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14 pages, 249 KiB

Open AccessArticle

by Ximu Wang, Chongxia Zhang and Liangyun Zhang

Mathematics 2024, 12(11), 1671; https://doi.org/10.3390/math12111671 - 27 May 2024

Cited by 1 | Viewed by 1173

In this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as

(Z, +, \circ)

and

(Z / (4), +, \circ)

. [...] Read more.

In this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as

(Z, +, \circ)

and

(Z / (4), +, \circ)

. We also present a necessary and sufficient condition for a skew brace to be a co-inverse skew brace. Additionally, we describe some constructions of Rota–Baxter quasiskew braces, and demonstrate that every Rota–Baxter skew brace can induce a quasigroup and a Rota–Baxter quasiskew brace. Full article

5 pages, 244 KiB

Open AccessProceeding Paper

New Structure of Skew Braces and Their Ideals

by Mehsin Jabel Atteya

Comput. Sci. Math. Forum 2023, 7(1), 54; https://doi.org/10.3390/IOCMA2023-14601 - 15 May 2023

Viewed by 971

Abstract

The primary motivation of this article is to introduce the definition of a strongly belonging element of a brace, presenting some results concerning the case where I acts as an ideal of a brace,

(A, +, \circ)

. We [...] Read more.

The primary motivation of this article is to introduce the definition of a strongly belonging element of a brace, presenting some results concerning the case where I acts as an ideal of a brace,

(A, +, \circ)

. We find that an element

a \in I

acts as strongly belonging element of I when

a + b

yields a for all b in I. Full article

(This article belongs to the Proceedings of The 1st International Online Conference on Mathematics and Applications)

17 pages, 353 KiB

Open AccessArticle

Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules

by Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa and Adolfo Ballester-Bolinches

Computation 2023, 11(3), 43; https://doi.org/10.3390/computation11030043 - 21 Feb 2023

Cited by 1 | Viewed by 1749

Abstract

The Kronecker algebra

K

is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over

K

are said to be Kronecker [...] Read more.

The Kronecker algebra

K

is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over

K

are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form

A x = B x^{'}

, where A and B are

m \times n

F

-matrices with

F

an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over

K

called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications in Computing)

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