scholarly journals On the arithmetic of Krull monoids with finite Davenport constant

2009 ◽  
Vol 321 (4) ◽  
pp. 1256-1284 ◽  
Author(s):  
Alfred Geroldinger ◽  
David J. Grynkiewicz
Keyword(s):  
Davenport Constant ◽  
Krull Monoids

scholarly journals THE CATENARY DEGREE OF KRULL MONOIDS II

2014 ◽  
Vol 98 (3) ◽  
pp. 324-354 ◽  
Author(s):  
ALFRED GEROLDINGER ◽  
QINGHAI ZHONG
Keyword(s):  
Class Group ◽  
Algebraic Function ◽  
Algebraic Number Field ◽  
Class Groups ◽  
Davenport Constant ◽  
Algebraic Function Field ◽  
Krull Monoids ◽  
Krull Monoid ◽  
Catenary Degree

Let$H$be a Krull monoid with finite class group$G$such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree$\mathsf{c}(H)$of$H$is the smallest integer$N$with the following property: for each$a\in H$and each pair of factorizations$z,z^{\prime }$of$a$, there exist factorizations$z=z_{0},\dots ,z_{k}=z^{\prime }$of$a$such that, for each$i\in [1,k]$,$z_{i}$arises from$z_{i-1}$by replacing at most$N$atoms from$z_{i-1}$by at most$N$new atoms. To exclude trivial cases, suppose that$|G|\geq 3$. Then the catenary degree depends only on the class group$G$and we have$\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where$\mathsf{D}(G)$denotes the Davenport constant of$G$. The cases when$\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldingeret al.[‘The catenary degree of Krull monoids I’,J. Théor. Nombres Bordeaux23(2011), 137–169], we determine the class groups satisfying$\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of$\mathsf{c}(H)$is known for no further class groups.

scholarly journals Differences in sets of lengths of Krull monoids with finite class group

2005 ◽  
Vol 17 (1) ◽  
pp. 323-345 ◽  
Author(s):  
Wolfgang A. Schmid
Keyword(s):  
Class Group ◽  
Krull Monoids ◽  
Sets Of Lengths

scholarly journals On half-factoriality of transfer Krull monoids

2020 ◽  
Vol 49 (1) ◽  
pp. 409-420
Author(s):  
Weidong Gao ◽  
Chao Liu ◽  
Salvatore Tringali ◽  
Qinghai Zhong
Keyword(s):  
Krull Monoids

scholarly journals Correction to: Davenport constant for semigroups

2020 ◽  
Vol 101 (3) ◽  
pp. 786-794
Author(s):  
Guoqing Wang
Keyword(s):  
Davenport Constant

The Monotone Catenary Degree of Krull Monoids

2012 ◽  
Vol 63 (3-4) ◽  
pp. 999-1031 ◽  
Author(s):  
Alfred Geroldinger ◽  
Pingzhi Yuan
Keyword(s):  
Krull Monoids ◽  
Catenary Degree

THE CATENARY AND TAME DEGREES ON A NUMERICAL MONOID ARE EVENTUALLY PERIODIC

2014 ◽  
Vol 97 (3) ◽  
pp. 289-300 ◽  
Author(s):  
SCOTT T. CHAPMAN ◽  
MARLY CORRALES ◽  
ANDREW MILLER ◽  
CHRIS MILLER ◽  
DHIR PATEL
Keyword(s):  
Periodicity Condition ◽  
Numerical Monoid ◽  
Cancellative Monoid ◽  
Krull Monoids ◽  
Catenary Degree

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.

HIGHER-ORDER CLASS GROUPS AND BLOCK MONOIDS OF KRULL MONOIDS WITH TORSION CLASS GROUP

2010 ◽  
Vol 09 (03) ◽  
pp. 433-464 ◽  
Author(s):  
WOLFGANG A. SCHMID
Keyword(s):  
Class Group ◽  
Point Of View ◽  
Torsion Class ◽  
Class Groups ◽  
Detailed Understanding ◽  
Krull Monoids ◽  
Krull Monoid ◽  
Zero Sum ◽  
Block Monoids ◽  
Block Monoid

Extensions of the notion of a class group and a block monoid associated to a Krull monoid with torsion class group are introduced and investigated. Instead of assigning to a Krull monoid only one abelian group (the class group) and one monoid of zero-sum sequences (the block monoid), we assign to it a recursively defined family of abelian groups, the first being the class group, and do alike for the block monoid. These investigations are motivated by the aim of gaining a more detailed understanding of the arithmetic of Krull monoids, including Dedekind and Krull domains, both from a technical and conceptual point of view. To illustrate our method, some first arithmetical applications are presented.

scholarly journals An upper bound for the Davenport constant of finite groups

2014 ◽  
Vol 218 (10) ◽  
pp. 1838-1844 ◽  
Author(s):  
Weidong Gao ◽  
Yuanlin Li ◽  
Jiangtao Peng
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
Davenport Constant

scholarly journals Minimal zero-sequences and the strong Davenport constant

1999 ◽  
Vol 203 (1-3) ◽  
pp. 271-277 ◽  
Author(s):  
Scott T. Chapman ◽  
Michael Freeze ◽  
William W. Smith
Keyword(s):  
Davenport Constant
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