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- W2330356667 abstract "Let $M$ be a commutative cancellative monoid. The set $Delta(M)$, 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 $Delta(M) subseteq {1, dots, n-2}$. 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 ${1, dots, n-2}$ occur as the delta set of an individual element from $M$. We first prove for $x in M$ that if $n - 2 in Delta(x)$, then $Delta(x) = {n-2}$ (i.e., not all subsets of ${1,dots, n-2}$ 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 $|Delta(x)| ge m$." @default.
- W2330356667 created "2016-06-24" @default.
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- W2330356667 date "2016-09-09" @default.
- W2330356667 modified "2023-09-27" @default.
- W2330356667 title "On Delta Sets and their Realizable Subsets in Krull Monoids with Cyclic Class Groups" @default.
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