arXiv subjects

Ferran Cedó

Publications and source records attributed to Ferran Cedó.

9 recordsLinked to original sources

arXiv2022

Indecomposable solutions of the Yang-Baxter equation of square-free cardinality

Indecomposable involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of cardinality $p_1\cdots p_n$, for different prime numbers $p_1,\ldots, p_n$, are studied. It is proved that they are multipermutation solutions of level $\leq n$. In particular, there is no simple solution of a non-prime square-free cardinality. This solves a problem stated in [F. Cedó, J. Okniński, Constructing finite simple solutions of the Yang-Baxter equation, Adv. Math. 391 (2021), 107968] and provides a far reaching extension of several earlier results on indecomposability of solutions. The proofs are based on a detailed study of the brace structure on the permutation group $\mathcal G(X,r)$ associated to such a solution. It is proved that $p_1,\ldots, p_n$ are the only primes dividing the order of $\mathcal{G}(X,r)$. Moreover, the Sylow $p_i$-subgroups of $\mathcal{G}(X,r)$ are elementary abelian $p_i$-groups and if $P_i$ denotes the Sylow $p_i$-subgroup of the additive group of the left brace $\mathcal{G}(X,r)$, then there exists a permutation $σ\in S_n$ such that $P_{σ(1)}, \, P_{σ(1)}P_{σ(2)}, \dots , P_{σ(1)}P_{σ(2)}\cdots P_{σ(n)}$ are ideals of the left brace $\mathcal{G}(X,r)$ and $\mathcal{G}(X,r)=P_1P_2\cdots P_n$. In addition, indecomposable solutions of cardinality $p_1\cdots p_n$ that are multipermutation of level $n$ are constructed, for every nonnegative integer $n$.

math.QA↗

arXiv2022

Corrigendum and Addendum to "Structure monoids of set-theoretic solutions of the Yang--Baxter equation"

One of the results in our article, which appeared in Publ. Mat. 65 (2021), 499--528, is that the structure monoid $M(X,r)$ of a left non-degenerate solution $(X,r)$ of the Yang-Baxter Equation is a left semi-truss, in the sense of Brzeziński, with an additive structure monoid that is close to being a normal semigroup. Let $η$ denote the least left cancellative congruence on the additive monoid $M(X,r)$. It is then shown that $η$ also is a congruence on the multiplicative monoid $M(X,r)$ and that the left cancellative epimorphic image $\bar{M}=M(X,r)/η$ inherits a semi-truss structure and thus one obtains a natural left non-degenerate solution of the Yang-Baxter equation on $\bar{M}$. Moreover, it restricts to the original solution $r$ for some interesting classes, in particular if $(X, r)$ is irretractable. The proof contains a gap. In the first part of the paper we correct this mistake by introducing a new left cancellative congruence $μ$ on the additive monoid $M(X,r)$ and show that it also yields a left cancellative congruence on the multiplicative monoid $M(X,r)$ and we obtain a semi-truss structure on $M(X,r)/μ$ that also yields a natural left non-degenerate solution. In the second part of the paper we start from the least left cancellative congruence $ν$ on the multiplicative monoid $M(X,r)$ and show that it also is a congruence on the additive monoid $M(X,r)$ in case $r$ is bijective. If, furthermore, $r$ is left and right non-degenerate and bijective then $ν=η$, the least left cancellative congruence on the additive monoid $M(X,r)$, extending an earlier result of Jespers, Kubat and Van Antwerpen to the infinite case.

math.RA↗

arXiv2021

New simple solutions of the Yang-Baxter equation and solutions associated to simple left braces

Involutive non-degenerate set theoretic solutions of the Yang-Baxter equation are considered, with a focus on finite solutions. A rich class of indecomposable and irretractable solutions is determined and necessary and sufficient conditions are found in order that these solutions are simple. Then a link between simple solutions and simple left braces is established, that allows us to construct more examples of simple solutions. In particular, the results answer some problems stated in the recent paper [13].

math.QA↗

arXiv2020

Constructing finite simple solutions of the Yang-Baxter equation

We study involutive non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation on a finite set X. The emphasis is on the case where (X,r) is indecomposable, so the associated permutation group acts transitively on X. One of the major problems is to determine how such solutions are built from the imprimitivity blocks; and also how to characterize these blocks. We focus on the case of so called simple solutions, which are of key importance. Several infinite families of such solutions are constructed for the first time. In particular, a broad class of simple solutions of order p^2, for any prime p, is completely characterized.

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↗

arXiv2018

An abundance of simple left braces with abelian multiplicative Sylow subgroups

Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can be considered as building blocks for the general theory. This program recently has been initiated by Bachiller and the authors. In this paper we study the simple finite left braces such that the Sylow subgroups of their multiplicative groups are abelian. We provide several new families of such simple left braces. In particular, they lead to the main, surprising result, that shows that there is an abundance of such simple left braces.

math.QA↗

arXiv2017

Asymmetric product of left braces and simplicity; new solutions of the Yang-Baxter equation

The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite left braces is fundamental in order to describe all finite such solutions of the Yang-Baxter equation. In this paper we continue the study of finite simple left braces with the emphasis on the application of the asymmetric product of left braces in order to construct new classes of simple left braces. We do not only construct new classes but also we interpret all previously known constructions as asymmetric products. Moreover, a construction is given of finite simple left braces with a multiplicative group that is solvable of arbitrary derived length.

math.QA↗

arXiv2017

Braces and symmetric groups with special conditions

We study symmetric groups and left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like $\textbf{Raut}$ and $\textbf{lri}$ on the properties of the symmetric group and its associated brace. We show that the symmetric group $G=G(X,r)$ associated to a nontrivial solution $(X,r)$ has multipermutation level $2$ if and only if $G$ satisfies $\textbf{lri}$. In the special case of a two-sided brace we express each of the conditions $\textbf{lri}$ and $\textbf{Raut}$ as identities on the associated radical ring $G_*$. We apply these to construct examples of two-sided braces satisfying some prescribed conditions. In particular we construct a finite two-sided brace with condition $\textbf{Raut}$ which does not satisfy $\textbf{lri}$. (It is known that condition $\textbf{lri}$ implies $\textbf{Raut}$). We show that a finitely generated two-sided brace which satisfies \textbf{lri} has a finite multipermutation level which is bounded by the number of its generators.

math.QA↗

arXiv2016

On the Yang-Baxter equation and left nilpotent left braces

We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite brace and into a finite braided group. For a left brace A, we explore the close relation between the multipermutation level of the solution associated with it and the radical chain $A^{(n+1)}=A^{(n)}* A$ introduced by Rump.

math.GR↗