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ON THE SET OF BETTI ELEMENTS OF A PUISEUX MONOID

Published online by Cambridge University Press:  20 May 2024

SCOTT T. CHAPMAN*
Affiliation:
Department of Mathematics, Sam Houston State University, Huntsville, TX 77341, USA
JOSHUA JANG
Affiliation:
Oxford Academy, Cypress, CA 90630, USA e-mail: joshdream01@gmail.com
JASON MAO
Affiliation:
The Academy for Mathematics, Science and Engineering, Rockaway, NJ 07866, USA e-mail: jmao142857@gmail.com
SKYLER MAO
Affiliation:
Saratoga School, Saratoga, CA 95070, USA e-mail: skylermao@gmail.com

Abstract

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.

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

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