Local tameness of v-noetherian monoids

Let H be a v-noetherian monoid, e.g., the multiplicative monoid Rset minus{0} of a noetherian domain R. We show that, for every bset membership, variantH, there exists a constant image having the following property: If image and a1,…,anset membership, variantH such that b divides the product a1dot operator…dot operatoran, then b already divides a subproduct of a1dot operator…dot operatoran consisting of at most ω(H,b) factors. Using the ω(H,dot operator)-quantities we derive a new characterization of local tameness–a crucial finiteness property in the theory of non-unique factorizations.