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Fall 2020 The elasticity of Puiseux monoids
Felix Gotti, Christopher O’Neill
J. Commut. Algebra 12(3): 319-331 (Fall 2020). DOI: 10.1216/jca.2020.12.319

Abstract

Let M be an atomic monoid and let x be a non-unit element of M. The elasticity of x, denoted by ρ(x), is the ratio of its largest factorization length to its shortest factorization length, and it measures how far x is from having all its factorizations of the same length. The elasticity ρ(M) of M is the supremum of the elasticities of all non-unit elements of M. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of 0). We characterize, in terms of the atoms, which Puiseux monoids M have finite elasticity, giving a formula for ρ(M) in this case. We also classify when ρ(M) is achieved by an element of M. When M is a primary Puiseux monoid (that is, a Puiseux monoid whose atoms have prime denominator), we describe the topology of the set of elasticities of M, including a characterization of when M is a bounded factorization monoid. Lastly, we give an example of a Puiseux monoid that is bifurcus (that is, every nonzero element has a factorization of length at most 2).

Citation

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Felix Gotti. Christopher O’Neill. "The elasticity of Puiseux monoids." J. Commut. Algebra 12 (3) 319 - 331, Fall 2020. https://doi.org/10.1216/jca.2020.12.319

Information

Received: 24 March 2017; Revised: 1 December 2017; Accepted: 17 December 2017; Published: Fall 2020
First available in Project Euclid: 5 September 2020

zbMATH: 07246822
MathSciNet: MR4146363
Digital Object Identifier: 10.1216/jca.2020.12.319

Subjects:
Primary: 13F15 , 20M14

Keywords: factorization , numerical monoid , Puiseux monoid

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.12 • No. 3 • Fall 2020
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