Aperiodic Flows on Finite Semigroups: Foundations and First Examples
Stuart Margolis, John Rhodes
TL;DR
This paper lays the foundations for a systematic study of aperiodic flows on finite semigroups, providing a unified Presentation Lemma via the Slice Theorem and establishing when a GM semigroup has a flow over the trivial semigroup in terms of division by a direct product with its $RLM(S)$ image. It situates flow theory within inverse semigroup concepts, notably through 1-point flows that correspond to $S_{II}\cap I(S)$ being aperiodic and to $E$-unitary covers, while connecting Rhodes and set-partition lattices to flow dynamics. The authors develop comprehensive frameworks, including flow descriptions to $SP(G\times B)$ and $Rh_{B}(G)$, the Flow Decomposition Theorem, and unified slice-based proofs, and they illustrate the machinery with small monoid examples and GM semigroups derived from Abelian group character tables. The work demonstrates both conceptual coherence and practical computability, with explicit constructions and a clear program for Flow II and Flow III, complemented by a MasterList of further examples. This advances understanding of Krohn–Rhodes complexity through aperiodic flow techniques and ties flows to classical inverse semigroup results.
Abstract
The theory of flows was used as a crucial tool in the recent proof by Margolis, Rhodes and Schilling that Krohn-Rhodes complexity is decidable. In this paper we begin a systematic study of aperiodic flows. We give the foundations of the theory of flows and give a unified approach to the Presentation Lemma and its relations to flows and the Slice Theorem. We completely describe semigroups having a flow over the trivial semigroup and connect this to classical results in inverse semigroup theory. We reinterpret Tilson's Theorem on the complexity of small monoids in terms of flows. We conclude with examples of semigroups built from the character table of Abelian Groups that have an aperiodic flows.