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

Scott T. ChapmanFelix GottiRoberto Pelayo — 2014

Colloquium Mathematicae

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...

On the Delta set of a singular arithmetical congruence monoid

Paul BaginskiScott T. ChapmanGeorge J. Schaeffer — 2008

Journal de Théorie des Nombres de Bordeaux

If a and b are positive integers with a b and a 2 a mod b , then the set M a , b = { x : x a mod b or x = 1 } is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid M with units M × and any x M M × we say that t is a factorization length of x if and only if there exist irreducible elements y 1 , ... , y t of M and x = y 1 y t . Let ( x ) = { t 1 , ... , t j } be the set of all such lengths (where t i < t i + 1 whenever i < j ). The Delta-set of the element x is defined as the set of gaps in ( x ) : Δ ( x ) = { t i + 1 - t i : 1 i < k } and the Delta-set of the monoid M ...

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