Winter 2022 On atomic density of numerical semigroup algebras
Austin A. Antoniou, Ranthony A. C. Edmonds, Bethany Kubik, Christopher O’Neill, Shannon Talbott
J. Commut. Algebra 14(4): 455-470 (Winter 2022). DOI: 10.1216/jca.2022.14.455


A numerical semigroup S is a cofinite, additively closed subset of the nonnegative integers that contains 0. We initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring 𝔽q[x] is zero for any finite field 𝔽q; we prove that the numerical semigroup algebra 𝔽q[S] also has atomic density zero for any numerical semigroup S. We also examine the particular algebra 𝔽2[x2,x3] in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field 𝔽q.


Download Citation

Austin A. Antoniou. Ranthony A. C. Edmonds. Bethany Kubik. Christopher O’Neill. Shannon Talbott. "On atomic density of numerical semigroup algebras." J. Commut. Algebra 14 (4) 455 - 470, Winter 2022.


Received: 21 August 2020; Revised: 3 March 2021; Accepted: 5 March 2021; Published: Winter 2022
First available in Project Euclid: 15 November 2022

MathSciNet: MR4509402
zbMATH: 1516.13001
Digital Object Identifier: 10.1216/jca.2022.14.455

Primary: 12E05 , 13A05 , 20M14

Keywords: atomic density , finite field , numerical semigroup

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


This article is only available to subscribers.
It is not available for individual sale.

Vol.14 • No. 4 • Winter 2022
Back to Top