, Volume 31, Issue 2, pp 271–278 | Cite as

Lattice Classification by Cut-through Coding



Inspired by engineering of high-speed switching with quality of service, this paper introduces a new approach to classify finite lattices by the concept of cut-through coding. An n-ary cut-through code of a finite lattice encodes all lattice elements by distinct n-ary strings of a uniform length such that for all j, the initial j encoding symbols of any two elements x and y determine the initial j encoding symbols of the meet and join of x and y. In terms of lattice congruences, some basic criteria are derived to characterize the n-ary cut-through codability of a finite lattice. N-ary cut-through codability also gives rise to a new classification of lattice varieties and in particular, defines a chain of ideals in the lattice of lattice varieties.


Cut-through coding Finite lattice Lattice congruence Lattice variety 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aït-Kaci, H., Boyer, R., Lincoln, P., Nasr, R.: Efficient implementation of lattice operations. ACM Trans. Program. Lang. Syst. 11(3), 115–146 (1989)CrossRefGoogle Scholar
  2. 2.
    Davey, B.A., Priestly, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press (2002)Google Scholar
  3. 3.
    Freese, R., Ježek, J., Nation, J.B.: Free Lattices. Mathematical Surveys and Monographs, vol. 42. American Mathematical Society (1995)Google Scholar
  4. 4.
    Grätzer, G.: General Lattice Theory, 2 edn. Birkhäuser (1998)Google Scholar
  5. 5.
    Grätzer, G.: The Congruences of a Finite Lattice: A Proof-by-Picture Approach. Birkhäuser (2006)Google Scholar
  6. 6.
    Habib, M., Nourine, L.: Bit-vector encoding for partially ordered sets. In: Proc. Int. Workshop Orders, Algorithms, Appl., Lyon, France (1994)Google Scholar
  7. 7.
    Jónsson, B.: Algebras whose congruence lattices are distributive. Math. Scand. 21, 110–121 (1967)MATHMathSciNetGoogle Scholar
  8. 8.
    Jónsson, B., Rival, I.: Lattice varieties covering the smallest non-modular variety. Pacific J. Math. 82, 463–478 (1979)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Li, S.Y.R.: Unified algebraic theory of sorting, routing, multicasting, and concentration networks. IEEE Trans. Comm. 58, 247–256 (2010)CrossRefGoogle Scholar
  10. 10.
    McKenzie, R.: Equational bases and nonmodular lattice varieties. Trans. Am. Math. Soc. 174, 1–43 (1972)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Nation, J.B.: Finite sublattices of a free lattice. Trans. Am. Math. Soc. 269, 311–337 (1982)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Zhu, J., Li, S.Y.R.: Optimizing switching element for minimal latency. U.S. Patent No. 7,609,695 (2009)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institute of Network Coding (Shenzhen), Shenzhen Research InstituteThe Chinese University of Hong KongShenzhenChina
  2. 2.Department of Information Engineering and Institute of Network CodingThe Chinese University of Hong KongNew TerritoriesHong Kong

Personalised recommendations