Skip to main content
SpringerLink
Log in

Order affine completeness of lattices with Boolean congruence lattices

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices L easily reduces to the case when L is a subdirect product of two simple lattices L 1 and L 2. Our main result claims that such a lattice is locally order affine complete iff L 1 and L 2 are tolerance trivial and one of the following three cases occurs:

  1. 1)

    L = L 1 × L 2

  2. 2)

    L is a maximal sublattice of the direct product

  3. 3)

    L is the intersection of two maximal sublattices, one containing 〈0, 1〉 and the other 〈1, 0〉.

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. I. Chajda, B. Zelinka: Tolerance relation on lattices. Čas. Pěst. Mat. 99 (1974), 394–399.

    MathSciNet  Google Scholar 

  2. K. Kaarli: On varieties generated by functionally complete algebras. Algebra Univers. 29 (1992), 495–502.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Czédli, E. Horváth, and S. Radeleczki: On tolerance lattices of algebras in congruence modular lattices. Acta Math. Hung. 100 (2003), 9–17.

    Article  MATH  Google Scholar 

  4. P. Hegedüs, P. P. Pálfy: Finite modular congruence lattices. Algebra Univers. 54 (2005), 105–120.

    Article  MATH  Google Scholar 

  5. K. Kaarli, A. F. Pixley: Polynomial Completeness in Algebraic Systems. Chapman & Hall/CRC, Boca Raton, 2000.

    Google Scholar 

  6. M. Kindermann: Über die Äquivalenz von Ordnungspolynomvollständigkeit und Toleranzeinfachheit endlicher Verbände. Contributions to General Algebra (Proc. Klagenfurt Conf. 1978). Verlag J. Heyn, Klagenfurt, 1979, pp. 145–149.

    Google Scholar 

  7. D. Schweigert: Über die endliche, ordnungspolynomvollständige Verbände. Monatsh. Math. 78 (1974), 68–76.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Wille: Eine Characterisierung endlicher, ordnungspolynomvollständiger Verbände. Arch. Math. (Basel) 28 (1977), 557–560.

    MATH  MathSciNet  Google Scholar 

  9. R. Wille: Über endliche, ordnungaffinvollständige Verbände. Math. Z. 155 (1977), 103–107.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Pure Mathematics, University of Tartu, 50090, Tartu, Estonia

    Kalle Kaarli & Vladimir Kuchmei

Authors
  1. Kalle Kaarli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Kuchmei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kalle Kaarli.

Additional information

Research supported by the Estonian Science Foundation grant number 5368.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kaarli, K., Kuchmei, V. Order affine completeness of lattices with Boolean congruence lattices. Czech Math J 57, 1049–1065 (2007). https://doi.org/10.1007/s10587-007-0113-1

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-007-0113-1

Keywords

Search

Navigation