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

Abstract

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).

14 pages