We study the arithmetic of seminormal v-noetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us to give precise arithmetical results analogous to the well-known results for Krull monoids having finite class group and prime divisors in each class. This allows us to show, for example, that unions of sets of lengths are intervals and to provide a characterization of half-factoriality.2010 Mathematics Subject Classification. 13A05, 13F05, 13F15, 13F45, 20M13.

Abstract. Let H be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every k ∈ N, let U k (H) denote the set of all ℓ ∈ N with the property that there are atomsis the union of all sets of lengths containing k).The Structure Theorem for Unions states that, for all sufficiently large k, the sets U k (H) are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds.This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

Communicated by E.M. Friedlander MSC: 13A05; 13F05; 20M13 a b s t r a c tLet H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b ∈ H, let ω(H, b) be the smallest N ∈ N 0 ∪ {∞} having the following property: if n ∈ N and a 1 , . . . , a n ∈ H are such that b divides a 1 · . . . · a n , then b already divides a subproduct of a 1 · . . . · a n consisting of at most N factors. The monoid H is called tame if sup{ω(H, u) | u is an atom of H} < ∞. 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.

Abstract. Let H be a Krull monoid with infinite class group and such that each divisor class of H contains a prime divisor. We show that for each finite set L of integers ≥ 2 there exists some h ∈ H such that the following are equivalent:

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