## About

16

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Introduction

Set-theoretical solutions of the Yang-Baxter equation, algebraic structures linked to solutions (in particular, braces, racks, cycle sets, etc.), regular subgroups of the affine group, self-distributive systems (quandles, racks, biracks).

**Skills and Expertise**

Additional affiliations

Education

March 2014 - July 2017

December 2008 - October 2012

September 2005 - December 2008

## Publications

Publications (16)

To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the category of semitrusses as introduced by Brzeziński. Fundamental examples of YB-semitrusses are structure monoids...

To determine and analyze arbitrary left non-degenerate set-theoretic solutions of the Yang-Baxter equation (not necessarily bijective), we introduce an associative algebraic structure, called a YB-semitruss, that forms a subclass of the category of semitrusses as introduced by Brzezi\'nski. Fundamental examples of YB-semitrusses are structure monoi...

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 or...

A set-theoretic solution of the Pentagon Equation on a non-empty set S is a map s:S2→S2 such that s23s13s12=s12s23, where s12=s×id, s23=id×s and s13=(τ×id)(id×s)(τ×id) are mappings from S3 to itself and τ:S2→S2 is the flip map, i.e., τ(x,y)=(y,x). We give a description of all involutive solutions, i.e., s2=id. It is shown that such solutions are de...

The distributive laws of ring theory are fundamental equalities in algebra. However, recently in the study of the Yang-Baxter equation, many algebraic structures with alternative “distributive” laws were defined. In an effort to study these “left distributive” laws and the interaction they entail on the algebraic structures, Brzeziński introduced s...

A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s\colon S^2\to S^2$ such that $s_{23}s_{13}s_{12}=s_{12}s_{23}$, where $s_{12}=s\times{\id}$, $s_{23}={\id}\times s$ and $s_{13}=(\tau\times{\id})({\id}\times s)(\tau\times{\id})$ are mappings from $S^3$ to itself and $\tau\colon S^2\to S^2$ is the flip map, i.e., $\...

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 ord...

In this work, we focus on the set-theoretical solutions of the Yang–Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product...

To study set-theoretic solutions of the Yang-Baxter equation several authors introduced algebraic structures. Rump and Ced\'o, Jespers and Okni\'nski introduced braces, Guarnieri and Vendramin introduced skew braces and Catino, Colazzo and Stefanelli and Jespers and Van Antwerpen introduced semi-braces. All these objects are subclasses of (semi-)tr...

We investigate the matched product of solutions associated with right and left shelves. First, we prove that the requirements to provide the matched product of solutions that come from shelves can be simplified. Then we give conditions for left non-degeneracy of the matched product. Later, we compute the structure shelf of the matched product of so...

In this work, we develop a novel construction technique for set-theoretical solutions of the Yang-Baxter equation. Our technique, named the matched product, is an innovative tool to construct new classes of involutive solutions as the matched product of two involutive solutions is still involutive, and vice versa. This method produces new examples...

In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product...

We describe the class of all skew left braces with non-trivial annihilator through ideal extension of a skew left brace. The ideal extension of skew left braces is a generalization to the non-abelian case of the extension of left braces provided by Bachiller in [D. Bachiller, Extensions, matched products, and simple braces, J. Pure Appl. Algebra 22...

In this paper we obtain new solutions of the Yang-Baxter equation that are left non-degenerate through left semi-braces, a generalization of braces introduced by Rump. In order to provide new solutions we introduce the asymmetric product of left semi-braces, a generalization of the semidirect product of braces, that allows us to produce several exa...

In this paper we introduce the asymmetric product of radical braces, a construction which extends the semidirect product of radical braces. This new construction allows to obtain rather systematic constructions of regular subgroups of the affine group and, in particular, our approach allows to put in a more general context the regular subgroups con...

Catino and Rizzo [‘Regular subgroups of the affine group and radical circle algebras’, Bull. Aust. Math. Soc.79 (2009), 103–107] established a link between regular subgroups of the affine group and the radical brace over a field on the underlying vector space. We propose new constructions of radical braces that allow us to obtain systematic constru...

## Projects

Projects (2)

Finding new set-theoretical solutions of the Yang-Baxter equation.