The pentagon equation (PE) is a basic equation of Mathematical Physics. 
Recent developments in this field of investigations can be found in [1]. In this talk, we show a new application of the PE. In particular, the PE turns out to be a useful tool to find set-theoretical solutions of the well-known Yang-Baxter equation. 

[1] F. Catino, M. Mazzotta, M.M. Miccoli: Set-theoretical solutions of the åpentagon equation on groups, Preprint available at arXiv:1902.04310.

In this talk, we focus on the matched product of solutions associated with shelves, and we simplify the conditions that allow obtaining this product. Furthermore, we study the properties of the shelf related to the matched product of two left non-degenerate solutions.

In 2011 Gateva-Ivanova and Cameron [2], following a suggestion due by P. Martin, posed the following question:

Question. For each positive integer denote by the minimal integer so that there exists a square-free involutive multipermutational solution of order and with . How does depend on ?

They showed [2] that for every and, some years later, Lebed and Vendramin improve [3] the estimation showing indirectly that for every . In this talk we will present some recent results obtained in [1] that allow us to improve the estimation of the sequence .

Partendo da alcuni esempi, si indicheranno alcune piste di lavoro per la costruzione di nuove soluzioni insiemistiche dell’equazione di Yang-Baxter.

The problem of finding set-theoretical solutions of the Yang-Baxter equation [2] (shortly solutions) has been dealt with from different points of view in the last years.
In [1] a novel construction technique for solutions was developed, the matched product. In this talk, we will display a method to construct new classes of particular solutions through the matched product of other solutions in the same class. As a particular case, starting from involutive (resp. idempotent) solutions the matched product is still involutive (resp. idempotent), and vice versa.

[1] F. Catino, I. Colazzo, P. Stefanelli, The matched product of set-theoretical solutions of the Yang-Baxter equation, submitted.
[2] V.G. Drinfeld, On some unsolved problems in quantum group theory. Lecture Notes in Math. 1510 (1992) 1-8.

In order to study the indecomposable cycle sets, a possible approach involves the permutation group associated to the cycle sets.

In this talk we consider the particular case of indecomposable cycle sets with cyclic permutation group. We will introduce, in this particular case, the definition of the Socle of a cycle sets and we will see how an indecomposable cyle sets with cyclic permutation group can be pictured.

[1] M. Castelli, G. Pinto, Indecomposable involutive set-theoretic solutions with cyclic permutation group, work in progress.

In this talk we will discuss on a particular class of left cycle sets called quasi-linear. We will see briefly the basic theory and we will discuss on a problem posed by Rump in [2] and the related partial results obtained in [1].

[1] M. Castelli, F. Catino, M.M. Miccoli and G. Pinto, Dynamical extensions of quasi-linear left cycle sets and the Yang-Baxter equation, Journal Algebra Appl. (to appear).
[2] W. Rump, Quasi-Linear Cycle Sets and the Retraction Problem for Set-Theoretic Solutions of the Quantum Yang-Baxter Equation, Algebra Colloq. 23 (1) (2016) 149-166.

The set-theoretical Yang-Baxter equation presents a form, called two-parameter set-theoretical Yang-Baxter equation, where the solutions depends precisely from two parameters. This formulation can be used in the study of systems of particles which collide among themselves, since from the conservation laws of this system we can derive the velocities after the collisions in terms of their masses (the parameters) and their velocities before the collisions and obtain this way a solution of said equation. This fact allow us to examine the integrability (that is the solvability) of a particular system periodic sequences of collisions.

In this talk we will discuss the structure monoid of a finite left non-degenerate solution of the Yang–Baxter equation and its associated monoid algebra. Using a realization of Lebed and Vendramin of as a regular submonoid in the semidirect product , we will prove that is a finite module over a central affine subalgebra. In particular, it is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension. Furthermore, we will discuss some results on prime ideals of and we will discuss the positive answer to a conjecture of Gateva–Ivanova on cancellativity of .

Finding solutions of the Yang-Baxter equation has been the subject of many studies in pure mathematics and theoretical physics. In particular, as proposed by Drinfeld [1], the study of set-theoretic solutions has been very attractive over the past few years. One of the tools to study these solutions, called braces, were introduced by Rump [2]. In this talk, we want to show another useful application of braces. In particular we describe a relation between braces and units of group rings [3]. In joint work with Lukasz Kubat, we try to generalize this result for skew braces, a generalization of braces. To do so, we define group near-rings, a generalization of group rings.

[1] V.G. Drinfeld. On some unsolved problems in quantum group theory. Lecture Notes in Math, 1510:1-8, 1992.
[2] W. Rump. Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra, 307:153-170, 2007.
[3] Y.P. Sysak. Product of groups and the quantum Yang-Baxter equation. Notes of a talk in Advances in Group Theory and Applications, 2011, Porto Cesareo.