Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.15591

On right units of special inverse monoids

Igor Dolinka·Robert D. Gray
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the class of monoids that arise as the submonoid of right units of finitely presented special inverse monoids (SIMs). Gray and Ruškuc (2024) gave the first example of a finitely presented SIM whose submonoid of right units does not admit a decomposition into a free product of the group of units and a finite rank free monoid. In the first part of this paper we prove a general result which shows that the only instances where the right units of a finitely presented SIM can admit such a free product decomposition is when their group of units is finitely presented. In showing this, we establish some general results about finite generation and presentability of subgroups of SIMs. In particular, we give an exact characterisation of when an arbitrary subgroup is finitely generated in terms of connectedness properties of unions of its cosets in its $\mathscr{R}$-class, and also a characterisation of when an arbitrary subgroup is finitely presented. We also give a sufficient condition for finite generation and presentability of an arbitrary subgroup given in terms of a geometric finiteness property called boundary width. As a consequence, we show that the classes of monoids of right units of finitely presented SIMs and prefix monoids of finitely presented groups are independent. In the second part of the paper, we show that every finitely generated submonoid of a finitely RC-presented monoid is isomorphic to a submonoid $N$ of a finitely presented SIM $M$ such that $N$ is a submonoid of the right units of $M$, and $N$ contains the group of units of $M$. This result generalises and extends the classification of groups of units of finitely presented SIMs recently obtained by Gray and Kambites (2025). From this, we derive a number of surprising properties of RC-presentations for right cancellative monoids contrasting the classical theory of monoid presentations.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Igor Dolinka, Robert D. Gray. 2025-12-17. On right units of special inverse monoids. https://arxiv.org/abs/2512.15591

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Haagerup property for groups and for tracial von Neumann algebras in terms of invariant and mixing states

The aim of the article is to provide two characterizations of the Haagerup property: one for locally compact, second countable groups, and the other one for finite von Neumann algebras. Both are expressed in terms of approximations of some non-ergodic invariant states by mixing ones for actions on unital $C^*$-algebras on the one hand, and for pairs of tracial von Neumann algebras by mixing binormal states on the other hand.

math.GR↗
arXiv2026

The McCullough-Miller complex for right angled Artin groups

McCullough and Miller constructed a contractible complex on which the pure symmetric automorphism group of a free group acts with free abelian stabilizers. This complex has been used for computations such as the cohomological dimension of these groups, their cohomology rings, and results about $\ell^2$-Betti numbers or BNRS-invariants. We generalize this construction to pure symmetric automorphism groups of arbitrary RAAGs and exhibit applications of this generalization.

math.GR↗
arXiv2026

The quantitative non-unique-product landscape at the global minimum: the Nielsen-Soelberg groups

Nielsen and Soelberg proved that a finite subset $A$ of a torsion-free group with $A\cdot A$ having no unique product satisfies $|A|\ge 8$, and exhibited two groups, here $G_1$ and $G_2$, attaining the bound. Nothing quantitative was known about these extremal configurations. We construct exact, independently verified models of both groups and compute the first quantitative invariants at the global minimum. In $G_1$ no $8$-element symmetric witness lies in the radius-$6$ ball ($933$ elements, certified infeasible), while the Nielsen-Soelberg witness lies in the radius-$7$ ball: the global minimum is spread out. In $G_2$, with its natural eight-generator metric, the witness and its inverse are the only two non-UP $8$-sets in the radius-$1$ ball, and the unique-product staircase takes the value $0$ at $n=8$ but $1$ at $n=9$ -- the first known minimizer whose square has exactly one uniquely represented element, so the simultaneous failure of t.u.p. and u.p. seen in the Promislow group is not universal. No $(7,9)$ two-sided witness exists in the searched balls, so the Nielsen-Soelberg profile bound may not be sharp. Finally we treat the universal group $G_3$. Its structure is known -- Soelberg's thesis identifies an index-$8$ Heisenberg subgroup of step $8$ and proves torsion-freeness, and Gardam, studying the same group as an amalgam of Klein bottle groups, shows it to be virtually nilpotent but not virtually abelian -- and what we add is a model in search coordinates in which balls can be enumerated. In it we reproduce the Nielsen-Soelberg two-sided pair and exhibit a symmetric $15$-element witness whose trivial-coset singleton generates the centre of that Heisenberg subgroup. It is rigid and rare: within $B(5)$ the size $15$ is exactly minimal, the coset profile is forced, and exactly four such witnesses exist in $B(4)$, one orbit. Hence $m_1(G_3)\in[8,15]$ against $m_2(G_3)=16$.

math.GR↗