# Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces

@article{Catino2020SettheoreticST, title={Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces}, author={Francesco Catino and Ilaria Colazzo and Paola Stefanelli}, journal={Forum Mathematicum}, year={2020}, volume={33}, pages={757 - 772} }

Abstract 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 that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the… Expand

#### 5 Citations

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#### References

SHOWING 1-10 OF 58 REFERENCES

Skew lattices and set-theoretic solutions of the Yang-Baxter equation

- Mathematics
- 2019

In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that has not yet been related to the… Expand

Semi-braces and the Yang-Baxter equation

- Mathematics
- 2017

Abstract 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… Expand

The matched product of set-theoretical solutions of the Yang-Baxter equation

- Mathematics
- 2020

Abstract 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… Expand

Set-theoretic solutions of the Yang-Baxter equation, graphs and computations

- Mathematics, Computer Science
- J. Symb. Comput.
- 2007

We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation… Expand

Set-theoretic solutions of the Yang–Baxter equation, RC-calculus, and Garside germs

- Mathematics
- 2014

Abstract Building on a result by W. Rump, we show how to exploit the right-cyclic law ( x y ) ( x z ) = ( y x ) ( y z ) in order to investigate the structure groups and monoids attached with… Expand

Braces and the Yang–Baxter Equation

- Mathematics
- 2012

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation are discussed and many consequences are derived. In particular, for each… Expand

Actions of skew braces and set-theoretic solutions of the reflection equation

- Mathematics
- 2018

A skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion… Expand

Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings

- Mathematics
- 2019

Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-...

A method of construction of finite-dimensional triangular semisimple Hopf algebras

- Mathematics
- 1998

The goal of this paper is to give a new method of constructing finite-dimensional semisimple triangular Hopf algebras, including minimal ones which are non-trivial (i.e. not group algebras). The… Expand

Braces, radical rings, and the quantum Yang–Baxter equation

- Mathematics
- 2007

Abstract Non-degenerate cycle sets are equivalent to non-degenerate unitary set-theoretical solutions of the quantum Yang–Baxter equation. We embed such cycle sets into generalized radical rings… Expand