  An atomic monoid $M$ is called a length-factorial monoid (or an
other-half-factorial monoid) if for each non-invertible element $x \in M$ no
two distinct factorizations of $x$ have the same length. The notion of
length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of
the well-studied notion of half-factoriality. They proved that in the setting
of integral domains, length-factoriality can be taken as an alternative
definition of a unique factorization domain. However, being a length-factorial
monoid is in general weaker than being a factorial monoid (i.e., a unique
factorization monoid). Here we further investigate length-factoriality. First,
we offer two characterizations of a length-factorial monoid $M$, and we use
such characterizations to describe the set of Betti elements and obtain a
formula for the catenary degree of $M$. Then we study the connection between
length-factoriality and purely long (resp., purely short) irreducibles, which
are irreducible elements that appear in the longer (resp., shorter) part of any
unbalanced factorization relation. Finally, we prove that an integral domain
cannot contain purely short and a purely long irreducibles simultaneously, and
we construct a Dedekind domain containing purely long (resp., purely short)
irreducibles but not purely short (resp., purely long) irreducibles.
