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.
- N. R. Baeth and F. Gotti: Factorizations in upper triangular matrices over information semialgebras, J. Algebra 562 (2020) 466–496.
- P. M. Cohn: Bezout rings and and their subrings, Proc. Cambridge Philos. Soc. 64 (1968) 251–264.
- J. Coykendall and W. W. Smith: On unique factorization domains, J. Algebra 332 (2011) 62–70.
- A. Geroldinger and Q. Zhong: A characterization of length-factorial Krull monoids, New York J. Math. 27 (2021) 1347–1374.
- A. Geroldinger and Q. Zhong: Factorization theory in commutative monoids, Semigroup Forum 100 (2020) 22–51.
- F. Gotti: On semigroup algebras with rational exponents, Comm. Algebra 50 (2022) 3–18.
- F. Gotti and H. Polo: On the arithmetic of polynomial semidomains. Preprint on arXiv: https://arxiv.org/pdf/2203.11478.pdf
- A. Zaks: Half-factorial domains, Bull. Amer. Math. Soc. 82 (1976) 721–723.
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