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On length densities

  • Scott T. Chapman ORCID logo EMAIL logo , Christopher O’Neill ORCID logo and Vadim Ponomarenko ORCID logo
Published/Copyright: December 1, 2021
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 2

Abstract

For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x∈M, denoted LD⁡(x), and the entire monoid M, denoted LD⁡(M). This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L⁢(x), ℓ⁢(x), and ρ⁢(x). We consider some general properties of LD⁡(x) and LD⁡(M) and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD⁡(M) is rational and there is a nonunit element x∈M with LD⁡(M)=LD⁡(x) (such a monoid is said to have accepted length density). While it is well known that the much studied asymptotic versions of L⁢(x), ℓ⁢(x), and ρ⁢(x) (denoted L¯⁢(x), ℓ¯⁢(x), and ρ¯⁢(x)) always exist, we show the somewhat surprising result that LD¯⁢(x)=limn→∞⁡LD⁡(xn) may not exist. We also give some finiteness conditions on M that force the existence of LD¯⁢(x).

Keywords: Non-unique factorization; length density; elasticity of factorization; tame degree; catenary degree
MSC 2010: 13F15; 20M14; 11R27

Communicated by Manfred Droste

Acknowledgements

The authors wish to thank the referee for many comments and suggestions that greatly improved our paper.

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Received: 2020-10-13
Revised: 2021-11-10
Published Online: 2021-12-01
Published in Print: 2022-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
https://doi.org/10.1515/forum-2021-0149
Keywords for this article
Non-unique factorization; length density; elasticity of factorization; tame degree; catenary degree

Articles in the same Issue

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
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