  A molecule is a nonzero non-unit element of an integral domain (resp.,
commutative cancellative monoid) having a unique factorization into
irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as
well as the molecules of their corresponding semigroup algebras, which we call
Puiseux algebras. We begin by presenting, in the context of numerical
semigroups, some results on the possible cardinalities of the sets of molecules
and the sets of reducible molecules (i.e., molecules that are not
irreducibles/atoms). Then we study the molecules in the more general context of
Puiseux monoids. We construct infinitely many non-isomorphic atomic Puiseux
monoids all whose molecules are atoms. In addition, we characterize the
molecules of Puiseux monoids generated by rationals with prime denominators.
Finally, we turn to investigate the molecules of Puiseux algebras. We provide a
characterization of the molecules of the Puiseux algebras corresponding to
root-closed Puiseux monoids. Then we use such a characterization to find an
infinite class of Puiseux algebras with infinitely many non-associated
reducible molecules.
