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

Volume 137 / 2014

Scott T. Chapman, Felix Gotti, Roberto Pelayo Colloquium Mathematicum 137 (2014), 137-146 MSC: Primary 20M13; Secondary 20M14, 11R27, 13F05. DOI: 10.4064/cm137-1-10

Abstract

Let $M$ be a commutative cancellative monoid. The set $\varDelta (M)$, 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 $n \ge 3$, then it is well-known that $\varDelta (M)\subseteq \{1, \ldots , n-2\}$. 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 $\{1, \ldots , n-2\}$ occur as the delta set of an individual element from $M$. We first prove for $x\in M$ that if $n-2\in \varDelta (x)$, then $\varDelta (x)=\{n-2\}$ (i.e., not all subsets of $\{1, \ldots , n-2\}$ 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 $|\varDelta (x)| \ge m$.

Authors

  • Scott T. ChapmanDepartment of Mathematics
    Sam Houston State University
    Box 2206
    Huntsville, TX 77341, U.S.A.
    e-mail
  • Felix GottiDepartment of Mathematics
    University of Florida
    Gainesville, FL 32611, U.S.A.
    e-mail
  • Roberto PelayoMathematics Department
    University of Hawai`i at Hilo
    Hilo, HI 96720, U.S.A.
    e-mail

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