Chains of Factorizations and Factorizations with Successive Lengths

Abstract

ABSTRACT Let H be a commutative cancellative monoid. H is called atomic if every nonunit a ∈ H decomposes (in general in a highly nonunique way) into a product of irreducible elements (atoms) u i of H. The integer n is called the length of (†), and L(a) = {n ∈ ℕ | a decomposes into n irreducible elements of H} is called the set of lengths of a. Two integers k < l are called successive lengths of a if L(a) ∩ {m ∈ ℕ | k ≤ m ≤ l} = {k, l}. For a ∈ H we denote by Z n (a) the set of factorizations of a with length n. Suppose now that H is one of the following monoids: i. A congruence monoid in a Dedekind domain with finite residue fields. ii. H = D\\ {0}, where D is a Noetherian domain having the following properties: is a Krull domain with finite divisor class group, and is a finite ring. iii. H = D\\ {0}, where D is a one-dimensional Noetherian domain with finite normalization and finite Picard group (but possibly infinite residue fields). Let a ∈ H. In the present article, we investigate the structure of concatenating chains in Z n (a) as well as the relation between Z k (a) and Z l (a) if k and l are successive lengths of a. The work continues earlier investigations in Foroutan (2003), Foroutan and Geroldinger (2004), and Hassler (to appear).