Length-factoriality in commutative monoids and integral domains

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