Length-factoriality and pure irreducibility

Abstract

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 of irreducible factors, counting repetitions). The notion of length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under the term “other-half-factoriality”: they used length-factoriality to provide a characterization of unique factorization domains. In this paper, we study length-factoriality in the more general context of commutative, cancellative monoids. In addition, we study factorization properties related to length-factoriality, namely, the PLS property (recently introduced by Chapman et al.) and bi-length-factoriality in the context of semirings.

Keywords:

• Atomicity
• factorization
• finite-rank monoid
• length-factoriality
• monoid algebra
• other-half-factoriality
• pure irreducible
• semidomain
• unique factorization

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13F15
• 13A05
• Secondary: 16Y60

Acknowledgments

We thank our mentor, Dr. Felix Gotti, for guiding our research project and making valuable suggestions, and the MIT PRIMES organizers for making our research possible. We would also like to thank the Reviewer for their thoughtful comments and suggestions.