Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-23T03:33:17.291Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  20 May 2024

Department of Mathematics, Sam Houston State University, Huntsville, TX 77341, USA
Oxford Academy, Cypress, CA 90630, USA e-mail:
The Academy for Mathematics, Science and Engineering, Rockaway, NJ 07866, USA e-mail:
Saratoga School, Saratoga, CA 95070, USA e-mail:


Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of $b \in M$ is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.

Research Article
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Aguilera, C., Gotti, M. and Hamelberg, A. F., ‘Factorization in reciprocal Puiseux monoids’, Preprint, 2021, arXiv:2112.04048.Google Scholar
Ajran, K., Bringas, J., Li, B., Singer, E. and Tirador, M., ‘Factorization in additive monoids of evaluation polynomial semirings’, Comm. Algebra 51 (2023), 43474362.CrossRefGoogle Scholar
Anderson, D. F. and Gotti, F., ‘Bounded and finite factorization domains’, in: Rings, Monoids and Module Theory, Springer Proceedings in Mathematics and Statistics, 382 (eds. Badawi, A. and Coykendall, J.) (Springer, Singapore, 2022), 757.Google Scholar
Chapman, S. T., ‘A tale of two monoids: a friendly introduction to the theory of non-unique factorizations’, Math. Mag. 87 (2014), 163173.CrossRefGoogle Scholar
Chapman, S. T., Coykendall, J., Gotti, F. and Smith, W. W., ‘Length-factoriality in commutative monoids and integral domains’, J. Algebra 578 (2021), 186212.CrossRefGoogle Scholar
Chapman, S. T., García-Sánchez, P., Llena, D., Ponomarenko, V. and Rosales, J., ‘The catenary and tame degree in finitely generated commutative cancellative monoids’, Manuscripta Math. 120 (2006), 253264.CrossRefGoogle Scholar
Chapman, S. T., García-Sánchez, P. A., Llena, D., Malyshev, A. and Steinberg, D., ‘On the delta set and the Betti elements of a BF-monoid’, Arab. J. Math. 1 (2012), 5361.CrossRefGoogle Scholar
Chapman, S. T., Gotti, F. and Gotti, M., ‘When is a Puiseux monoid atomic?’, Amer. Math. Monthly 128 (2021), 302321.CrossRefGoogle Scholar
Correa-Morris, J. and Gotti, F., ‘On the additive structure of algebraic valuations of polynomial semirings’, J. Pure Appl. Algebra 226 (2022), Article no. 107104.CrossRefGoogle Scholar
Coykendall, J., Dobbs, D. E. and Mullins, B., ‘On integral domains with no atoms’, Comm. Algebra 27 (1999), 58135831.CrossRefGoogle Scholar
Coykendall, J. and Zafrullah, M., ‘AP-domains and unique factorization’, J. Pure Appl. Algebra 189 (2004), 2735.CrossRefGoogle Scholar
García-Sánchez, P., Ojeda, I. and Rosales, J., ‘Affine semigroups having a unique Betti element’, J. Algebra Appl. 12 (2013), Article no. 1250177.CrossRefGoogle Scholar
García-Sánchez, P. A. and Ojeda, I., ‘Uniquely presented finitely generated commutative monoids’, Pacific J. Math. 248 (2010), 91105.CrossRefGoogle Scholar
García-Sánchez, P. A. and Rosales, J. C., Numerical Semigroups, Developments in Mathematics, 20 (Springer-Verlag, Dordrecht, 2009).Google Scholar
Geroldinger, A., Gotti, F. and Tringali, S., ‘On strongly primary monoids, with a focus on Puiseux monoids’, J. Algebra 567 (2021), 310345.CrossRefGoogle Scholar
Geroldinger, A. and Halter-Koch, F., Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, 278 (Chapman and Hall/CRC, Boca Raton, FL, 2006).CrossRefGoogle Scholar
Geroldinger, A. and Schmid, W. A., ‘A realization theorem for sets of lengths in numerical monoids’, Forum Math. 30 (2018), 11111118.CrossRefGoogle Scholar
Geroldinger, A. and Zhong, Q., ‘Factorization theory in commutative monoids’, Semigroup Forum 100 (2020), 2251.CrossRefGoogle Scholar
Gotti, F., ‘On the atomic structure of Puiseux monoids’, J. Algebra Appl. 16 (2017), Article no. 1750126.CrossRefGoogle Scholar
Gotti, F., ‘Increasing positive monoids of ordered fields are FF-monoids’, J. Algebra 518 (2019). 4056.CrossRefGoogle Scholar
Gotti, F., ‘Systems of sets of lengths of Puiseux monoids’, J. Pure Appl. Algebra 223 (2019), 18561868.CrossRefGoogle Scholar
Gotti, F., ‘Irreducibility and factorizations in monoid rings’, in: Numerical Semigroups, Springer INdAM Series, 40 (eds. Barucci, V., Chapman, S. T., D’Anna, M. and Fröberg, R.) (Springer, Cham, 2020), 129139.CrossRefGoogle Scholar
Gotti, F., ‘On semigroup algebras with rational exponents’, Comm. Algebra 50 (2022), 318.CrossRefGoogle Scholar
Gotti, F. and Gotti, M., ‘Atomicity and boundedness of monotone Puiseux monoids’, Semigroup Forum 96 (2018), 536552.CrossRefGoogle Scholar
Gotti, F. and Gotti, M., ‘On the molecules of numerical semigroups, Puiseux monoids, and monoid algebras’, in: Numerical Semigroups, Springer INdAM Series, 40 (eds. Barucci, V., Chapman, S. T., D’Anna, M. and Fröberg, R.) (Springer, Cham, 2020), 141161.CrossRefGoogle Scholar
Gotti, F. and Li, B., ‘Atomic semigroup rings and the ascending chain condition on principal ideals’, Proc. Amer. Math. Soc. 151 (2023), 22912302.Google Scholar
Grams, A., ‘Atomic rings and the ascending chain condition for principal ideals’, Proc. Cambridge Philos. Soc. 75 (1974), 321329.CrossRefGoogle Scholar
Halter-Koch, F., ‘Finiteness theorems for factorizations’, Semigroup Forum 44 (1992), 112117.CrossRefGoogle Scholar
Miller, E. and Sturmfels, B., Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227 (Springer-Verlag, New York, 2004).Google Scholar