  There are several families of atomic monoids whose arithmetical invariants
have received a great deal of attention during the last two decades. The
factorization theory of finitely generated monoids, strongly primary monoids,
Krull monoids, and C-monoids are among the most systematically studied. Puiseux
monoids, which are additive submonoids of $\mathbb{Q}_{\ge 0}$ consisting of
nonnegative rational numbers, have only been studied recently. In this paper,
we provide evidence that this family comprises plenty of monoids with a
basically unexplored atomic structure. We do this by showing that the
arithmetical invariants of the well-studied atomic monoids mentioned earlier
cannot be transferred to most Puiseux monoids via homomorphisms that preserve
atomic configurations, i.e., transfer homomorphisms. Specifically, we show that
transfer homomorphisms from a non-finitely generated atomic Puiseux monoid to a
finitely generated monoid do not exist. We also find a large family of Puiseux
monoids that fail to be strongly primary. In addition, we prove that the only
nontrivial Puiseux monoid that accepts a transfer homomorphism to a Krull
monoid is $\mathbb{N}_0$. Finally, we classify the Puiseux monoids that happen
to be C-monoids.
