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2014 | 137 | 1 | 137-146
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On delta sets and their realizable subsets in Krull monoids with cyclic class groups

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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.
Słowa kluczowe
  • Department of Mathematics, Sam Houston State University, Box 2206, Huntsville, TX 77341, U.S.A.
  • Department of Mathematics, University of Florida, Gainesville, FL 32611, U.S.A.
  • Mathematics Department, University of Hawai`i at Hilo, Hilo, HI 96720, U.S.A.
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