Skip to main content

Infinite Examples of Cancellative Monoids That Do Not Always Have Least Common Multiple

Abstract

We will study the presentations of fundamental groups of the complement of complexified real affine line arrangements that do not contain two parallel lines. By Yoshinaga’s minimal presentation, we can give positive homogeneous presentations of the fundamental groups. We consider the associated monoids defined by the presentations. It turns out that, in some cases, left (resp. right) least common multiple does not always exist. Hence, the monoids are neither Garside nor Artin. Nevertheless, we will show that they carry certain particular elements similar to the fundamental elements in Artin monoids and that, by improving the classical method in combinatorial group theory, they are cancellative monoids. As a result, we will show that the word problem can be solved and the center of them is determined.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. Brieskorn, E.: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math. 12, 57–61 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  2. Bessis, D., Michel, J.: Explicit presentations for exceptional braid groups. Exp. Math. 13, 257–266 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  3. Brieskorn, E., Saito, K.: Artin-Gruppen und Coxeter-Gruppen. Invent. Math. 17, 245–271 (1972). English trans. by Coleman, C., Corran, R., Crisp, J., Easdown, D., Howlett, R., Jackson, D., Ram, A. at the University of Sydney (1996)

    MathSciNet  Article  MATH  Google Scholar 

  4. Cheniot, D.: Une démonstration du théorème de Zariski sur les sectionshyperplanes d‘une hypersurface projective et du théorème de van Kampen sur le groupe fondamental du complémentaire d’une courbe projective plane. Compos. Math. 27, 141–158 (1973)

    MathSciNet  MATH  Google Scholar 

  5. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups. Mathematical Surveys and Monographs, vol. 7. American Mathematical Society, Providence (1961)

    MATH  Google Scholar 

  6. Dehornoy, P.: Groupes de Garside. Ann. Sci. Éc. Norm. Super. 35, 267–306 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Dehornoy, P.: Complete positive group presentations. J. Algebra 268, 156–197 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  8. Dehornoy, P.: The subword reversing method. Int. J. Algebra Comput. 21, 71–118 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  9. Deligne, P.: Les immeubles des tresses généralizé. Invent. Math. 17, 273–302 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  10. Dehornoy, P., Paris, L.: Gaussian groups and Garside groups, two generalizations of Artin groups. Proc. Lond. Math. Soc. (3) 79, 569–604 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  11. ElRifai, E., Morton, H.: Algorithms for positive braids. Q. J. Math. Oxf. Ser. 2(45), 479–497 (1994)

    MathSciNet  Article  Google Scholar 

  12. Garside, F.A.: The braid groups and other groups. Q. J. Math. Oxf. Ser. 2(20), 235–254 (1969)

    MathSciNet  Article  Google Scholar 

  13. Gebhardt, V.: A new approach to the conjugacy problem in Garside groups. J. Algebra 292, 282–302 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  14. Ishibe, T.: On the monoid in the fundamental group of type \(\mathrm {B_{ii}}\). Hiroshima Math. J. 42, 1–142 (2012)

    MathSciNet  Google Scholar 

  15. Ishibe, T.: The skew growth functions N M,deg(t) for the monoid of type B ii and others. Preprint

  16. Oka, M., Sakamoto, K.: Product theorem of the fundamental group of a reducible curve. J. Math. Soc. Jpn. 30, 599–602 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  17. Picantin, M.: The conjugacy problem in small Gaussian groups. Commun. Algebra 29, 1021–1039 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  18. Saito, K., Ishibe, T.: Monoids in the fundamental groups of the complement of logarithmic free divisors in \({\mathbb{C}}^{3}\). J. Algebra 344, 137–160 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  19. Tokunaga, H., Shimada, I.: Algebraic curves and singularities. Part I. Fundamental groups and singularities (2001). Kyoritsu (in Japanese )

    Google Scholar 

  20. Yoshinaga, M.: Hyperplane arrangements and Lefschetz hyperplane section theorem. Kodai Math. J. 30, 157–194 (2007)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The author was very glad to participate in the conference JSPS-VAST Japan–Vietnam Bilateral Joint Projects “Topology of Singularities and Related Topics, III” Dalat, Vietnam, 26–30 March 2012. The author thanks all the organizers: Nguyen Viet Dung, Ta Le Loi, Pham Tien Son, Mutsuo Oka and Masaharu Ishikawa. The conference was supported by Japan Society for the Promotion of Science, Vietnamese Academy of Science and Technology, GDRI Singularities France–Japan–Vietnam (CNRS), National Foundation for Science Technology Development. The author is grateful to Kyoji Saito for very interesting discussions and encouragement. This research is supported by JSPS Fellowships for Young Scientists (23⋅10023). This research is also supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tadashi Ishibe.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ishibe, T. Infinite Examples of Cancellative Monoids That Do Not Always Have Least Common Multiple. Vietnam J. Math. 42, 305–326 (2014). https://doi.org/10.1007/s10013-014-0062-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-014-0062-6

Keywords

  • Fundamental group
  • The word problem

Mathematics Subject Classification (2010)

  • 20F05