Currently displaying 1 – 4 of 4

Showing per page

Order by Relevance | Title | Year of publication

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

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

Journal de Théorie des Nombres de Bordeaux

If $a$ and $b$ are positive integers with $a\le b$ and ${a}^{2}\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}b$, then the set ${M}_{a,b}=\left\{x\in ℕ:x\equiv a\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}b\phantom{\rule{4pt}{0ex}}\text{or}\phantom{\rule{4pt}{0ex}}x=1\right\}$ is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units ${M}^{×}$ and any $x\in M\setminus {M}^{×}$ we say that $t\in ℕ$ is a factorization length of $x$ if and only if there exist irreducible elements ${y}_{1},...,{y}_{t}$ of $M$ and $x={y}_{1}\cdots {y}_{t}$. Let $ℒ\left(x\right)=\left\{{t}_{1},...,{t}_{j}\right\}$ 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 $ℒ\left(x\right)$: $\Delta \left(x\right)=\left\{{t}_{i+1}-{t}_{i}:1\le i<k\right\}$ and the Delta-set of the monoid $M$...

Page 1