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

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 S,+,∘ and prove that all elements generating deformed solutions belong precisely to the set 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 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 ·, with ∘ representing the circle or adjoint operation. In this context, (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_α,ϕ_α,β] of skew braces B_α, we establish that a deformed solution forms a semilattice of solutions on each skew brace B_α if and only if the semilattice Y is bounded by an element 1 and the deforming element z lies in B₁.