The catenary degree of Krull monoids I

Alfred Geroldinger[1]; David J. Grynkiewicz[1]; Wolfgang A. Schmid[2]

  • [1] Institut für Mathematik und Wissenschaftliches Rechnen Karl–Franzens–Universität Graz Heinrichstraße 36 8010 Graz, Austria
  • [2] CMLS École polytechnique 91128 Palaiseau cedex, France

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 1, page 137-169
  • ISSN: 1246-7405

Abstract

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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 c ( H ) of H is the smallest integer N with the following property: for each a ∈ H and each two factorizations z , z ′ of a , there exist factorizations z = z 0 , ... , z k = z ′ of a such that, for each i ∈ [ 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. Under a very mild condition on the Davenport constant of G , we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c ( H ) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c ( H ) and characterize when c ( H ) ≤ 4 .

How to cite

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Geroldinger, Alfred, Grynkiewicz, David J., and Schmid, Wolfgang A.. "The catenary degree of Krull monoids I." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 137-169. <http://eudml.org/doc/219687>.

@article{Geroldinger2011,
abstract = {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 two factorizations $z, z^\{\prime\}$ of $a$, there exist factorizations $z = z_0, \ldots , z_k = z^\{\prime\}$ 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. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf c (H)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf c(H)$ and characterize when $\mathsf c(H)\le 4$.},
affiliation = {Institut für Mathematik und Wissenschaftliches Rechnen Karl–Franzens–Universität Graz Heinrichstraße 36 8010 Graz, Austria; Institut für Mathematik und Wissenschaftliches Rechnen Karl–Franzens–Universität Graz Heinrichstraße 36 8010 Graz, Austria; CMLS École polytechnique 91128 Palaiseau cedex, France},
author = {Geroldinger, Alfred, Grynkiewicz, David J., Schmid, Wolfgang A.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {non-unique factorizations; Krull monoids; catenary degree; zero-sum sequence; Davenport constant},
language = {eng},
month = {3},
number = {1},
pages = {137-169},
publisher = {Société Arithmétique de Bordeaux},
title = {The catenary degree of Krull monoids I},
url = {http://eudml.org/doc/219687},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Geroldinger, Alfred
AU - Grynkiewicz, David J.
AU - Schmid, Wolfgang A.
TI - The catenary degree of Krull monoids I
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 137
EP - 169
AB - 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 two factorizations $z, z^{\prime}$ of $a$, there exist factorizations $z = z_0, \ldots , z_k = z^{\prime}$ 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. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf c (H)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf c(H)$ and characterize when $\mathsf c(H)\le 4$.
LA - eng
KW - non-unique factorizations; Krull monoids; catenary degree; zero-sum sequence; Davenport constant
UR - http://eudml.org/doc/219687
ER -

