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Paola Stefanelli

Publications and source records attributed to Paola Stefanelli.

At least 19 recordsLinked to original sources
arXiv2026

Action accessibility in the variety of skew braces

In this paper, we answer negatively to a question posed in the context of the 2025 Oberwolfach Mini-Workshop ``The Yang-Baxter Equation and Representations of Braid Groups'' regarding the existence of split extensions classifiers in the category of skew braces. To this end, we show that the variety of skew braces is not action accessible by investigating the categorical notion of centraliser in the sense of Huq. In light of their intrinsic relationship with skew braces, an analogous result is proven for the variety of post-Lie algebras over a fixed field.

math.CT↗
arXiv2026

Right groups, left quasigroups, and right heaps

A right group is a semigroup $(S,\cdot)$ in which, for every $a,b\in S$, there is a unique $x\in S$ such that $a\cdot x=b$. In this article, we develop the theory of heaps starting not from groups, but from right groups. We thus get a natural definition of right heap. It is even possible to develop part of the theory starting from a left quasigroup, which is the non-associative analogue of a right group. Our motivation for this study is the investigation of left non-degenerate set-theoretic solutions of the Yang--Baxter equation. Thus, we are led to an analogue of the skew left trusses introduced by T.~Brzeziński.

math.GR↗
arXiv2026

Right groups and the set-theoretic Yang-Baxter equation

In this paper, we provide techniques to obtain left non-degenerate set-theoretic solutions of the Yang-Baxter equation, drawing on the class of right groups. To this end, we introduce the new algebraic structures of left $RG$-semibraces, which include left (cancellative) semibraces as a proper subclass.

math.GR↗
arXiv2026

Parametric reflection maps: an algebraic approach

We study solutions of the parametric set-theoretic reflection equation from an algebraic perspective by employing recently derived generalizations of the familiar shelves and racks, called parametric (p)-shelves and racks. Generic invertible solutions of the set-theoretic reflection equation are also obtained by a suitable parametric twist. The twist leads to considerably simplified constraints compared to the ones obtained from general set-theoretic reflections. In this context, novel algebraic structures of (skew) p-braces that generalize the known (skew) braces and are suitable for the parametric Yang-Baxter equation are introduced. The p-rack Yang-Baxter and reflection operators as well as the associated algebraic structures are defined setting up the frame for formulating the p-rack reflection algebra.

math.RA↗
arXiv2026

Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation

We introduce a novel algebraic structure called di-skew brace by which we show that generalized digroups systematically yield bijective, non-degenerate solutions to the set-theoretic Yang-Baxter equation. We study the structural properties of these solutions with a particular focus on their left derived shelves, which belong to the class of conjugation racks. Consistently, we show that these solutions belong to a broader class that includes skew brace solutions. In particular, we prove that each such solution can be decomposed as a hemi-semidirect product of a skew brace solution endowed with a certain compatible action on the idempotents of the associated di-skew brace structure. Finally, we provide concrete instances of these solutions through a suitable notion of averaging operators on groups.

math.QA↗
arXiv2025

Quasi racks, quasi bijective and quasi non-degenerate set-theoretic solutions of the Yang-Baxter equation

This work initiates a systematic study of the class of quasi bijective and quasi non-degenerate solutions to the set-theoretic Yang-Baxter equation. The motivation stems from the observation that solutions that arise from dual weak braces belong to these classes. The notions of quasi rack and derived solution are introduced and examined, extending the classical definitions. Additionally, a family of quasi left non-degenerate solutions is described in terms of quasi racks and g-twists, analogous to the left non-degenerate case. Furthermore, we completely characterize a class of quasi racks that are Plonka sum of racks.

math.QA↗
arXiv2025

Reflections to set-theoretic solutions of the Yang-Baxter equation

The main aim of this paper is to determine reflections to bijective and non-degenerate solutions of the Yang-Baxter equation, by exploring their connections with their derived solutions. This is motivated by a recent description of left non-degenerate solutions in terms of a family of automorphisms of their associated left rack. In some cases, we show that the study of reflections for bijective and non-degenerate solutions can be reduced to those of derived type. Moreover, we extend some results obtained in the literature for reflections of involutive non-degenerate solutions to more arbitrary solutions. Besides, we provide ways for defining reflections for solutions obtained by employing some classical construction techniques of solutions. Finally, we gather some numerical data on reflections for bijective non-degenerate solutions associated with skew braces of small order.

math.QA↗
arXiv2024

Deformed solutions of the Yang-Baxter equation associated to dual weak braces

A dual weak brace is an algebraic structure $\left(S,\,+,\,\circ\right)$ including skew braces and giving rise to a set-theoretic solution of the Yang-Baxter equation. We show that such a map belongs to a family of set-theoretic solutions, called deformed solutions, that are defined on $S$ and depending on certain parameters. We prove these elements are exactly those belonging to the distributor of $S$, i.e., $\mathcal{D}_r(S)=\{z \in S \, \mid \, \forall \, a,b \in S \quad (a+b) \circ z=a\circ z-z+b \circ z\}$, that is a full inverse subsemigroup of $\left(S, \circ\right)$. Regarding $S$ as a strong semilattice $[Y, B_α, ϕ_{α,β}]$ of skew braces $B_α$, we analyze when $\mathcal{D}_r(S)=\mathop{\dot{\bigcup}}\limits_{α\in Y} \mathcal{D}_r(B_α)$ and in which cases a deformed solution is the strong semilattices of deformed solutions.

