Erdős-Zaks all divisor sets

Let Zn be the finite cyclic group of order n and S ⊆ Zn. We examine the factorization properties of the Block Monoid B(Zn, S) when S is constructed using a method inspired by a 1990 paper of Erdős and Zaks. For such a set S, we develop an algorithm in Section 2 to produce and order a set {Mi} i=1 which contains all the non-primary irreducible Blocks (or atoms) of B(Zn, S). This construction yields a weakly half-factorial Block Monoid (see [9]). After developing some basic properties of the set {Mi} i=1 , we examine in Section 3 the connection between these irreducible blocks and the Erdős-Zaks notion of “splittable sets.” In particular, the Erdős-Zaks notion of “irreducible” does not match the classic notion of “irreducible” for the commutative cancellative monoids B(Zn, S). We close in Sections 4 and 5 with a detailed discussion of the special properties of the blocks M1 with an emphasis on the case where the exponents of M1 take on extreme values. The work of Section 5 allows us to offer alternate arguments for two of the main results of the original paper by Erdős and Zaks. 1. Background and Introduction The roots of the study of non-unique factorizations lie in questions involving the factorizations of algebraic integers. It was not until the papers [14] and [15] of Zaks that such questions were considered in more general structures such as Dedekind or Krull domains. If D is a Dedekind or Krull domain with divisor class group G, then the pioneering work of Zaks and Skula [10] showed that the behavior of factorizations of elements into products of irreducibles (or atoms) can be determined by the study of the Block Monoid B(G,S) associated to D (where S is the subset of divisor classes of G which contain height-one prime ideals). This observation was the basis of many papers which have appeared in the literature over the past 20 years and led to the close examination of the arithmetical structure of Block Monoids, especially in the case where the abelian group G is finite or torsion. The study of this arithmetic involves both combinatorial group theory and number theory and such investigations led to a joint paper by Zaks and Erdős [4] in 1990 which examines the behavior of what they termed splittable sets of integers. While the atomic structure of a Block Monoid can be complicated, the ideas developed in [4] can be used to produce Block Monoids of the form B(Zn, S) whose atoms can be enumerated and analyzed. We will refer to the sets S derived in this process as Erdős-Zaks All Divisor Sets (or EZADS). In this paper we will study Block Monoids which arise during the EZADS process and the arithmetic of their associated sets of atoms. Before a precise description of our results, we review the necessary notation and definitions. If G is an abelian group with S ⊆ G, then let F(G,S) be the free abelian monoid with basis S. Each element B ∈ F(G,S) has a unique representation of the form B = ∏ g∈S gg where each vg(B) is a nonnegative integer. We define the length of B by | B |= ∑ g∈S vg(B) and the sum of B by 2000 Mathematics Subject Classification. 11D68, 13F15, 20M13.