2024/08/12 by Carl‐Fredrik Nyberg‐Brodda, Nyberg-Brodda, Carl-Fredrik
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2408.05986
openalex publication_date 2024/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The free regular ⋆-monoid of rank r is the freest r-generated regular monoid Fr^⋆ in which every element m has a distinguished pseudo-inverse m^⋆ satisfying mm^⋆ m = m and (m^⋆)^⋆ = m. We study the growth rate of the monogenic regular ⋆-monoid F1^⋆, and prove that this growth rate is intermediate. In particular, we deduce that Fr^⋆ is not rational or automatic for any r ≥ 1, yielding the analogue of a result of Cutting & Solomon for free inverse monoids. Next, for all ranks r ≥ 1 we determine the integral homology groups H_∗(Fr^⋆, ℤ), and by constructing a collapsing scheme prove that they vanish in dimension 3 and above. As a corollary, we deduce that the free regular ⋆-monoid Fr^⋆ of rank r ≥ 1 does not have the homological finiteness property FP2, yielding the analogue of a result of Gray & Steinberg for free inverse monoids.