  Given two linear transformations, with representing matrices $A$ and $B$ with
respect to some bases, it is not clear, in general, whether the Tracy-Singh
product of the matrices $A$ and $B$ corresponds to a particular operation on
the linear transformations. Nevertheless, it is not hard to show that in the
particular case that each matrix is a square matrix of order of the form $n^2$,
$n>1$, and is partitioned into $n^2$ square blocks of order $n$, then their
Tracy-Singh product, $A \boxtimes B$, is similar to $A \otimes B$, and the
change of basis matrix is a permutation matrix. In this note, we prove that in
the special case of linear operators induced from set-theoretic solutions of
the Yang-Baxter equation, the Tracy-Singh product of their representing
matrices is the representing matrix of the linear operator obtained from the
direct product of the set-theoretic solutions.

Combinatorics;Rings and Algebras