Aperiodic Flows on Finite Semigroups: Foundations and First Examples
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| Format: | Preprint |
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2024
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| _version_ | 1866916594482937856 |
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| author | Margolis, Stuart Rhodes, John |
| author_facet | Margolis, Stuart Rhodes, John |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_06668 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Aperiodic Flows on Finite Semigroups: Foundations and First Examples Margolis, Stuart Rhodes, John Dynamical Systems Group Theory 20M20, 20M35, 20M18 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. |
| title | Aperiodic Flows on Finite Semigroups: Foundations and First Examples |
| topic | Dynamical Systems Group Theory 20M20, 20M35, 20M18 |
| url | https://arxiv.org/abs/2410.06668 |