Factorizations in evaluation monoids of Laurent semirings

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.

16 pages