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

TL;DR

This paper investigates the atomic and elastic properties of additive monoids derived from algebraic valuations of cyclic free semirings, revealing that atoms are maximally non-prime when the algebraic number is less than one and confirming full elasticity for rational cases.

Contribution

It provides new insights into the primality and elasticity of algebraic valuation monoids, especially proving full elasticity for rational algebraic numbers.

Findings

Atoms are maximally non-prime when bc<1.

Elasticity of M_bc is full for rational bc.

Results deepen understanding of atomic decompositions in algebraic valuation monoids.

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).