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A cancellative commutative monoid is atomic if every non-invertible elementfactors into irreducibles. Under certain mild conditions on a positivealgebraic number $\alpha$, the additive monoid $M_\alpha$ of the evaluationsemiring $\mathbb{N}_0[\alpha]$ is atomic. The atomic structure of both theadditive and the multiplicative monoids of $\mathbb{N}_0[\alpha]$ has been thesubject of several recent papers. Here we focus on the monoids $M_\alpha$, andwe study its omega-primality and elasticity, aiming to better understand somefundamental 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 asthey 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).

On the primality and elasticity of algebraic valuations of cyclic free semirings