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Longest and shortest factorizations in embedding dimension three

Baian Liu and JiaYan Yap

Vol. 16 (2023), No. 4, 673–688
Abstract

For a numerical monoid n1,,nk minimally generated by n1,,nk , with n1 < < nk, the longest and shortest factorization lengths of an element x, denoted as L(x) and (x), respectively, follow the identities L(x + n1) = L(x) + 1 and (x + nk) = (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