  An (additive) commutative monoid is called atomic if every given
non-invertible element can be written as a sum of atoms (i.e., irreducible
elements), in which case, such a sum is called a factorization of the given
element. The number of atoms (counting repetitions) in the corresponding sum is
called the length of the factorization. Following Geroldinger and Zhong, we say
that an atomic monoid $M$ is a length-finite factorization monoid if each $b
\in M$ has only finitely many factorizations of any prescribed length. An
additive submonoid of $\mathbb{R}_{\ge 0}$ is called a positive monoid.
Factorizations in positive monoids have been actively studied in recent years.
The main purpose of this paper is to give a better understanding of the
non-unique factorization phenomenon in positive monoids through the lens of the
length-finite factorization property. To do so, we identify a large class of
positive monoids which satisfy the length-finite factorization property. Then
we compare the length-finite factorization property to the bounded and the
finite factorization properties, which are two properties that have been
systematically investigated for more than thirty years.