math.QA↗
arXiv2024

Quandles as pre-Lie skew braces, set-theoretic Hopf algebras & universal R-matrices

We present connections between left non-degenerate solutions of the set-theoretic braid equation and left shelves using Drinfel'd homomorphisms. We generalize the notion of affine quandle, by using heap endomorphisms and metahomomorphisms, and identify the underlying Yang-Baxter algebra for solutions of the braid equation associated to a given quandle. We introduce the notion of the pre-Lie skew brace and identify certain affine quandles that give rise to pre-Lie skew braces. Generalisations of the braiding of a group, associated to set-theoretic solutions of the braid equation are also presented. These generalized structures encode part of the underlying Hopf algebra. We then introduce the quasi-triangular (quasi) Hopf algebras and universal R-matrices for rack and set-theoretic algebras. Generic set-theoretic solutions coming from heap endomorphisms are also identified.

math.QA↗
arXiv2023

Solutions of the Yang-Baxter equation and strong semilattices of skew braces

We prove that any set-theoretic solution of the Yang-Baxter equation associated to a dual weak brace is a strong semilattice of non-degenerate bijective solutions. This fact makes use of the description of any dual weak brace $S$ we provide in terms of strong semilattice $Y$ of skew braces $B_α$, with $α\in Y$. Additionally, we describe the ideals of $S$ and study its nilpotency by correlating it to that of each skew brace $B_α$.

math.QA↗
arXiv2023

Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation

In this paper, we introduce the theory of Rota-Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples $\left(S,+,\circ\right)$ where $\left(S,+\right)$ and $\left(S,\circ\right)$ are Clifford semigroups such that $a\circ\left(b+c\right) = a\circ b - a +a\circ c$ and $a\circ a^- = -a+a$, for all $a,b,c\in S$. To each algebraic structure is associated a set-theoretic solution of the Yang-Baxter equation that has a behaviour near to the bijectivity and non-degeneracy. Drawing from the theory of Clifford semigroups, we provide methods for constructing dual weak braces and deepen some structural aspects, including the notion of ideal.

math.QA↗
arXiv2023

Set-theoretical solutions of the pentagon equation on Clifford semigroups

Given a set-theoretical solution of the pentagon equation $s:S\times S\to S\times S$ on a set $S$ and writing $s(a, b)=(a\cdot b,\, θ_a(b))$, with $\cdot$ a binary operation on $S$ and $θ_a$ a map from $S$ into itself, for every $a\in S$, one naturally obtains that $\left(S,\,\cdot\right)$ is a semigroup. In this paper, we focus on solutions on Clifford semigroups $\left(S,\,\cdot\right)$ satisfying special properties on the set of the idempotents $E(S)$. Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which $θ_a$ remains invariant in $E(S)$, for every $a\in S$. Moreover, considering $(S,\,\cdot)$ as a disjoint union of groups, we construct a family of idempotent-fixed solutions, i.e., those solutions for which $θ_a$ fixes every element in $E(S)$, for every $a\in S$, starting from a solution on each group.

math.GR↗
arXiv2022

Affine structures on groups and semi-braces

We introduce affine structures on groups and show they form a category equivalent to that of semi-braces. In particular, such a new description of semi-braces includes that presented by Rump for braces. By specific affine structures, we provide several instances of bi-skew braces, including some that are not $λ$-homomorphic. Finally, we give a method for determining affine structures on the Zappa product of two groups both endowed with affine structures and prove that such a construction allows for obtaining semi-braces that are not matched product of semi-braces.

math.GR↗
arXiv2022

Set-theoretic solutions of the Yang-Baxter equation associated to weak braces

We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang-Baxter equation. Specifically, a weak (left) brace is a non-empty set $S$ endowed with two binary operations $+$ and $\circ$ such that both $(S,+)$ and $(S, \circ)$ are inverse semigroups and they hold \begin{align*} a \circ \left(b+c\right) = a\circ b - a +a\circ c \qquad \text{and} \qquad a\circ a^- = - a + a, \end{align*} for all $a,b,c \in S$, where $-a$ and $a^-$ are the inverses of $a$ with respect to $+$ and $\circ$, respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution $r$ associated to an arbitrary weak brace $S$ has a behavior close to bijectivity, namely $r$ is a completely regular element in the full transformation semigroup on $S\times S$. In addition, we provide some methods to construct weak braces.

math.QA↗
arXiv2021

Simplicity and finite primitive level of indecomposable set-theoretic solutions of the Yang-Baxter equation

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we especially study two classes of current interest: the simple solutions and those having finite primitive level. In particular, we provide two group-theoretic characterizations of these solutions, involving their permutation groups. Finally, we deal with some open questions.

math.QA↗
arXiv2020

Inverse semi-braces and the Yang-Baxter equation

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple $(S,+, \cdot)$ with $(S,+)$ a semigroup and $(S, \cdot)$ an inverse semigroup satisfying the relation $a \left(b + c\right) = a b + a\left(a^{-1} + c\right)$, for all $a,b,c \in S$, where $a^{-1}$ is the inverse of $a$ in $(S, \cdot)$. In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known.

math.QA↗
arXiv2020

Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces

This paper aims to introduce a construction technique of set-theoretic solutions of the Yang-Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.

math.QA↗