## Journal of Commutative Algebra

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

### The elasticity of Puiseux monoids

Felix Gotti and Christopher O’Neill

#### Abstract

Let $M$ be an atomic monoid and let $x$ be a non-unit element of $M$. The elasticity of $x$, denoted by $\rho \left(x\right)$, 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 $\rho \left(M\right)$ 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 ${\mathbb{Q}}_{\ge 0}$). We characterize, in terms of the atoms, which Puiseux monoids $M$ have finite elasticity, giving a formula for $\rho \left(M\right)$ in this case. We also classify when $\rho \left(M\right)$ 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

**Source**

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

**Dates**

Received: 24 March 2017

Revised: 1 December 2017

Accepted: 17 December 2017

First available in Project Euclid: 5 September 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.jca/1599271221

**Digital Object Identifier**

doi:10.1216/jca.2020.12.319

**Mathematical Reviews number (MathSciNet)**

MR4146363

**Zentralblatt MATH identifier**

07246822

**Subjects**

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

**Keywords**

numerical monoid Puiseux monoid factorization

#### Citation

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. https://projecteuclid.org/euclid.jca/1599271221