arXiv subjects

Andrea Albano

Publications and source records attributed to Andrea Albano.

5 recordsLinked to original sources

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

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 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

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

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↗