This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

Abstract. Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths. In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid.

Let H be a Krull monoid with finite class group G. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set L(a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ N such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ ∆ * (H), where ∆ * (H) denotes the set of minimal distances of H. We show that max ∆ * (H) ≤ max{exp(G) − 2, r(G) − 1} and that equality holds if every class of G contains a prime divisor, which holds true for holomorphy rings in global fields.