Abstract.
Let H be an atomic monoid. For \(k \in {\Bbb N}\) let \({\cal V}_k (H)\) denote the set of all \(m \in {\Bbb N}\) with the following property: There exist atoms (irreducible elements) u 1, …, u k , v 1, …, v m ∈ H with u 1· … · u k = v 1 · … · v m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets \({\cal V}_k (H)\) are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every \(k \in {\Bbb N}\), max \({\cal V}_{2k+1} (H) = k \vert G\vert + 1\) which settles Problem 38 in [4].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J Amos ST Chapman N Hine J Paixao (2007) ArticleTitleSets of lengths do not characterize numerical monoids Integers 7 8 Occurrence Handle2373112
Baginski P, Chapman ST, Hine N, Paixao J (2008) On the asymptotic behavior of unions of sets of lengths in atomic monoids. Involve, to appear
ST Chapman M Freeze WW Smith (1999) ArticleTitleMinimal zero sequences and the strong Davenport constant Discrete Math 203 271–277 Occurrence Handle0944.20013 Occurrence Handle10.1016/S0012-365X(99)00027-8 Occurrence Handle1696250
ST Chapman S Glaz (2000) One hundred problems in commutative ring theory ST Chopman (Eds) et al. Non-Noetherian Commutative Ring Theory Kluwer Dordrecht 459–476
ST Chapman WW Smith (1990) ArticleTitleFactorization in Dedekind domains with finite class group Israle J Math 71 65–95 Occurrence Handle0717.13014 Occurrence Handle10.1007/BF02807251 Occurrence Handle1074505
ST Chapman WW Smith (1993) ArticleTitleOn lengths of factorizations of elements in an algebraic number ring J Number Theory 43 24–30 Occurrence Handle0765.11043 Occurrence Handle10.1006/jnth.1993.1003 Occurrence Handle1200805
ST Chapman WW Smith (1998) ArticleTitleGeneralized sets of lengths J Algebra 200 449–471 Occurrence Handle0929.13010 Occurrence Handle10.1006/jabr.1997.7273 Occurrence Handle1610656
ST Chapman WW Smith (2005) ArticleTitleA characterization of minimal zero-sequences of index one in finite cyclic groups Integers 5 IssueID1 5 Occurrence Handle2192246
Foroutan A, Geroldinger A (2005) Monotone chains of factorizations in C-monoids. In: Chapman ST (ed) Arithmetical Properties of Commutative Rings and Monoids, Lect Notes Pure Appl Math 241, pp 99–113. Boca Raton, FL: Chapman & Hall/CRC
Freeze M, Geroldinger A (2008) Unions of sets of lengths. Funct Approximotio, Comment Math, to appear
W Gao (2000) ArticleTitleZero sums in finite cyclic groups Integers 0 9
W Gao A Geroldinger (2006) ArticleTitleZero-sum problems in finite abelian groups: a survey Expo Math 24 337–369 Occurrence Handle1122.11013 Occurrence Handle2313123
A Geroldinger F Halter-Koch (2006) Non-Unique Factorizations SeriesTitleAlgebraic, Combinatorial and Analytic Theory Chapman & Hall/CRC Boca Raton, FL Occurrence Handle1113.11002
A Geroldinger W Hassler (2008) ArticleTitleArithmetic of Mori domains and monoids J Algebra 319 3419–3463 Occurrence Handle05300816 Occurrence Handle10.1016/j.jalgebra.2007.11.025 Occurrence Handle2408326
A Geroldinger W Hassler (2008) ArticleTitleLocal tameness of v-noetherian monoids J Pure Appl Algebra 212 1509–1524 Occurrence Handle1133.20047 Occurrence Handle10.1016/j.jpaa.2007.10.020 Occurrence Handle2391663
A Geroldinger G Lettl (1990) ArticleTitleFactorization problems in semigroups Semigroup Forum 40 23–38 Occurrence Handle0693.20063 Occurrence Handle10.1007/BF02573249 Occurrence Handle1014223
PA Grillet (2001) Commutative Semigroups Kluwer Dordrecht Occurrence Handle1040.20048
F Halter-Koch (1998) Ideal systems SeriesTitleAn Introduction to Multiplicative Ideal Theory Marcel Dekker New York Occurrence Handle0953.13001
W Hassler (2004) ArticleTitleFactorization in finitely generated domains J Pure Appl Algebra 186 151–168 Occurrence Handle1061.13002 Occurrence Handle10.1016/S0022-4049(03)00142-7 Occurrence Handle2025595
F Kainrath (2005) ArticleTitleElasticity of finitely generated domains Houston J Math 31 43–64 Occurrence Handle1089.13001 Occurrence Handle2123000
S Savchev F Chen (2007) ArticleTitleLong zero-free sequences in finite cyclic groups Discrete Math 307 2671–2679 Occurrence Handle1148.20038 Occurrence Handle10.1016/j.disc.2007.01.012 Occurrence Handle2351696
Schmid WA (2007) A realization theorem for sets of lengths. Manuscript
P Yuan (2007) ArticleTitleOn the index of minimal zero-sum sequences over finite cyclic groups J Comb Theory Ser A 114 1545–1551 Occurrence Handle1128.11011 Occurrence Handle10.1016/j.jcta.2007.03.003
Author information
Authors and Affiliations
Additional information
Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria
Rights and permissions
About this article
Cite this article
Gao, W., Geroldinger, A. On products of k atoms. Monatsh Math 156, 141–157 (2009). https://doi.org/10.1007/s00605-008-0547-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-008-0547-z