Journal of Commutative Algebra

The elasticity of Puiseux monoids

Felix Gotti and Christopher O’Neill

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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).

Article information

J. Commut. Algebra, Volume 12, Number 3 (2020), 319-331.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20M14: Commutative semigroups 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05]

numerical monoid Puiseux monoid factorization


Gotti, Felix; O’Neill, Christopher. The elasticity of Puiseux monoids. J. Commut. Algebra 12 (2020), no. 3, 319--331. doi:10.1216/jca.2020.12.319.

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