Searcharxiv⌕ Search

arXiv subjects

Maria Ferrara

Publications and source records attributed to Maria Ferrara.

13 recordsLinked to original sources
arXiv2026

A probabilistic approach to the Yang--Baxter equation and skew braces

We investigate finite non-degenerate set-theoretic solutions to the Yang--Baxter equation and skew braces using a probabilistic approach. We introduce four probabilities that measure how far a solution is from being a flip, but in different ways. Our main results state that for solutions arising from skew braces, these probabilities exhibit a rigid behaviour --- apart from a finite list of exceptional values (which we show to occur by means of explicit examples), they admit an upper bound that is slightly above $\frac{1}{2}$. In the skew brace setting, these probabilities measure how far the underlying skew brace is from being a trivial brace (in two different ways: one via the annihilator and the other via the socle), a trivial skew brace, and an almost trivial skew brace. We also introduce a probability that is related to the indecomposable components of a solution, and a probability that measures how close an arbitrary bijective non-degenerate map is to being a solution. In contrast to our main results, these two probabilities do not exhibit a discrete behaviour near $1$ --- they can be made arbitrarily close to $1$.

math.GR↗
arXiv2026

Finite-index Problems in Skew Braces

We investigate finite-index problems in skew braces. For every sub-skew brace \(A\) of a skew brace \(B\), we prove that finite additive index is equivalent to finite multiplicative index; whenever these indices are finite, they coincide. This answers Question~3.7 of \cite{CPV} affirmatively. We then construct a left brace with a strong left ideal of index \(3\) containing no finite-index ideal, giving a negative answer to Question~3.6 of \cite{CPV}. We also show that finite additive and multiplicative conjugacy classes, together with a finite \(\lambda\)-orbit, force an element to be an \((s)\)-element, thereby answering Question~5.21 of \cite{CPV}. Finally, for every \(n\geq 3\), a one-generated free right-nilpotent skew brace of class \(n\), introduced in \cite{Free}, has an index-\(2\) ideal that is not finitely generated as a skew brace, although it is finitely generated as an ideal.

math.GR↗
arXiv2026

A Finite Skew Brace with Perfect Additive Group and Almost Simple Multiplicative Group

We construct a finite skew brace whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. This gives an affirmative answer to Problem 20.109 in the twenty-first edition of the Kourovka Notebook. The additive and multiplicative groups of our example are \[ \PSU_5(64)\times A_5 \quad\text{and}\quad \Aut(\PSU_5(64)), \] respectively. The construction combines a skew brace with additive group \(A_5\) and multiplicative group \((C_5\rtimes C_4)\times C_3\), the splitting of the automorphism extension of \(\PSU_5(64)\), and a semidirect product of skew braces.

math.GR↗
arXiv2026

Chain conditions on skew braces and solutions of the Yang-Baxter Equation

Classical works of Hall and McLain show that solubility and local nilpotency play a key role in deriving finite generation in groups from maximal or minimal conditions on normal subgroups. In this work, brace-theoretical analogues of Hall's and McLain's results are analysed for skew braces satisfying the maximal or minimal condition on ideals. We also introduce finiteness and chain conditions on non-degenerate set-theoretic solutions of the Yang-Baxter equation, and their impact on associated structure and permutation skew braces of solutions is also described.

math.GR↗
arXiv2025

On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation

The aim of this paper is to provide purely arithmetical characterisations of those natural numbers $n$ for which every non-degenerate set-theoretic solution of cardinality $n$ of the Yang--Baxter equation arising from a skew brace (sb-solution for short) satisfies some relevant properties, such as being a flip or being involutive. For example, it turns out that every sb-solution of cardinality $n$ has finite multipermutation level if and only if its prime factorisation $n= p_1^{\alpha_1} \ldots p_t^{\alpha_t}$ is cube-free, namely $\alpha_i\leq 2$ for every $i$, and $p_i$ does not divide $p_j^{\alpha_j}-1$ for $i\neq j$. Two novel constructions of skew braces will play a central role in our proofs. We shall also introduce the notion of supersoluble solution and show how this concept is related to that of supersoluble skew brace. In doing so, we have spotted an irreparable mistake in the proof of Theorem C [Ballester-Bolinches et al., Adv. Math. 455 (2024)], which characterizes soluble solutions in terms of soluble skew braces.

math.GR↗
arXiv2025

Skew Braces from a model-theoretic point of view 1

Skew braces are one of the main algebraic tools controlling the structure of a non-degenerate bijective set-theoretic solution of the Yang-Baxter equation. The aim of this paper is to study model-theoretically tame skew braces, with particular attention to the notions of solubility and nilpotency.

math.GR↗
arXiv2024

TimelinePTC: Development of a unified interface for pathways to care collection, visualization, and collaboration in first episode psychosis

