  As highlighted in a series of recent papers by Tringali and the author,
fundamental aspects of the classical theory of factorization can be
significantly generalized by blending the languages of monoids and preorders.
Specifically, the definition of a suitable preorder on a monoid allows for the
exploration of decompositions of its elements into (more or less) arbitrary
factors. We provide an overview of the principal existence theorems in this new
theoretical framework. Furthermore, we showcase additional applications beyond
classical factorization, emphasizing its generality. In particular, we recover
and refine a classical result by Howie on idempotent factorizations in the full
transformation monoid of a finite set.
