On the additive structure of algebraic valuations of cyclic free semirings

In this paper, we study factorizations in the additive monoids of positive algebraic valuations $mathbb{N}_0[alpha]$ of the semiring of polynomials $mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when $mathbb{N}_0[alpha]$ is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid $mathbb{N}_0[alpha]$, the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for $mathbb{N}_0[alpha]$ to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of $alpha$. The properties of being finitely generated, half-factorial, and other-half-factorial are also investigated along the way.