Advertisement

Algebra universalis

, 79:85 | Cite as

On the number of essential arguments of homomorphisms between products of median algebras

  • Miguel Couceiro
  • Gerasimos C. Meletiou
Article
  • 7 Downloads

Abstract

In this paper we characterize classes of median-homomorphisms between products of median algebras, that depend on a given number of arguments, by means of necessary and sufficent conditions that rely on the underlying algebraic and on the underlying order structure of median algebras. In particular, we show that a median-homomorphism that take values in a median algebra that does not contain a subalgebra isomorphic to the m-dimensional Boolean algebra as a subalgebra cannot depend on more than \(m-1\) arguments. In view of this result, we also characterize the latter class of median algebras. We also discuss extensions of our framework on homomorphisms over median algebras to wider classes of algebras.

Keywords

Median algebra Median-homomorphism Essential argument Hypercube-freeness Congruence distributivity 

Mathematics Subject Classification

06A12 06D99 06F99 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her thorough review and insightful remarks that improved the current paper and that led to the extension of our results to wider classes of algebras.

References

  1. 1.
    Arrow, K.J.: A difficulty in the concept of social welfare. J. Polit. Econ. 58(4), 328–346 (1950)CrossRefGoogle Scholar
  2. 2.
    Bandelt, H.J., Hedlíková, J.: Median algebras. Discrete math. 45, 1–30 (1983)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bandelt, H.J., Janowitz, M., Meletiou, G.C.: \(n\)-Median semilattices. Order 8, 185–195 (1991)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bandelt, H.J., Meletiou, G.C.: An algebraic setting for near-unanimity consensus. Order 7, 169–178 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bandelt, H .J., Meletiou, G .C.: The zero-completion of a median algebra. Czech. Math. J. 43(118), 3, 409–417 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Birkhoff, G., Kiss, S.A.: A ternary operation in distributive lattices. Bull. Am. Math. Soc. 53, 749–752 (1947)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Birkhoff, G.: Lattice Theory, volume 25 of American Mathematical Society Colloquium Publications, revised edition. American Mathematical Society, New York (1948)Google Scholar
  8. 8.
    Burris, S., Sankappanavar, H. P.: A course in universal algebra, Millenium edition. www.math.uwaterloo.ca/~snburris
  9. 9.
    Chajda, I., Goldstern, M., Länger, H.: A note on homomorphisms between products of algebras. Algebra Univer. 79, 25 (2018).  https://doi.org/10.1007/s00012-018-0517-9 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Couceiro, M., Marichal, J.-L., Teheux, B.: Conservative median algebras and semilattices. Order 33(1), 121–132 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Couceiro, M., Foldes, S., Meletiou, G.C.: On homomorphisms between products of median algebras. Algebra Univ. 78(4), 545–553 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Couceiro, M., Meletiou, G.C.: On a special class of median algebras. Miskolc Math. Notes 18(1), 167–171 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)zbMATHGoogle Scholar
  14. 14.
    Isbell, J.R.: Median algebra. Trans. Am. Math. Soc. 260(2), 319–362 (1980)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sholander, M.: Trees, lattices, order, and betweenness. Proc. Am. Math. Soc. 3(3), 369–381 (1952)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sholander, M.: Medians, lattices, and trees. Proc. Am. Math. Soc. 5(5), 808–812 (1954)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Van De Vel, M.L.J.: Theory of Convex Structures. Theory of convex structures. North-Holland Mathematical Library, vol. 50. Elsevier Science Publishers B.V., Amsterdam (1993)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of Lorraine, CNRS, Inria, LORIANancyFrance
  2. 2.TEI of EpirusArtaGreece

Personalised recommendations