  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
$A=A(K,X,r)=K\langle X\mid xy=uv \mbox{ whenever }r(x,y)=(u,v)\rangle$. Note
that $A=\oplus_{n\geq 0} A_n$ is a graded algebra, where $A_n$ is the linear
span of all the elements $x_1\cdots x_n$, for $x_1,\dots ,x_n\in X$. One of the
known results asserts that the maximal possible value of $\dim (A_2)$
corresponds to involutive solutions and implies several deep and important
properties of $A(K,X,r)$. Following recent ideas of Gateva-Ivanova
\cite{GI2018}, we focus on the minimal possible values of the dimension of
$A_2$. 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 in
\cite{GI2018} are solved.
