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 α, the additive monoid M α of the evaluation semiring N 0 [α] is atomic. The atomic structure of both the additive and the multiplicative monoids of 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 N 0 [α] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of N 0 [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).