arXiv subjects

Nóra Szakács

Publications and source records attributed to Nóra Szakács.

16 recordsLinked to original sources

arXiv2026

On the simplicity of Katsura algebras

We give a complete characterization of the (purely infinite) simplicity of Katsura algebras and the associated Steinberg algebras. This is achieved by characterizing when the singular ideals vanish via the self-similar groupoid model derived from Exel and Pardo. Analogous results are given for the algebras arising from the faithful quotient of the self-similar action. Finally, we describe polynomial-space algorithms to determine if each singular ideal vanishes and provide the first non-Hausdorff examples of non-contracting self-similar groupoids for which simplicity is algorithmically decidable.

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arXiv2026

Algebraic singular functions are not always dense in the ideal of $C^*$-singular functions

We give the first examples of étale (non-Hausdorff) groupoids $\mathcal G$ whose $C^*$-algebras contain singular elements that cannot be approximated by singular elements in $\mathcal C_c(\mathcal G)$. We provide two examples: one is a bundle of groups, and the other a minimal and effective groupoid constructed from a self-similar action on an infinite alphabet. Moreover, we also prove that the Baum--Connes assembly map for the first example is not surjective, not even on the level of its essential $C^*$-algebra.

math.OA↗

arXiv2025

The large scale geometry of inverse semigroups and their maximal group images

The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of non-commutative $C^*$-algebras. We study the relationship between the geometry of an inverse semigroup and that of its maximal group image, and in particular the geometric \textit{distortion} of the natural map from the former to the latter. This turns out to have both implications for semigroup theory and potential relevance for operator algebras associated to inverse semigroups. Along the way, we also answer a question of Lledó and Martínez by providing a more direct proof that an $E$-unitary inverse semigroup has Yu's Property A if its maximal group image does.

math.GR↗

arXiv2024

The growth of free inverse monoids

We compute the rate of exponential growth of the free inverse monoid of rank $r$ (and hence an upper bound on the corresponding rate for all $r$-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to $\sqrt{e(2k-1)}$ as $k \to \infty$.

math.GR↗

arXiv2024

Pretzel monoids

We introduce an interesting class of left adequate monoids which we call pretzel monoids. These, on the one hand, are monoids of birooted graphs with respect to a natural `glue-and-fold' operation, and on the other hand, are shown to be defined in the category of left adequate monoids by a natural class of presentations. They are also shown to be the free idempotent-pure expansions of right cancellative monoids, making them, in some sense, the left adequate analogues of Margolis-Meakin expansions for inverse monoids. The construction recovers the second author's geometric model of free left adequate monoids when the right cancellative monoid is free.

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arXiv2024

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

Given any quasi-countable, in particular any countable inverse semigroup $S$, we introduce a way to equip $S$ with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on $S$. This allows us to unambiguously define the uniform Roe algebra of $S$, which we prove can be realized as a canonical crossed product of $\ell^\infty(S)$ and $S$. We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension $0$. Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups, and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by $S$ being locally $\mathcal L$-finite, and equivalently sparse as a metric space.

math.OA↗

arXiv2023

$E$-unitary and $F$-inverse monoids, and closure operators on group Cayley graphs

We show that the category of $X$-generated $E$-unitary inverse monoids with greatest group image $G$ is equivalent to the category of $G$-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of $G$. Analogously, we study $F$-inverse monoids in the extended signature $(\cdot, 1, ^{-1}, ^\mathfrak{m})$, and show that the category of $X$-generated $F$-inverse monoids with greatest group image $G$ is equivalent to the category of $G$-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of $G$. As an application, we show that presentations of $F$-inverse monoids in the extended signature can be studied by tools analogous to Stephen's procedure in inverse monoids, in particular, we introduce the notions of $F$-Schützenberger graphs and $P$-expansions.

math.GR↗

arXiv2022

Algorithmic properties of inverse monoids with hyperbolic and tree-like Schützenberger graphs

We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.

math.GR↗

arXiv2022

Inverse monoids and immersions of $Δ$-complexes

An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between $Δ$-complexes is a $Δ$-map that induces injections from star sets of $\mathcal D$ to star sets of $\mathcal C$. We study immersions between finite-dimensional connected $Δ$-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersions into $2$-dimensional $CW$-complexes.

math.GR↗

arXiv2021

On the simplicity of Nekrashevych algebras of contracting self-similar groups

Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych $C^\ast$-algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the $C^\ast$-algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Basilica group, Gupta-Sidki groups, GGS-groups, multi-edge spinal groups, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk-Erschler group and Fabrykowski-Gupta group) and self-replicating spinal automaton groups.

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arXiv2020

Simplicity of inverse semigroup and étale groupoid algebras

In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an étale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential $C^*$-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic $0$. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.

math.RA↗

arXiv2020

Inverse monoids of partial graph automorphisms

A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well.

math.CO↗

arXiv2016

On F-inverse covers of finite-above inverse monoids

Finite-above inverse monoids are a common generalization of finite inverse monoids and Margolis--Meakin expansions of groups. Given a finite-above $E$-unitary inverse monoid $M$ and a group variety $\mathit{U}$, we find a condition for $M$ and $\mathit{U}$, involving a construction of descending chains of graphs, which is equivalent to $M$ having an $F$-inverse cover via $\mathit{U}$. In the special case where $\mathit{U}=\mathit{Ab}$, the variety of Abelian groups, we apply this condition to get a simple sufficient condition for $M$ to have no $F$-inverse cover via $\mathit{Ab}$, formulated by means of the natural parial order and the least group congruence of $M$.

math.GR↗

arXiv2015

On the graph condition regarding the $F$-inverse cover problem

In their paper titled "On $F$-inverse covers of inverse monoids", Auinger and Szendrei have shown that every finite inverse monoid has an $F$-inverse cover if and only if each finite graph admits a locally finite group variety with a certain property. We study this property and prove that the class of graphs for which a given group variety has the required property is closed downwards in the minor ordering, and can therefore be described by forbidden minors. We find these forbidden minors for all varieties of Abelian groups, thus describing the graphs for which such a group variety satisfies the above mentioned condition.

math.GR↗

arXiv2015

Inverse monoids and immersions of 2-complexes

It is well known that under mild conditions on a connected topological space $\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal X$. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order to classify immersions into a 2-dimensional CW-complex $\mathcal C$, we need to replace the fundamental group of $\mathcal C$ by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex.

math.GR↗

arXiv2012

On the number of slim, semimodular lattices

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Czédli and E.T. Schmidt as the duals of the lattices consisting of the intersections of the members of two composition series in a group. Our main result determines the number of (isomorphism classes of) these lattices of a given size in a recursive way. The corresponding planar diagrams, up to similarity, are also enumerated. We prove that the number of diagrams of slim, distributive lattices of a given length b is the n-th Catalan number. Besides lattice theory, the paper includes some combinatorial arguments on permutations and their inversions.

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