## Journal of Commutative Algebra

### The elasticity of Puiseux monoids

#### 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 $ℚ≥0$). 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

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

Dates
Revised: 1 December 2017
Accepted: 17 December 2017
First available in Project Euclid: 5 September 2020

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

#### 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

#### References

• J. Amos, S. T. Chapman, N. Hine, and J. a. Paixão, “Sets of lengths do not characterize numerical monoids”, Integers 7 (2007), art.,id.,A50.
• P. Baginski, S. T. Chapman, C. Crutchfield, K. G. Kennedy, and M. Wright, “Elastic properties and prime elements”, Results Math. 49:3-4 (2006), 187–200.
• S. T. Chapman, M. T. Holden, and T. A. Moore, “Full elasticity in atomic monoids and integral domains”, Rocky Mountain J. Math. 36:5 (2006), 1437–1455.
• S. T. Chapman, F. Gotti, and R. Pelayo, “On delta sets and their realizable subsets in Krull monoids with cyclic class groups”, Colloq. Math. 137:1 (2014), 137–146.
• S. Colton and N. Kaplan, “The realization problem for delta sets of numerical semigroups”, J. Commut. Algebra 9:3 (2017), 313–339.
• A. Geroldinger and F. Halter-Koch, Non-unique factorizations: algebraic, combinatorial and analytic theory, Pure and Applied Mathematics 278, Chapman & Hall/CRC, Boca Raton, FL, 2006.
• A. Geroldinger and W. A. Schmid, “A realization theorem for sets of distances”, J. Algebra 481 (2017), 188–198.
• F. Gotti, “On the atomic structure of Puiseux monoids”, J. Algebra Appl. 16:7 (2017), art.,id.,1750126.
• F. Gotti and M. Gotti, “Atomicity and boundedness of monotone Puiseux monoids”, Semigroup Forum 96:3 (2018), 536–552.
• P. A. Grillet, Commutative semigroups, Advances in Mathematics 2, Kluwer, Dordrecht, 2001.