Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let sans-serifD(G) be the Davenport constant of G. Then a product of two atoms of H can be written as a product of at most sans-serifD(G) atoms. We study this extremal case and consider the set scriptU{2,D(G)}(H) defined as the set of all l∈double-struckN with the following property: there are two atoms u,v∈H such that uv can be written as a product of l atoms as well as a product of sans-serifD(G) atoms. If G is cyclic, then scriptU{2,D(G)}(H)={2,D(G)}. If G has rank two, then we show that (apart from some exceptional cases) scriptU{2,D(G)}(H)=[2,D(G)]∖{3}. This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two. As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank 2 prime power order group uniquely characterizes the group.

a b s t r a c t Let M be a Krull monoid with divisor class group Z, and let S ⊆ Z denote the set of divisor classes of M which contain prime divisors. We find conditions on S equivalent to the finiteness of both ∆(M), the Delta set of M, and c(M), the catenary degree of M. In the finite case, we obtain explicit upper bounds on max ∆(M) and c(M). Our methods generalize and complement a previous result concerning the elasticity of M.

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Abstract. In a commutative, cancellative, atomic monoid M , the elasticity of a non-unit x is defined to be ρ(x) = L(x)/l(x), where L(x) is the supremum of the lengths of factorizations of x into irreducibles and l(x) is the corresponding infimum. The elasticity ρ(M ) of M is given as the supremum of the elasticities of the nonzero non-units in the domain. We call ρ(M ) accepted if there exists a non-unit x ∈ M with ρ(M ) = ρ(x). In this paper, we show for a monoid M with accepted elasticity thatif M has a prime element. We develop the ideas of taut and flexible elements to study the set {ρ(x) | x a non-unit of M } when M does not possess a prime element. Mathematics Subject Classification (2000). 20M14, 13F20, 13F15.

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