On two-sided skew braces

Trappeniers, Senne

doi:10.1016/j.jalgebra.2023.05.003

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2024

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Cited by 3 publications

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Skew Braces: A Brief Survey

Vendramin

2024

Trends in Mathematics

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Skew Braces: A Brief Survey

Vendramin

2024

Trends in Mathematics

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Deformed solutions of the Yang–Baxter equation associated to dual weak braces

Mazzotta,

Rybołowicz,

Stefanelli

2024

Annali di Matematica

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A recent method for acquiring new solutions of the Yang–Baxter equation involves deforming the classical solution associated with a skew brace. In this work, we demonstrate the applicability of this method to a dual weak brace $$\left( S,+,\circ \right) $$ S , + , ∘ and prove that all elements generating deformed solutions belong precisely to the set $$\mathcal {D}_r(S)=\{z \in S \mid \forall a,b \in S \, \, (a+b) \circ z = a\circ z-z+b \circ z\}$$ D r ( S ) = { z ∈ S ∣ ∀ a , b ∈ S ( a + b ) ∘ z = a ∘ z - z + b ∘ z } , which we term the distributor of S. We show it is a full inverse subsemigroup of $$\left( S, \circ \right) $$ S , ∘ and prove it is an ideal for certain classes of braces. Additionally, we express the distributor of a brace S in terms of the associativity of the operation $$\cdot $$ · , with $$\circ $$ ∘ representing the circle or adjoint operation. In this context, $$(\mathcal {D}_r(S),+,\cdot )$$ ( D r ( S ) , + , · ) constitutes a Jacobson radical ring contained within S. Furthermore, we explore parameters leading to non-equivalent solutions, emphasizing that even deformed solutions by idempotents may not be equivalent. Lastly, considering S as a strong semilattice $$[Y, B_\alpha , \phi _{\alpha ,\beta }]$$ [ Y , B α , ϕ α , β ] of skew braces $$B_\alpha $$ B α , we establish that a deformed solution forms a semilattice of solutions on each skew brace $$B_\alpha $$ B α if and only if the semilattice Y is bounded by an element 1 and the deforming element z lies in $$B_1$$ B 1 .

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Rota–Baxter Operators on Skew Braces

Wang,

Zhang,

Zhang

2024

Mathematics

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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,+,∘) and (Z/(4),+,∘). 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.

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On two-sided skew braces

Cited by 3 publications

References 23 publications

Skew Braces: A Brief Survey

Skew Braces: A Brief Survey

Deformed solutions of the Yang–Baxter equation associated to dual weak braces

Rota–Baxter Operators on Skew Braces

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