On delta sets and their realizable subsets in Krull monoids with cyclic class groups

Scott T. Chapman, Felix Gotti, and Roberto Pelayo. "On delta sets and their realizable subsets in Krull monoids with cyclic class groups." Colloquium Mathematicae 137.1 (2014): 137-146. <http://eudml.org/doc/284011>.

@article{ScottT2014,
abstract = {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.},
author = {Scott T. Chapman, Felix Gotti, Roberto Pelayo},
journal = {Colloquium Mathematicae},
keywords = {commutative cancellative monoids; factorization lengths; nonunique factorizations; Krull monoids; block monoids; delta sets},
language = {eng},
number = {1},
pages = {137-146},
title = {On delta sets and their realizable subsets in Krull monoids with cyclic class groups},
url = {http://eudml.org/doc/284011},
volume = {137},
year = {2014},
}

TY - JOUR
AU - Scott T. Chapman
AU - Felix Gotti
AU - Roberto Pelayo
TI - On delta sets and their realizable subsets in Krull monoids with cyclic class groups
JO - Colloquium Mathematicae
PY - 2014
VL - 137
IS - 1
SP - 137
EP - 146
AB - 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.
LA - eng
KW - commutative cancellative monoids; factorization lengths; nonunique factorizations; Krull monoids; block monoids; delta sets
UR - http://eudml.org/doc/284011
ER -