Erdős-Zaks all divisor sets

Abstract

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=1n−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=1n−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.