Anna Deng | Searcharxiv

arXiv2026

Factorizations in rational monogenic semidomains

For $\alpha \in \mathbb{C}$, the monogenic semidomain generated by $\alpha$ is the smallest subsemiring $S_\alpha$ of the complex field $\mathbb{C}$ containing $\alpha$. We initiate a systematic study of the arithmetic and factorizations of the monogenic semidomains $S_q$ generated by rational parameters $q$. After some preliminaries, we introduce and investigate the monoid of technical fractions $T_q$, which is a divisor-closed submonoid of the multiplicative monoid of $S_q$ that encodes a significant amount of arithmetic information about $S_q$. We then study several fundamental factorization properties of $S_q$: the bounded factorization (BF) and finite factorization (FF) properties, the unique factorization (UF) property, and the half-factorial (HF) property. First, we prove that $S_q$ satisfies the UF property if and only if it satisfies the HF property, which happens when $q \in \mathbb{N} \cup \mathbb{N}^{-1}$. We determine all the positive rational values of the parameter $q$ for which $S_q$ satisfies the FF property. Then we show that, over the class of rational monogenic semidomains, the BF property is equivalent to the ascending chain condition on principal ideals. Finally, we prove that $S_q$ is a Krull semidomain if and only if it is root-closed, which happens precisely when $S_q$ satisfies the UF property.

math.AC↗

arXiv2025

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

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.

math.AC↗