Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids

Víctor Blanco[1]; Pedro A. García-Sánchez; Alfred Geroldinger[2]

  • [1] Departamento de Álgebra, Universidad de Granada, Granada 18071, Espana
  • [2] Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria

Actes des rencontres du CIRM (2010)

  • Volume: 2, Issue: 2, page 95-98
  • ISSN: 2105-0597

Abstract

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Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids.We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.

How to cite

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Blanco, Víctor, García-Sánchez, Pedro A., and Geroldinger, Alfred. "Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids." Actes des rencontres du CIRM 2.2 (2010): 95-98. <http://eudml.org/doc/196279>.

@article{Blanco2010,
abstract = {Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids.We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.},
affiliation = {Departamento de Álgebra, Universidad de Granada, Granada 18071, Espana; Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria},
author = {Blanco, Víctor, García-Sánchez, Pedro A., Geroldinger, Alfred},
journal = {Actes des rencontres du CIRM},
keywords = {presentations for semigroups; catenary degree; tame degree; sets of lengths; numerical monoid; Krull monoid},
language = {eng},
number = {2},
pages = {95-98},
publisher = {CIRM},
title = {Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids},
url = {http://eudml.org/doc/196279},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Blanco, Víctor
AU - García-Sánchez, Pedro A.
AU - Geroldinger, Alfred
TI - Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids
JO - Actes des rencontres du CIRM
PY - 2010
PB - CIRM
VL - 2
IS - 2
SP - 95
EP - 98
AB - Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids.We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.
LA - eng
KW - presentations for semigroups; catenary degree; tame degree; sets of lengths; numerical monoid; Krull monoid
UR - http://eudml.org/doc/196279
ER -

References

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  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. V. Blanco, P. A. García-Sánchez, and A. Geroldinger, Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids, manuscript. Zbl1279.20072
  3. 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
  4. 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
  5. M. Delgado, P.A. García-Sánchez, and J. Morais, “numericalsgps”: a gap package on numerical semigroups, (http://www.gap-system.org/Packages/numericalsgps.html). Zbl1169.20309
  6. M. Freeze and A. Geroldinger, Unions of sets of lengths, Funct. Approximatio, Comment. Math. 39 (2008), 149 – 162. Zbl1228.20046MR2490095
  7. W. Gao and A. Geroldinger, On products of k atoms, Monatsh. Math. 156 (2009), 141 – 157. Zbl1184.20051MR2488859
  8. P.A. García-Sánchez and I. Ojeda, Uniquely presented finitely generated commutative monoids, Pacific J. Math. 248 (2010), 91 – 105. Zbl1208.20052MR2734166
  9. 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
  10. 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
  11. A. Geroldinger and F. Kainrath, On the arithmetic of tame monoids with applications to Krull monoids and Mori domains, J. Pure Appl. Algebra 214 (2010), 2199 – 2218. Zbl1207.20055MR2660909
  12. F. Kainrath, Arithmetic of Mori domains and monoids : the Global Case, manuscript. Zbl1207.20055
  13. A. Philipp, A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degree and applications to semigroup rings, Semigroup Forum. Zbl1213.20059
  14. —, A precise result on the arithmetic of non-principal orders in algebraic number fields, manuscript. Zbl1303.11126
  15. —, A characterization of arithmetical invariants by the monoid of relations, Semigroup Forum 81 (2010), 424 – 434. Zbl1213.20059MR2735659
  16. J.C. Rosales and P.A. García-Sánchez, Finitely Generated Commutative Monoids, Nova Science Publishers, 1999. Zbl0966.20028MR1694173
  17. W.A. Schmid, A realization theorem for sets of lengths, J. Number Theory 129 (2009), 990 – 999. Zbl1191.11031MR2516967

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