This paper presents TimelinePTC, a web-based tool developed to improve the collection and analysis of Pathways to Care (PTC) data in first episode psychosis (FEP) research. Accurately measuring the duration of untreated psychosis (DUP) is essential for effective FEP treatment, requiring detailed understanding of the patient's journey to care. However, traditional PTC data collection methods, mainly manual and paper-based, are time-consuming and often fail to capture the full complexity of care pathways. TimelinePTC addresses these limitations by providing a digital platform for collaborative, real-time data entry and visualization, thereby enhancing data accuracy and collection efficiency. Initially created for the Specialized Treatment Early in Psychosis (STEP) program in New Haven, Connecticut, its design allows for straightforward adaptation to other healthcare contexts, facilitated by its open-source codebase. The tool significantly simplifies the data collection process, making it more efficient and user-friendly. It automates the conversion of collected data into a format ready for analysis, reducing manual transcription errors and saving time. By enabling more detailed and consistent data collection, TimelinePTC has the potential to improve healthcare access research, supporting the development of targeted interventions to reduce DUP and improve patient outcomes.

cs.HC↗
arXiv2024

Finite skew braces of square-free order and supersolubility

The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers, and that in an arbitrary supersoluble skew brace $B$ many relevant skew brace-theoretical properties are easier to identify: for example, a centrally nilpotent ideal of $B$ is $B$-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, $B$ has finite multipermutational level if and only if $(B,+)$ is nilpotent. Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite non-degenerate solution of the Yang--Baxter Equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

math.GR↗
arXiv2023

A note on right-nil and strong-nil skew braces

The aim of this short note is to completely answer Questions 2.34 and 2.35 of arXiv:1806.01127. In particular, we show that a finite strong-nil skew brace $B$ of abelian type need not be right-nilpotent, but that this is the case if~$B$ is of nilpotent type and $b\ast b=0$ for all $b\in B$ (our examples show that this is the best possible result).

math.GR↗
arXiv2023

Central nilpotency of left skew braces and solutions of the Yang-Baxter equation

Nipotency of skew braces is related to certain types of solutions of the Yang-Baxter equation. This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory (Section 4.1) and we introduce an "index" for subbraces (Section 4.2), but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent (Example B), a rather peculiar fact. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a {\it good} Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient in our work is the characterisation of the commutator of ideals in terms of absorbing polynomials (Section 3); this solves Problem 3.4 of arXiv:2109.04389. Moreover, we provide an example (Example A) showing that the idealiser of a subbrace (as defined in arXiv:2205.01572v2) does not exist in general.

math.GR↗
arXiv2023

Joins of $\sigma$-subnormal subgroups

Let $\sigma=\{\sigma_j\,:\, j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup $X$ of a finite group $G$ is~\textit{$\sigma$-subnormal} in $G$ if there exists a chain of subgroups $$X=X_0\leq X_1\leq\ldots\leq X_n=G$$ such that, for each $1\leq i\leq n-1$, $X_{i-1}\trianglelefteq X_i$ or $X_i/(X_{i-1})_{X_i}$ is a $\sigma_{j_i}$-group for some $j_i\in J$. Skiba~[12] studied the main properties of $\sigma$-subnormal subgroups in finite groups and showed that the set of all $\sigma$-subnormal subgroups plays a relevant role in the structure of a finite soluble group. In [5], we laid the foundation of a general theory of $\sigma$-subnormal subgroups (and $\sigma$-series) in locally finite groups. It turns out that the main difference between the finite and the locally finite case concerns the behaviour of the join of $\sigma$-subnormal subgroups: in finite groups, $\sigma$-subnormal subgroups form a sublattice of the lattice of all subgroups [3], but this is no longer true for arbitrary locally finite groups. This is similar to what happens with subnormal subgroups, so it makes sense to study the class $\mathfrak{S}_\sigma^\infty$ (resp. $\mathfrak{S}_\sigma$) of locally finite groups in which the join of (resp. of finitely many) $\sigma$-subnormal subgroups is $\sigma$-subnormal. Our aim is to study how much one can extend a group in one of these classes before going outside the same class (see for example Theorems~3.6, 3.8, 5.5 and 5.7). Also, $\sigma$-subnormality criteria for the join of $\sigma$-subnormal subgroups are obtained: similarly to a celebrated theorem of Williams (see [15]), we give a necessary and sufficient conditions for a join of two $\sigma$-subnormal subgroups to always be $\sigma$-subnormal; consequently, we show that the join of two orthogonal $\sigma$-subnormal subgroups is $\sigma$-subnormal (extending a result of Roseblade [11]).

math.GR↗
arXiv2022

On derived-indecomposable solutions of the Yang--Baxter equation

If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.

math.GR↗