Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Lattice generation of small equivalences of a countable set

  • 18 Accesses

  • 5 Citations

Abstract

Given a countable set A, let Equ(A) denote the lattice of equivalences of A. We prove the existence of a four-generated sublattice Q of Equ(A) such that Q contains all atoms of Equ(A). Moreover, Q can be generated by four equivalences such that two of them are comparable. Our result is a reasonable generalization of Strietz [5, 6] from the finite case to the countable one; and in spite of its essentially simpler proof it asserts more for the countable case than [2, 3].

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

References

  1. 1.

    Chajda, I. and Czédli, G. (1996) How to generate the involution lattice of quasiorders?, Studia Sci. Math. (Budapest), to appear.

  2. 2.

    Czédli, G. (1996) Four-generated large equivalence lattices, Acta Sci. Math. (Szeged), to appear.

  3. 3.

    Czédli, G. (1996) (1+1+2)-generated equivalence lattices, in preparation.

  4. 4.

    RivalI. and StanfordM. (1992) Algebraic aspects of partition lattices, in Matroids and Applications, Cambridge Univ. Press, Cambridge, pp. 106–122.

  5. 5.

    Strietz, H. (1975) Finite partition lattices are four-generated, in Proc. Lattice Th. Conf. Ulm, pp. 257–259.

  6. 6.

    StrietzH. (1977) Über Erzeugendenmengen endlicher Partitionverbände, Studia Sci. Math. Hungarica 12, 1–17.

  7. 7.

    ZádoriZ. (1986) Generation of finite partition lattices, in Colloquia Math. Soc. J. Bolyai 43, Lectures in Universal Algebra (Proc. Conf. Szeged), (1983), North-Holland, Amsterdam, New York, pp. 573–586.

Download references

Author information

Additional information

Dedicated to George Grätzer on his 60th birthday

This research was supported by the NFSR of Hungary (OTKA), grant no. T7442.

Communicated by I. Rival

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Czédli, G. Lattice generation of small equivalences of a countable set. Order 13, 11–16 (1996). https://doi.org/10.1007/BF00383964

Download citation

Mathematics Subject Classifications (1985)

  • Primary 06B99
  • Secondary 06C10

Key words

  • lattice
  • equivalence
  • equivalence lattice
  • generating set