Factorizations in rational monogenic semidomains

Abstract

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.