Skip to main content
SpringerLink
Log in

A characterization of modular law and distributive law of lattices by the Yang–Baxter maps

  • Original Paper
  • Area 1
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this article, we generalize a bubble sort map to a map on the lattice, and characterize the modular law and the distributive law of lattices from the viewpoint of the Yang–Baxter maps.

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. Adler, V.E., Bobenko, A.I., Suris, Y.B.: Geometry of Yang–Baxter maps: pencils of conics and quadrirational mappings. Commun. Anal. Geom. 12(5), 967–1007 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baxter, R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press Inc, London (1982)

    MATH  Google Scholar 

  4. Baxter, R.J.: Solvable eight-vertex model on an arbitrary planar lattice. Phil. Trans. Royal Soc. Lond. 289, 315–346 (1978)

    Article  MathSciNet  Google Scholar 

  5. Bukhshtaber, V.M.: Yang–Baxter mappings. Uspekhi Mat. Nauk 53(6), 324, 241–242 [translation. Russian Math. Surveys 53(6), 1343–1345 (1998)]

  6. Drinfel’d, V.G.: On Some Unsolved Problems in Quantum Group Theory. Quantum Groups (Leningrad, 1990), Lecture Notes in Math., vol. 1510, pp. 1–8. Springer, Berlin (1992)

  7. Grätzer, G.: General Lattice Theory, Second edn. Birkhäuser, Basel (2003)

    MATH  Google Scholar 

  8. Ikegami, T., Takahashi, D., Matsukidaira, J.: On solutions to evolution equations defined by lattice operators. Jpn. J. Indus. Appl. Math. 31, 211–230 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. McGuire, J.B.: Study of exactly soluble one-dimensional N-body problems. J. Math. Phys. 5, 622–636 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  10. Matsumoto, D.K., Shibukawa, Y.: Quantum Yang–Baxter equation, braided semigroups, and dynamical Yang–Baxter maps. Tokyo J. Math. 38(1), 227–237 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nagai, A., Takahashi, D., Tokihiro, T.: Soliton cellular automaton, Toda molecule equation and sorting algorithm. Phys. Lett. A 255, 265–271 (1999)

    Article  Google Scholar 

  12. Nagai, A., Tokihiro, T., Satsuma, J.: Ultra-discrete Toda molecule equation. Phys. Lett. A 244, 383–388 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shibukawa, Y.: Dynamical Yang–Baxter maps with an invariance condition. Publ. Res. Inst. Math. Sci. 43(4), 1157–1182 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Veselov, A.P.: Yang–Baxter maps and integrable dynamics. Phys. Lett. A 314(3), 214–221 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yang, C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19, 1312–1315 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, C.N.: \(S\) matrix for the one-dimensional N-body problem with repulsive or attractive \(\delta \)-function interaction. Phys. Rev. 168, 1920–1923 (1968)

    Article  Google Scholar 

Download references

Acknowledgements

The author wish to express thanks to Professor Kimio Ueno for his advices. The author wish to express thanks to his family Rieko, Rui and Riku.

Author information

Authors and Affiliations

  1. Center for Promotion of Educational Innovation, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama, Saitama, 337-8570, Japan

    Diogo Kendy Matsumoto

Authors
  1. Diogo Kendy Matsumoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diogo Kendy Matsumoto.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matsumoto, D.K. A characterization of modular law and distributive law of lattices by the Yang–Baxter maps. Japan J. Indust. Appl. Math. 34, 335–341 (2017). https://doi.org/10.1007/s13160-017-0248-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-017-0248-x

Keywords

Mathematics Subject Classification

Search

Navigation