A hierarchical structure of transformation semigroups with applications to probability limit measures

TL;DR

This paper introduces a hierarchical framework for transformation semigroups on finite sets, enabling efficient analysis of probability limit measures and structural properties like kernel rank and right group conditions.

Contribution

It develops a hierarchy of semigroups and kernels that simplifies computations related to probability limits and structural analysis in transformation semigroups.

Findings

Hierarchy of semigroups and kernels aids in probability limit computations

Determines the rank of kernels in transformation semigroups from graph colorings

Characterizes kernels of rank one less than the number of vertices as right groups

Abstract

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels. This kernel hierarchy produces a set of tools that provides direct access to computations of interest in probability limit theorems; in particular, finding certain factors of idempotent limit measures. In addition, when considering transformation semigroups that arise naturally from edge colorings of directed graphs, as in the road-coloring problem, the hierarchy produces simple techniques to determine the rank of the kernel and to decide when a given kernel is a right group. In particular, it is shown that all kernels of rank one less than the number of vertices must be right groups and their structure for the case of two generators is described.