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Length-Factoriality and Pure Irreducibility

Alan Bu, Joseph Vulakh, Alex Zhao

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.

Length-Factoriality and Pure Irreducibility

Abstract

An atomic monoid is called length-factorial if for every non-invertible element , no two distinct factorizations of 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.
Paper Structure (11 sections, 12 theorems, 17 equations)

This paper contains 11 sections, 12 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Setting of Monoids
  4. Connection to the Finite Factorization Property
  5. Inheritance
  6. Attainability of Master Factorization Relations
  7. Monoid Domains
  8. Semidomains
  9. What Is a Semidomain?
  10. Pure Irreducibles
  11. Restrictions on Bi-LFSs

Key Result

Proposition 2.1

An atomic monoid $M$ is a proper LFM if and only if it admits an unbalanced master factorization relation $(z_1, z_2)$, in which case the only master factorizations are $(z_1, z_2)$ and $(z_2, z_1)$.

Theorems & Definitions (26)

  • Proposition 2.1: CCGS21
  • Proposition 2.2: CCGS21
  • Proposition 3.1
  • proof
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 16 more