Longest and shortest factorizations in embedding dimension three

Liu, Baian; Yap, JiaYan

doi:10.2140/involve.2023.16.673

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Longest and shortest factorizations in embedding dimension three

Baian Liu and JiaYan Yap

Vol. 16 (2023), No. 4, 673–688

DOI: 10.2140/involve.2023.16.673

Abstract

For a numerical monoid $⟨ n_{1}, \dots, n_{k} ⟩$ minimally generated by $n_{1}, \dots, n_{k} \in ℕ$ , with $n_{1} < \dots < n_{k}$ , the longest and shortest factorization lengths of an element $x$ , denoted as $L (x)$ and $ℓ (x)$ , respectively, follow the identities $L (x + n_{1}) = L (x) + 1$ and $ℓ (x + n_{k}) = ℓ (x) + 1$ for sufficiently large elements $x$ . We characterize when these identities hold for all elements of numerical monoids of embedding dimension three.

Keywords

numerical monoid, factorization length, Betti element

Mathematical Subject Classification

Primary: 11D75, 20M13, 20M14

Milestones

Received: 15 June 2022

Revised: 24 September 2022

Accepted: 10 October 2022

Published: 31 October 2023

Communicated by Scott T. Chapman

Authors

Baian Liu
Department of Mathematics The Ohio State University Columbus, OH United States

JiaYan Yap
Department of Mathematics The Ohio State University Columbus, OH United States

Vol. 16, No. 4, 2023