The Schützenberger groups and maximal subgroups of tropical matrices

2025/01/16 by Thomas Aird, Aird, Thomas
Engineering · Mathematics · #12K10 #20H99 (Primary) 16Y60 #20M10 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings and Algebras (math.RA) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2501.09501

openalex publication_date 2025/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify the Schützenberger groups of the category of matrices over the tropical semiring, M(\mathbbT), in doing so, we obtain a classification for the Schützenberger groups of the semigroupoid of matrices over the finitary tropical semiring, M(\mathbbF\mathbbT). We then classify the maximal subgroups of the monoid of n × n matrices over the tropical semiring, M_n(\mathbbT), for all n ∈ ℕ; generalising a result in the literature and correcting an erroneous proof. We proceed to show that for some n ∈ ℕ there exists a group which appears as a Schützenberger group of M_n(\mathbbT) but does not appear as a maximal subgroup.

Related

• Tropical matrix groups
• Tropical matrices and group representations
• Semigroup identities of tropical matrices through matrix ranks
• Identities of tropical matrices and plactic monoids
• Generating sets, presentations, and growth of tropical matrix monoids