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

Scott T. Chapman ¹   ; Felix Gotti ²   ; Roberto Pelayo ³

¹ Department of Mathematics Sam Houston State University Box 2206 Huntsville, TX 77341, U.S.A.
² Department of Mathematics University of Florida Gainesville, FL 32611, U.S.A.
³ Mathematics Department University of Hawai`i at Hilo Hilo, HI 96720, U.S.A.

Colloquium Mathematicum, Tome 137 (2014) no. 1, pp. 137-146

Résumé

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 $|\varDelta (x)| \ge m$.

Détail

DOI : 10.4064/cm137-1-10

Keywords: commutative cancellative monoid set vardelta which consists positive integers which distances between consecutive factorization lengths elements widely studied object theory nonunique factorizations krull monoid cyclic class group order well known vardelta subseteq ldots n moreover equality holds containment each class contains prime divisor note consider question determining which subsets ldots n occur delta set individual element first prove n vardelta vardelta n subsets ldots n realized delta sets individual elements close proving archimedean type property delta sets krull monoids finite cyclic class group every natural number there exist krull monoid finite cyclic class group has element vardelta