Factorizations in evaluation monoids of Laurent semirings

TL;DR

This paper investigates the factorization properties of a semiring constructed from Laurent polynomials over nonnegative integers evaluated at a positive real number, providing characterizations of atomicity, ACCP, UFP, and elasticity.

Contribution

It offers new characterizations of factorization properties in semirings derived from Laurent polynomial semirings evaluated at real numbers.

Findings

Characterized when the semiring is atomic.

Determined conditions for the ascending chain condition on principal ideals.

Identified when the semiring has unique factorization and when it has infinite elasticity.

Abstract

For a positive real number $\alpha$, let $\mathbb{N}_0[\alpha,\alpha^{-1}]$ be the semiring of all real numbers $f(\alpha)$ for $f(x)$ lying in $\mathbb{N}_0[x,x^{-1}]$, which is the semiring of all Laurent polynomials over the set of nonnegative integers $\mathbb{N}_0$. In this paper, we study various factorization properties of the additive structure of $\mathbb{N}_0[\alpha, \alpha^{-1}]$. We characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ is atomic. Then we characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ satisfies the ascending chain condition on principal ideals in terms of certain well-studied factorization properties. Finally, we characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ satisfies the unique factorization property and show that, when this is not the case, $\mathbb{N}_0[\alpha, \alpha^{-1}]$ has infinite elasticity.