  A cancellative commutative monoid is atomic if every non-invertible element
factors into irreducibles. Under certain mild conditions on a positive
algebraic number $\alpha$, the additive monoid $M_\alpha$ of the evaluation
semiring $\mathbb{N}_0[\alpha]$ is atomic. The atomic structure of both the
additive and the multiplicative monoids of $\mathbb{N}_0[\alpha]$ has been the
subject of several recent papers. Here we focus on the monoids $M_\alpha$, and
we study its omega-primality and elasticity, aiming to better understand some
fundamental questions about their atomic decompositions. We prove that when
$\alpha$ is less than 1, the atoms of $M_\alpha$ are as far from being prime as
they can possibly be. Then we establish some results about the elasticity of
$M_\alpha$, including that when $\alpha$ is rational, the elasticity of
$M_\alpha$ is full (this was previously conjectured by S. T. Chapman, F. Gotti,
and M. Gotti).

Commutative Algebra