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

• Autorzy: Scott T. Chapman , Felix Gotti , Roberto Pelayo
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 20M14: Commutative semigroups
• 20M13: Arithmetic theory of monoids
• 13F05: Dedekind, Pr\"ufer, Krull and Mori rings and their generalizations
• 11R27: Units and factorization

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2014
• Tom: 137
• Numer: 1
• Strony: 137-146

Daty

wydano

2014

Twórcy

autor

Scott T. Chapman

• Department of Mathematics, Sam Houston State University, Box 2206, Huntsville, TX 77341, U.S.A.

autor

Felix Gotti

• Department of Mathematics, University of Florida, Gainesville, FL 32611, U.S.A.

autor

Roberto Pelayo

• Mathematics Department, University of Hawai`i at Hilo, Hilo, HI 96720, U.S.A.

Identyfikatory

DOI

10.4064/cm137-1-10