On conductor submonoids of factorial monoids
Alfred Geroldinger, Weihao Yan, Qinghai Zhong
TL;DR
This work investigates submonoids of factorial monoids whose non-invertible elements lie in the conductor, introducing conductor submonoids as a unifying framework that generalizes ideal extension and gap absorbing notions. It establishes that a submonoid is conductor if and only if it is gap absorbing, resolving key conjectures and revealing connections to local domains and ideals. The authors derive sharp arithmetic invariants for conductor submonoids, notably showing a universal bound $\mathsf c(H)\le 3$ on catenary degree and describing length sets as intervals; they also develop transfer-theoretic methods to study conductor submonoids associated with Krull monoids, obtaining detailed behavior in both finite and infinite class group cases. The results advance structural and arithmetic understanding of submonoids within factorial contexts and yield concrete applications to rings and monoids of ideals, with implications for factorization theory in non-factorial settings.
Abstract
We study algebraic and arithmetic properties of submonoids (resp. subrings) of factorial monoids (resp. factorial domains) whose non-invertible elements all lie in the conductor. This continues earlier work of Baeth, Cisto, et al.. On our way we answer several conjectures, formulated in their papers in the affirmative ([1,Conjecture 4.16] and [6, Conjectures 2.3 and 2.10, and Section 9]).