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Francesco Catino

Publications and source records attributed to Francesco Catino.

13 recordsLinked to original sources
arXiv2026

Set-theoretic solutions of the Yang-Baxter equation from inverse braces

We introduce the algebraic structure of an inverse brace, namely, a triple $(S,+,\circ)$ such that both $(S,+)$ and $(S,\circ)$ are inverse semigroups and the following identity holds $a\circ(b+c)=a\circ b-a+a\circ c$, for all $ a, b, c \in S$, where $-a$ denotes the inverse of $a \in S$, with respect to $+$. In particular, every weak brace is an inverse brace. We investigate the fundamental properties of inverse braces and analyze the relationship between additive and multiplicative idempotents, characterizing the condition under which an inverse brace is a weak brace. Our main results concern the connection with set-theoretic solutions to the Yang-Baxter equation. Specifically, we provide a class of inverse braces that yield solutions and give several examples. Finally, we introduce constructions of inverse braces via the matched product and the strong semilattice of inverse braces. We show that these constructions preserve the conditions required to produce solutions, thereby providing a systematic method for generating new examples.

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↗
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↗
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↗
arXiv2020

Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and dynamical extensions of q-cycle sets

A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions obtained by Rump in [36] and answer in a positive way to a question posed by Cedó, Jespers, and Verwimp [19, Question 4.2]. Moreover, we develop a theory of extensions for left non-degenerate set-theoretic solutions of the Yang-Baxter equation that allows one to construct new families of set-theoretic solutions.

math.QA↗
arXiv2019

Set-theoretical solutions of the Yang-Baxter and pentagon equations on semigroups

The Yang-Baxter and pentagon equations are two well-known equations of Mathematical Physic. If $S$ is a set, a map $s:S\times S\to S\times S$ is said to be a set theoretical solution of the Yang-Baxter equation if $$ s_{23}\, s_{13}\, s_{12} = s_{12}\, s_{13}\, s_{23}, $$ where $s_{12}=s\times id_S$, $s_{23}=id_S\times s$, and $s_{13}=(id_S\times τ)\,s_{12}\,(id_S\times τ)$ and $τ$ is the flip map, i.e., the map on $S\times S$ given by $τ(x,y)=(y,x)$. Instead, $s$ is called a set-theoretical solution of the pentagon equation if $$ s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}. $$ The main aim of this work is to display how solutions of the pentagon equation turn out to be a useful tool to obtain new solutions of the Yang-Baxter equation. Specifically, we present a new construction of solutions of the Yang-Baxter equation involving two specific solutions of the pentagon equation. To this end, we provide a method to obtain solutions of the pentagon equation on the matched product of two semigroups, that is a semigroup including the classical Zappa product.

math.QA↗
arXiv2019

The matched product of set-theoretical solutions associated with shelves

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 solutions. Finally, we prove that the structure shelf of the matched product does not depend on the choice of the actions.

math.QA↗
arXiv2019

The matched product of the solutions to the Yang-Baxter equation of finite order

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 of two solutions $r_S$ and $r_T$ is of finite order if and only if $r_S$ and $r_T$ are. Furthermore, we show that with sufficient information on $r_S$ and $r_T$ we can precisely establish the order of the matched product. Finally, we prove that if $B$ is a finite semi-brace, then the associated solution $r$ satisfies $r^n=r$, for an integer $n$ closely linked with $B$.

math.QA↗
arXiv2019

About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equation

In this paper, we introduce a new sequence $\bar{N}_m$ to find a new estimation of the cardinality $N_m$ of the minimal involutive square-free solution of level $m$. As an application, using the first values of $\bar{N}_m$, we improve the estimations of $N_m$ obtained by Gateva-Ivanova and Cameron and by Lebed and Vendramin. Following the approach of the first part, in the last section we construct several new counterexamples to the Gateva-Ivanova's Conjecture.

math.QA↗
arXiv2019

Set-theoretical solutions of the pentagon equation on groups

Let $M$ be a set. A set-theoretical solution of the pentagon equation on $M$ is a map $s:M\times M\longrightarrow M\times M$ such that \begin{equation*} s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}, \end{equation*} where $s_{12}=s\times id_M$, $s_{23}=id_M \times s$ and $s_{13}=(id_M \times τ) s_{12}(id_M \times τ)$, and $τ$ is the flip map, i.e., the permutation on $M\times M$ given by $τ(x,y)=(y,x)$, for all $x,y\in M$. In this paper we give a complete description of the set-theoretical solutions of the form $s(x,y)=(x\cdot y , x\ast y)$ when either $(M,\cdot)$ or $(M,\ast)$ is a group; moreover, we raise some questions.

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