Paper detail

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

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