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