Factorizations in evaluation monoids of Laurent semirings

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