Length-factoriality and pure irreducibility

An atomic monoid $M$ is called length-factorial if for every non-invertible element $x \in M$, no two distinct factorizations of $x$ into irreducibles have the same length (i.e., number irreducible factors, counting repetitions). The notion length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under term 'other-half-factoriality': they used to provide a characterization unique factorization domains. In this paper, we study more general context commutative, cancellative monoids. addition, properties related length-factoriality, namely, PLS property (recently Chapman et al.) bi-length-factoriality semirings.