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Given a finite non-degenerate set-theoretic solution ( X ,  r ) of the Yang–Baxter equation and a field K , the structure K -algebra of ( X ,  r ) is $$A=A(K,X,r)=K\langle X\mid xy=uv \text{ whenever } r(x,y)=(u,v)\rangle $$ A = A ( K , X , r ) = K ⟨ X ∣ x y = u v whenever r ( x , y ) = ( u , v ) ⟩ . Note that $$A=\oplus _{n\ge 0} A_n$$ A = ⊕ n ≥ 0 A n is a graded algebra, where $$A_n$$ A n is the linear span of all the elements $$x_1\cdots x_n$$ x 1 ⋯ x n , for $$x_1,\dots ,x_n\in X$$ x 1 , ⋯ , x n ∈ X . One of the known results asserts that the maximal possible value of $$\dim (A_2)$$ dim ( A 2 ) corresponds to involutive solutions and implies several deep and important properties of A ( K ,  X ,  r ). Following recent ideas of Gateva-Ivanova (A combinatorial approach to noninvolutive set-theoretic solutions of the Yang–Baxter equation. arXiv:1808.03938v3 [math.QA], 2018), we focus on the minimal possible values of the dimension of $$A_2$$ A 2 . We determine lower bounds and completely classify solutions ( X ,  r ) for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed by Gateva-Ivanova (2018) are solved.

Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition