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

TL;DR

This paper investigates the atomic and strong atomic structures of additive monoids generated by algebraic numbers, exploring which pairs of such structures can be realized within these monoids.

Contribution

It introduces the concept of realizable pairs of atoms and strong atoms in additive monoids of algebraic numbers, aiming to classify all such pairs.

Findings

Defined realizable pairs of atoms and strong atoms.

Connected atomic structures with algebraic number theory.

Aimed to characterize all possible realizable pairs.

Abstract

Let $M$ be a cancellative and commutative monoid. A non-invertible element of $M$ is called an atom (or irreducible element) if it cannot be factored into two non-invertible elements, while an atom $a$ of $M$ is called strong if $a^n$ has a unique factorization in $M$ for every $n \in \mathbb{N}$. The monoid $M$ is atomic if every non-invertible element factors into finitely many atoms (repetitions allowed). For an algebraic number $\alpha$, we let $M_\alpha$ denote the additive monoid of the subsemiring $\mathbb{N}_0[\alpha]$ of $\mathbb{C}$. The atomic structure of $M_\alpha$ reflects intricate interactions between algebraic number theory and additive semigroup theory. For $m, n \in \mathbb{N}_0 \cup \{ \infty \}$ (with $m \le n$), the pair $(m,n)$ is called realizable if there exists an algebraic number $\alpha \in \mathbb{C}$ such that $M_\alpha$ has $m$ strong atoms and $n$ atoms. Our primary goal is to identify classes of realizable pairs with the long-term goal of providing a complete description of the full set of realizable pairs.