2021/03/04 by Robert D. Gray, Nik Ruškuc, Gray, Robert D. +1 · 2 citations
Computer Science · Mathematics · #20F05 #20M05 #20M18 #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras #semigroups and automata theory
paper · doi:10.48550/arxiv.2103.02995
openalex publication_date 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations where all the defining relations are of the form r=1. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.