, with stepwise transitions downward governed by hitting the critical generator. The hitting time $a_i^2 = a_i$2 to reach the zero element is distributed as a sum of $a_i^2 = a_i$3 independent geometric random variables, a strong claim supported by explicit calculation. Importantly, for uniform distributions over generators, the expected convergence time is $a_i^2 = a_i$4.

Ultrametric Geometry on $a_i^2 = a_i$5

Building upon the level function, a natural ultrametric is defined over $a_i^2 = a_i$6:

$a_i^2 = a_i$7

This metric renders $a_i^2 = a_i$8 a finite ultrametric space, as demonstrated by standard arguments. The balls and spheres are characterized concretely: the ball around $a_i^2 = a_i$9 of radius $a_i a_j a_i = a_j a_i a_j = a_i a_j$0 consists of all $a_i a_j a_i = a_j a_i a_j = a_i a_j$1 for which $a_i a_j a_i = a_j a_i a_j = a_i a_j$2. Spheres admit a combinatorial description in terms of the content map. The recursive structure of balls and spheres mirrors the recursive, layered structure inherent in Kiselman's semigroup.

Theoretical and Practical Implications

The interplay between the algebraic structure (deletion endomorphisms, idempotent stratification), dynamics (eventual constancy of partial and random products), and the metric geometry (ultrametric induced by the level function) imbues $a_i a_j a_i = a_j a_i a_j = a_i a_j$3 with a hierarchical nature. The results enable the analysis of convergence rates for product processes, with applications in noncommutative, combinatorial, and probabilistic semigroup theory, as well as in models where stabilization phenomena are of interest.

The analysis of random product stabilization rates, especially with the explicit geometric decomposition of hitting times, is directly relevant for the theory of random walks on semigroups and could be generalized to other, possibly infinite, idempotent semigroups. The ultrametric and combinatorial stratification potentially links this structure to p-adic and hierarchical models in other mathematical areas.

Conclusion

This paper provides a rigorous synthesis of algebraic, probabilistic, and metric properties in Kiselman's semigroup, unifying these aspects through the level function and its induced ultrametric. The explicit analysis of random and deterministic product stabilization and the fine description of metric balls and spheres exemplify how dynamic, algebraic, and geometric perspectives can converge in finite idempotent semigroups. The framework presented suggests directions for expansion to other classes of semigroups and further study of ultrametric structures in algebraic dynamics.