Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element bεH, let σ(H,b) be the smallest NεN0u∞ having the following property: if nεN and a1,...,anεH are such that b divides a1...........an, then b already divides a subproduct of a1...........an consisting of at most N factors. The monoid H is called tame if supωσ;(H,u)|uis an atom ofH∞. This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.
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