2011/03/01 by G. Budzban, Ph. Feinsilver, P. Feinsilver · 1 citation
Computer Science · Mathematics · #Commutative Algebra and Its Applications #Hierarchy #Idempotence #Kernel (algebra) #Limit (mathematics) #Limits and Structures in Graph Theory #Rank (graph theory) #Simple (philosophy) #Transformation (genetics) #math.DS #math.PR #msc:20M20 #msc:37A50 #msc:5C81 #msc:60B15 #msc:68Q70 #semigroups and automata theory
paper · pdf · doi:10.1080/10236198.2011.639365
published as J. Difference Equ. Appl., 18(8):1405--1434, 2012 · 35 pages, 4 figures
arxiv created 2011/03/01 · openalex publication_date 2012/01/23 · openalex created_date 2016/06/24 · arxiv updated 2016/12/02 · openalex updated_date 2026/08/05
The structure of transformation semigroups on a finite set is analysed 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 colourings of directed graphs, as in the road-colouring 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.