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 [Formula: see text], the additive monoid [Formula: see text] of the evaluation semiring [Formula: see text] is atomic. The atomic structure of both the additive and the multiplicative monoids of [Formula: see text] has been the subject of several recent papers. Here we focus on the monoids [Formula: see text], and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when [Formula: see text] is less than [Formula: see text], the atoms of [Formula: see text] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of [Formula: see text], including that when [Formula: see text] is rational, the elasticity of [Formula: see text] is full (this was previously conjectured by Chapman, Gotti and Gotti).