Local tameness of \(v\)-Noetherian monoids. (Q2477651)

The authors show that if \(H\) is a \(v\)-Noetherian monoid, then for every \(a\in H\) there exists a positive integer \(\omega(a,H)\) such that if \(a\) divides a product \(b_1\cdots b_n\) of elements of \(H\), then for some \(k\leq\omega(a,H)\) \(a\) divides the product of \(k\) \(b_i\)'s. In particular this applies to the multiplicative semigroup of non-zero elements of a Noetherian domain. In the case, when the conductor \((H:\widehat H)\) (\(\widehat H\) being the complete integral closure of \(H\)) is non-empty, an explicit upper bound for \(\omega(a,H)\) is given. It is proved (Th. 3.6) that one can characterize atomic locally tame monoids using \(\omega(a,H)\). Finally it is shown that if \(\alpha>1\) is an irrational number, then in the monoid \[ H_\alpha=\{(x,y)\in\mathbb{N}^2:y<\alpha x\}\cup\{(0,0)\}, \] with component-wise addition one has \(\omega(a,H_\alpha)=\infty\) for all \(a\in H_\alpha\).