Additive factorizations in finitely generated semidomains

Abstract

For finite $S \subseteq \mathbb{C}$, we study additive factorizations in $\mathbb{N}_0[S]$ and their behavior under adjoining generators. We construct rational and quadratic examples with infinitely many atoms but without atomicity, including a quadratic example in which every pure power of each generator is an atom. Adjoining generators algebraically independent over $\mathbb{Q}(S)$ preserves atomicity and the standard factorization properties, while unique factorization ascends from the individual monoids under joint linear disjointness for positive algebraic generators. Using the theorem of Evertse, Schlickewei, and Schmidt, we prove that positive real generators give finitely many factorizations of each prescribed length. Thus, bounded and finite factorization are equivalent, and half-factoriality implies finite factorization. We exhibit a set of two quadratic generators that separates half-factoriality from unique factorization and give an augmentation criterion for half-factoriality when every monomial is an atom. Lastly, we characterize complete integral closure for positive rational generators.