Master List of Examples in Complexity Theory of Finite Semigroup Theory
Stuart Margolis, John Rhodes
TL;DR
The paper assembles a master list of finite semigroups that are particularly informative for Krohn–Rhodes complexity, focusing on GM semigroups built from 0-simple Rees matrix structures and their translational hulls. It develops a standardized notation and workflow (including ${\rm RM}(B,G)$, ${\rm CM}(A,G)$, the translational hull $\Omega(I(S))$, the $\operatorname{RLM}$ image, and the Evaluation Transformation Semigroup $\mathcal{E}(L)$) and uses Flow Theory to compare $Sc$ with $\operatorname{RLM}(S)c$ via the Rhodes lattice $Rh_{B}(G)$ or the Set-Partition lattice $\operatorname{SP}(G\times B)$. The Master List then provides diverse, concrete examples (e.g., TF, TFA1, UTV, BIRIP, CBIRIP, RG1, RG2, S4, and S2) to illustrate when aperiodic flows exist, how complexity can exceed the $\operatorname{RLM}$-image, and how constructions from abelian character tables yield low-complexity GM semigroups. The work clarifies how to construct, analyze, and compare semigroups within the Krohn–Rhodes framework, offering a resource for both theoretical exploration and algorithmic decision procedures in complexity. The results demonstrate nuanced interactions between flow existence, $\operatorname{RLM}$ complexity, and overall $S_c$, underscoring the need to compute $\operatorname{RLM}(S)c$ separately from $Sc$.
Abstract
This document gives a list of finite semigroups that are interesting from the point of view of Krohn-Rhodes complexity theory. The list will be expanded and updates as "time goes by".