Skip to main content

Affine completeness of the algebra of full binary trees

Abstract

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of full binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non associative 1–affine complete algebra. As far as we know, it is the first example of such an algebra.

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

Fig. 1

References

  1. Cégielski, P., Grigorieff, S., Guessarian, I.: Newton representation of functions over natural integers having integral difference ratios. Int. J. Number Theory 11(7), 2019–2139 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  2. Cégielski, P., Grigorieff, S., Guessarian, I.: Congruence preserving functions on free monoids. Algebra Univ. Spring. Verlag 78(3), 389–406 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  3. Grätzer, G.: On Boolean functions (notes on lattice theory. II). Rev. Math. Pures Appl. (Académie de la République Populaire Roumaine) 7, 693–697 (1962)

    MathSciNet  MATH  Google Scholar 

  4. Grätzer, G.: Universal algebra, 2nd edn. Springer Verlag, New York (1979)

    MATH  Google Scholar 

  5. Grätzer, G.: Boolean functions on distributive lattices. Acta Mathematica Hungarica 15, 193–201 (1964)

    MathSciNet  MATH  Google Scholar 

  6. Kaarli, K., Pixley, A.F.: Polynomial Completeness in Algebraic Systems. Chapman & Hall/CRC (2001)

  7. Nöbauer, W.: Affinvollständige Moduln. Mathematische Nachrichten 86, 85–96 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  8. Haussmann, B.A., Ore, Ø.: Theory of quasi-groups. Am. J. Math. 59(4), 983–1004 (1937)

    MathSciNet  Article  MATH  Google Scholar 

  9. Ploščica, M., Haviar, M.: Congruence-preserving functions on distributive lattices. Algebra Universalis 59, 179–196 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  10. Serre, J.-P.: Lie Algebras and Lie Groups, 1964 lectures given at Harvard University. Lecture notes in mathematics, vol. 1500. Springer Verlag, New York (1992)

    MATH  Google Scholar 

  11. Werner, H.: Produkte von Kongruenzen Klassengeometrien Universeller Algebren. Mathematische Zeitschrift 121, 111–140 (1971)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referees for their extremely careful reading and comments which helped in shortening the proof of the main theorem and improving the presentation of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Irène Guessarian.

Additional information

Presented by M. Haviar.

To the memory of Kate Karagueuzian-Gibbons and Giliane Arnold.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Arnold, A., Cégielski, P., Grigorieff, S. et al. Affine completeness of the algebra of full binary trees. Algebra Univers. 81, 55 (2020). https://doi.org/10.1007/s00012-020-00690-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-020-00690-6

Keywords

  • Congruence
  • Affine completeness
  • Full binary trees

Mathematics Subject Classification

  • 06A99
  • 08A30
  • 08B20