  The theory of $\tau$-factorizations on integral domains was developed by
Anderson and Frazier. This theory characterized all the known factorizations
and opened the opportunity to create new ones. It can be visualized as a
restriction to the structure's multiplicative operation, by considering a
symmetric relation $\tau$ on the set of non-zero non-unit elements of an
integral domain.
  The main goal of this work is to study the $\tau$-factorization concept, when
$\tau$ is a composition of two or more relations. To achieve this, the specific
properties one can obtain from the given relations are verified and analyzed.
Some of the studied properties which are the most known include: reflexivity,
symmetry, transitivity, antisymmetry. And others related to the
$\tau$-factorization theory, like: divisive, associate-preserving and
multiplicative relations.
