Local tameness of vv-noetherian monoids

Let HH be a vv-noetherian monoid, e.g., the multiplicative monoid R∖{0}R∖{0} of a noetherian domain RR. We show that, for every b∈Hb∈H, there exists a constant ω(H,b)∈N0ω(H,b)∈N0 having the following property: If n∈Nn∈N and a1,…,an∈Ha1,…,an∈H such that bb divides the product a1⋅…⋅ana1⋅…⋅an, then bb already divides a subproduct of a1⋅…⋅ana1⋅…⋅an consisting of at most ω(H,b)ω(H,b) factors. Using the ω(H,⋅)ω(H,⋅)-quantities we derive a new characterization of local tameness–a crucial finiteness property in the theory of non-unique factorizations.