  In {\it Set-theoretical solutions to the quantum Yang-Baxter equation} (Duke
Math. J. {\bf 100} (1999), 169--209), Etingof, Schedler and Soloviev
introduced, for each non-degenerate involutive set-theoretical solution
$(X,\sigma,\tau)$ of the Yang-Baxter equation, the equivalence relation $\sim$
defined on the set $X$ and they considered a new non-degenerate involutive
induced \emph{retraction} solution defined on the quotient set $X^{\sim}$. It
is well known that translating set-theoretical non-degenerate solutions of the
Yang-Baxter equation into the universal algebra language we obtain an algebra
called a \emph{birack}. In the paper we introduce the \emph{generalized
retraction} relation $\approx$ on a birack, which is equal to $\sim$ in an
involutive case. We present a complete algebraic proof that the relation
$\approx$ is a congruence of the birack. Thus we show that the retraction of a
set-theoretical non-degenerate solution is well defined not only in the
involutive case but also in the case of all non-involutive solutions.
