Let M be a commutative cancellative monoid. The set Δ(M), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M, is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that Δ(M) ⊆ {1,..., n-2}. Moreover, equality holds for this containment when each class contains a prime divisor from M. In this note, we consider the question of determining which subsets of {1,..., n-2} occur as the delta set of an individual element from M. We first prove for x ∈ M that if n-2 ∈ Δ(x), then Δ(x) = {n-2} (i.e., not all subsets of {1, ..., n-2} can be realized as delta sets of individual elements). We close by proving an Archimedean-type property for delta sets from Krull monoids with finite cyclic class group: for every natural number m, there exist a Krull monoid M with finite cyclic class group such that M has an element x with |Δ(x)| ≥ m.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-10,
author = {Scott T. Chapman and Felix Gotti and Roberto Pelayo},
title = {On delta sets and their realizable subsets in Krull monoids with cyclic class groups},
journal = {Colloquium Mathematicae},
volume = {135},
year = {2014},
pages = {137-146},
zbl = {1323.20056},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-10}
}
Scott T. Chapman; Felix Gotti; Roberto Pelayo. On delta sets and their realizable subsets in Krull monoids with cyclic class groups. Colloquium Mathematicae, Tome 135 (2014) pp. 137-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-10/