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

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Chapman S., Gotti F., Pelayo R.

Colloquium Mathematicum

2014

tom 137

nr 1

137-146

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Chapman S.

1 Gotti F.

1 Pelayo R.

1 2014

Wyniki wyszukiwania

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