References

top
  1. J. Amos, S.T. Chapman, N. Hine, and J. Paixao, Sets of lengths do not characterize numerical monoids. Integers 7 (2007), Paper A50, 8p. Zbl1139.20056MR2373112
  2. D.D. Anderson, S.T. Chapman, F. Halter-Koch, and M. Zafrullah, Criteria for unique factorization in integral domains. J. Pure Appl. Algebra 127 (1998), 205–218. Zbl0949.13015MR1617195
  3. P. Baginski, S.T. Chapman, R. Rodriguez, G.J. Schaeffer, and Y. She, On the delta set and catenary degree of Krull monoids with infinite cyclic divisor class group. J. Pure Appl. Algebra 214 (2010), 1334 – 1339. Zbl1193.20071MR2593666
  4. G. Bhowmik and J.-C. Schlage-Puchta, Davenport’s constant for groups of the form ℤ 3 ⊕ ℤ 3 ⊕ ℤ 3 d . Additive Combinatorics (A. Granville, M.B. Nathanson, and J. Solymosi, eds.), CRM Proceedings and Lecture Notes, vol. 43, American Mathematical Society, 2007, pp. 307–326. Zbl1173.11012MR2359480
  5. C. Bowles, S.T. Chapman, N. Kaplan, and D. Reiser, On delta sets of numerical monoids. J. Algebra Appl. 5 (2006), 695–718. Zbl1115.20052MR2269412
  6. S.T. Chapman, J. Daigle, R. Hoyer, and N. Kaplan, Delta sets of numerical monoids using nonminimal sets of generators. Commun. Algebra 38 (2010), 2622–2634. Zbl1209.20058MR2674690
  7. S.T. Chapman, P.A. García-Sánchez, and D. Llena, The catenary and tame degree of numerical monoids. Forum Math. 21 (2009), 117 – 129. Zbl1177.20070MR2494887
  8. S.T. Chapman, P.A. García-Sánchez, D. Llena, and J. Marshall, Elements in a numerical semigroup with factorizations of the same length. Can. Math. Bull. 54 (2010), 39–43. Zbl1213.20056
  9. S.T. Chapman, P.A. García-Sánchez, D. Llena, V. Ponomarenko, and J.C. Rosales, The catenary and tame degree in finitely generated commutative cancellative monoids. Manuscr. Math. 120 (2006), 253–264. Zbl1117.20045MR2243561
  10. S.T. Chapman, R. Hoyer, and N. Kaplan, Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77 (2009), 273–279. Zbl1204.20078MR2520501
  11. Y. Edel, Sequences in abelian groups G of odd order without zero-sum subsequences of length exp ( G ) . Des. Codes Cryptography 47 (2008), 125–134. Zbl1196.11043MR2375461
  12. Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin, and L. Rackham, Zero-sum problems in finite abelian groups and affine caps. Quarterly. J. Math., Oxford II. Ser. 58 (2007), 159–186. Zbl1205.11028MR2334860
  13. Y. Edel, S. Ferret, I. Landjev, and L. Storme, The classification of the largest caps in A G ( 5 , 3 ) . J. Comb. Theory, Ser. A 99 (2002), 95 –110. Zbl1023.51007MR1911459
  14. M. Freeze and W.A. Schmid, Remarks on a generalization of the Davenport constant. Discrete Math. 310 (2010), 3373–3389. Zbl1228.05302MR2721098
  15. W. Gao and A. Geroldinger, On long minimal zero sequences in finite abelian groups. Period. Math. Hung. 38 (1999), 179–211. Zbl0980.11014MR1756238
  16. —, Zero-sum problems in finite abelian groups : a survey. Expo. Math. 24 (2006), 337–369. Zbl1122.11013MR2313123
  17. A. Geroldinger, Additive group theory and non-unique factorizations. Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Ruzsa, eds.), Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, 2009, pp. 1–86. Zbl1221.20045MR2522037
  18. A. Geroldinger, D.J. Grynkiewicz, and W.A. Schmid, The catenary degree of Krull monoids II. manuscript. Zbl1253.11101
  19. A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, 2006. Zbl1113.11002MR2194494
  20. A. Geroldinger and W. Hassler, Arithmetic of Mori domains and monoids. J. Algebra 319 (2008), 3419–3463. Zbl1195.13022MR2408326
  21. A. Geroldinger and J. Kaczorowski, Analytic and arithmetic theory of semigroups with divisor theory. J. Théor. Nombres Bordx. 4 (1992), 199–238. Zbl0780.11046MR1208863
  22. A. Geroldinger and R. Schneider, On Davenport’s constant. J. Comb. Theory, Ser. A 61 (1992), 147–152. Zbl0759.20008MR1178393
  23. R. Gilmer, Commutative Semigroup Rings. The University of Chicago Press, 1984. Zbl0566.20050MR741678
  24. P.A. Grillet, Commutative Semigroups. Kluwer Academic Publishers, 2001. Zbl1040.20048MR2017849
  25. F. Halter-Koch, Ideal Systems. An Introduction to Multiplicative Ideal Theory. Marcel Dekker, 1998. Zbl0953.13001MR1828371
  26. A. Iwaszkiewicz-Rudoszanska, On the distribution of coefficients of logarithmic derivatives of L -functions attached to certain arithmetical semigroups. Monatsh. Math. 127 (1999), 189–202. Zbl0940.11040MR1680523
  27. —, On the distribution of prime divisors in arithmetical semigroups. Funct. Approximatio, Comment. Math. 27 (1999), 109 – 116. Zbl0964.11036MR1746844
  28. H. Kim, The distribution of prime divisors in Krull monoid domains. J. Pure Appl. Algebra 155 (2001), 203–210. Zbl0971.13015MR1801415
  29. H. Kim and Y. S. Park, Krull domains of generalized power series. J. Algebra 237 (2001), 292–301. Zbl1039.13012MR1813891
  30. C.R. Leedham-Green, The class group of Dedekind domains. Trans. Am. Math. Soc. 163 (1972), 493–500. Zbl0231.13008MR292806
  31. M. Omidali, The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences. Forum Math., to appear. Zbl1252.20057MR2494887
  32. O. Ordaz, A. Philipp, I. Santos, and W.A. Schmid, On the Olson and the strong Davenport constants. J. Théor. Nombres Bordx., to appear. Zbl1252.11011
  33. A. Potechin, Maximal caps in AG ( 6 , 3 ) . Des. Codes Cryptography 46 (2008), 243–259. Zbl1187.51010MR2372838
  34. J.C. Rosales and P.A. García-Sánchez, Numerical Semigroups. Springer, 2009. Zbl1220.20047MR2549780
  35. W.A. Schmid, A realization theorem for sets of lengths. J. Number Theory 129 (2009), 990–999. Zbl1191.11031MR2516967

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