On the arithmetic of tame monoids with applications to Krull monoids and Mori domains. (Q989074)

Let \((H,\cdot)\) be an atomic monoid, i.e., a commutative, cancellative semigroup with unit element such that each non-invertible element is the product of irreducible elements. The multiplicative monoids of Noetherian domains are atomic monoids. Various aspects of the arithmetic of atomic monoids that are tame are studied. An atomic monoid is tame if there exists some \(\omega(H)\in\mathbb{N}\) such that whenever \(b,a_1,\dots,a_n\) are irreducibles such that \(b\mid a_1\cdots a_n\) then there exists a subset \(I\subset\{1,\dots n\}\) with \(|I|\leq\omega(H)\) such that \(b\mid\prod_{i\in I}a_i\). This is not the original definition of tameness, and \(\omega(H)\) is not the tame degree of \(H\); yet, this is a novel characterization of tameness obtained in the paper under review (as it is somewhat lengthy, we omit the original definition of tameness). Examples of tame monoids are, e.g., finitely generated monoids and Krull monoids with finite class group. In particular, it is proved that the Structure Theorem of Sets of Lengths holds for tame monoids. When restricted to Krull monoids, this is a generalization of a recent result of \textit{A. Geroldinger} and \textit{D. J. Grynkiewicz} [J. Algebra 321, No. 4, 1256-1284 (2009; Zbl 1196.20066)] where this was proved under the condition that the Davenport constant of the associated block monoid is finite. Indeed, in the present paper a Krull monoid is constructed such that its tame degree is only \(2\), yet the Davenport constant of the associated block monoid is infinite.