, Volume 20, Issue 3, pp 185–221 | Cite as

The Zig-Zag Property and Exponential Cancellation of Ordered Sets

  • Ralph McKenzie


This paper outlines a chapter in the theory of ordered sets that concerns decompositions of ordered sets using various natural operations introduced by Garrett Birkhoff in 1940 – chiefly, the operations of sum (disjoint union), Cartesian product, and exponentiation.

This area has seen two brief, four-year flowerings of research activity during the past sixty-four years. Ivan Rival, working with several collaborators around 1978, obtained beautiful results on exponentiation which attracted other researchers to this topic. Altogether, about a dozen people contributed a chain of significant new results between 1978–1982. The subject then lay dormant until a major contribution by Jonathan D. Farley appeared in 1996. This pre-figured the solution, four years later, of a central problem about exponentiation of finite ordered sets that Birkhoff had formulated in 1942.

The theory of operations on ordered sets is a beautiful chapter of mathematics. We concentrate on those portions that are most relevant for the solution of the Birkhoff problem, offering (hopefully) accessible proofs of many results, including some new results. We would like to see a new renaissance of activity in this area, which continues to offer many attractive open problems.

cancellation of exponents exponentiation posets 


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  1. 1.
    Bauer, H.: Garben und Automorphismen geordneter Mengen, Dissertation, Technische Hochschule Darmstadt, 1982.Google Scholar
  2. 2.
    Bergman, C., McKenzie, R. and Nagy, Sz.: How to cancel a linearly ordered exponent, Colloq. Math. Soc. János Bolyai 21 (1982), 87–94.MathSciNetGoogle Scholar
  3. 3.
    Birkhoff, G.: Extended arithmetic, Duke Math. J. 3 (1937), 311–316.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Birkhoff, G.: Generalized arithmetic, Duke Math. J. 9 (1942), 283–302.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Birkhoff, G.: Lattice Theory, 2nd edn, Colloquium Publications, Vol. 25, Amer. Math. Soc., Providence, RI.Google Scholar
  6. 6.
    Chang, C. C., Jónsson, B. and Tarski, A.: Refinement properties for relational structures, Fund. Math. 55 (1964), 249–281.zbMATHMathSciNetGoogle Scholar
  7. 7.
    Davey, B. A., Duffus, D., Quackenbush, R.W. and Rival, I.: Exponents of finite simple lattices, J. London Math. Soc. 17 (1978), 203–221.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Day, M. M.: Arithmetic of ordered systems, Trans. Amer. Math. Soc. 58 (1945), 1–43.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Duffus, D.: Powers of ordered sets, Order 1 (1982), 83–92.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Duffus, D. and Rival, I.: A logarithmic property for exponents of partially ordered sets, Canad. J. Math. 30 (1978), 797–807.zbMATHMathSciNetGoogle Scholar
  11. 11.
    Duffus, D., Jónsson, B. and Rival, I.: Structure results for function lattices, Canad. J. Math. 30 (1978), 392–400.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Duffus, D. and Wille, R.: A theorem on partially ordered sets of order-preserving mappings, Proc. Amer. Math. Soc. 76 (1979), 14–16.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Farley, J. D.: The automorphism group of a function lattice: A problem of Jónsson and McKenzie, Algebra Universalis 36 (1996), 8–45.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hashimoto, J.: On the product decomposition of partially ordered sets, Math. Japon. 1 (1948), 120–123.zbMATHMathSciNetGoogle Scholar
  15. 15.
    Hashimoto, J.: On direct product decomposition of partially ordered sets, Ann. of Math. (2) 54 (1951), 315–318.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Hashimoto, J. and Nakayama, T.: On a problem of G. Birkhoff, Proc. Amer. Math. Soc. 1 (1950), 141–142.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Jónsson, B.: Powers of partially ordered sets: The automorphism group, Math. Scand. 51 (1982), 121–141.zbMATHMathSciNetGoogle Scholar
  18. 18.
    Jónsson, B. and McKenzie, R.: Powers of partially ordered sets: Cancellation and refinement properties, Math. Scand. 51 (1982), 87–120.zbMATHMathSciNetGoogle Scholar
  19. 19.
    Lovasz, L.: Operations with structures, Acta Math. Acad. Sci. Hungar. 18 (1967), 337–339.Google Scholar
  20. 20.
    McKenzie, R.: Arithmetic of finite ordered sets: Cancellation of exponents, I, Order 16 (1999), 313–333.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    McKenzie, R.: Arithmetic of finite ordered sets: Cancellation of exponents, II, Order 17 (2000),309–332.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Novotný, M.: Ñber gewisse Eigenschaften von Kardinaloperationen, Spisy Priírod. Fak. Univ. Brno (1960), 465–484.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Ralph McKenzie
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleU.S.A.

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