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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. [ABSTRACT FROM AUTHOR]

Abstract: Let be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element , let be the smallest   having the following property: if   and   are such that divides  , then already divides a subproduct of consisting of at most factors. The monoid is called tame if . 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  such that the set of lengths of any element is an almost arithmetical multiprogression with bound . [Copyright &y& Elsevier]

Presents geometric notes on monoids. Proposition; Corollary; Proof.

Let 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 , let denote the set of all with the property that there are atoms such that (thus, is the union of all sets of lengths containing ). The Structure Theorem for Unions states that, for all sufficiently large , the sets 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. [ABSTRACT FROM AUTHOR]