On the additive structure of algebraic valuations of polynomial semirings

Abstract

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 length-factorial are also investigated along the way.

20 pages

References in corpus (4)

• On strongly primary monoids, with a focus on Puiseux monoids
• Factorizations in upper triangular matrices over information semialgebras
• Factorizations of polynomials with integral non-negative coefficients
• A characterization of length-factorial Krull monoids

Cited by in corpus (1)

• On the set of atoms and strong atoms in additive monoids of cyclic semidomains