Skip to main content
SpringerLink
Log in

Semigroup identities in the monoid of triangular tropical matrices

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

An Erratum to this article was published on 04 April 2016

Abstract

We show that the submonoid of all n×n triangular tropical matrices satisfies a nontrivial semigroup identity and provide a generic construction for classes of such identities. The utilization of the Fibonacci number formula gives us an upper bound on the length of these 2-variable semigroup identities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adjan, S.I.: Defining relations and algorithmic problems for groups and semigroups. In: Proceeding of the Steklov Institute of Mathematics, vol. 85. AMS, Providence (1967)

    Google Scholar 

  2. Akian, M., Bapat, R., Gaubert, S.: Max-plus algebra. In: Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R. (eds.) Handbook of Linear Algebra. Chapman & Hall, London (2006)

    Google Scholar 

  3. Butkovič, P.: Max-Linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London (2010)

    Book  MATH  Google Scholar 

  4. Gaubert, S., Katz, R.D.: Reachability problems for products of matrices in semirings. Int. J. Algebra Comput. 16(3), 603–627 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. IHÉS 53, 53–73 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry vol. 35. Birkhauser, Basel (2007). Oberwolfach seminars

    MATH  Google Scholar 

  7. Izhakian, Z., Margolis, S.W.: Semigroup identities in the monoid of 2-by-2 tropical matrices. Semigroup Forum 80(2), 191–218 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Pin, J.-E.: Tropical Semirings, vol. 11, pp. 50–69. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  9. Richter-Gebert, J., Sturmfels, B., Theobald, T.: First Steps in Tropical Geometry. Idempotent Mathematics and Mathematical Physics. Contemp. Math., vol. 377, pp. 289–317. AMS, Providence (2005)

    Book  MATH  Google Scholar 

  10. Shevrin, L.N., Volkov, M.V.: Identities of semigroups. Sov. Math. Izv. VUZ 29(11), 1–64 (1985). Izv. Vuzov. Math. 29(11), 3–47 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Shneerson, L.: Identities in finitely generated semigroups of polynomial growth. J. Algebra 154(1), 67–85 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Simon, I.: Recognizable sets with multiplicities in the tropical semiring. In: Chytil, M., Janiga, L., Koubek, V. (eds.) MFCS 88. Lecture Notes in Computer Science, vol. 324, pp. 107–120. Springer, Berlin (1988)

    Google Scholar 

Download references

Acknowledgements

The author thanks Glenn Merlet for the useful conversation in CIRM, Luminy, September 2011. The author would like to thank the referee for many valuable comments and helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bremen, 28359, Bremen, Germany

    Zur Izhakian

Authors
  1. Zur Izhakian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zur Izhakian.

Additional information

Communicated by Jean-Eric Pin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Izhakian, Z. Semigroup identities in the monoid of triangular tropical matrices. Semigroup Forum 88, 145–161 (2014). https://doi.org/10.1007/s00233-013-9507-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-013-9507-6

Keywords

Search

Navigation