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

Abstract 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 $$\left( S,+,\circ \right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>∘</mml:mo> </mml:mfenced> </mml:math> and prove that all elements generating deformed solutions belong precisely to the set $$\mathcal {D}_r(S)=\{z \in S \mid \forall a,b \in S \, \, (a+b) \circ z = a\circ z-z+b \circ z\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>z</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> <mml:mo>∣</mml:mo> <mml:mo>∀</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>S</mml:mi> <mml:mspace/> <mml:mspace/> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∘</mml:mo> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>z</mml:mi> <mml:mo>-</mml:mo> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>z</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which we term the distributor of S . We show it is a full inverse subsemigroup of $$\left( S, \circ \right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mo>∘</mml:mo> </mml:mfenced> </mml:math> 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 $$\cdot $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>·</mml:mo> </mml:math> , with $$\circ $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∘</mml:mo> </mml:math> representing the circle or adjoint operation. In this context, $$(\mathcal {D}_r(S),+,\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$[Y, B_\alpha , \phi _{\alpha ,\beta }]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> of skew braces $$B_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> , we establish that a deformed solution forms a semilattice of solutions on each skew brace $$B_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> if and only if the semilattice Y is bounded by an element 1 and the deforming element z lies in $$B_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> .