Abstract
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.
Similar content being viewed by others
References
Bauer, H.: Garben und Automorphismen geordneter Mengen, Dissertation, Technische Hochschule Darmstadt, 1982.
Bergman, C., McKenzie, R. and Nagy, Sz.: How to cancel a linearly ordered exponent, Colloq. Math. Soc. János Bolyai 21 (1982), 87–94.
Birkhoff, G.: Extended arithmetic, Duke Math. J. 3 (1937), 311–316.
Birkhoff, G.: Generalized arithmetic, Duke Math. J. 9 (1942), 283–302.
Birkhoff, G.: Lattice Theory, 2nd edn, Colloquium Publications, Vol. 25, Amer. Math. Soc., Providence, RI.
Chang, C. C., Jónsson, B. and Tarski, A.: Refinement properties for relational structures, Fund. Math. 55 (1964), 249–281.
Davey, B. A., Duffus, D., Quackenbush, R.W. and Rival, I.: Exponents of finite simple lattices, J. London Math. Soc. 17 (1978), 203–221.
Day, M. M.: Arithmetic of ordered systems, Trans. Amer. Math. Soc. 58 (1945), 1–43.
Duffus, D.: Powers of ordered sets, Order 1 (1982), 83–92.
Duffus, D. and Rival, I.: A logarithmic property for exponents of partially ordered sets, Canad. J. Math. 30 (1978), 797–807.
Duffus, D., Jónsson, B. and Rival, I.: Structure results for function lattices, Canad. J. Math. 30 (1978), 392–400.
Duffus, D. and Wille, R.: A theorem on partially ordered sets of order-preserving mappings, Proc. Amer. Math. Soc. 76 (1979), 14–16.
Farley, J. D.: The automorphism group of a function lattice: A problem of Jónsson and McKenzie, Algebra Universalis 36 (1996), 8–45.
Hashimoto, J.: On the product decomposition of partially ordered sets, Math. Japon. 1 (1948), 120–123.
Hashimoto, J.: On direct product decomposition of partially ordered sets, Ann. of Math. (2) 54 (1951), 315–318.
Hashimoto, J. and Nakayama, T.: On a problem of G. Birkhoff, Proc. Amer. Math. Soc. 1 (1950), 141–142.
Jónsson, B.: Powers of partially ordered sets: The automorphism group, Math. Scand. 51 (1982), 121–141.
Jónsson, B. and McKenzie, R.: Powers of partially ordered sets: Cancellation and refinement properties, Math. Scand. 51 (1982), 87–120.
Lovasz, L.: Operations with structures, Acta Math. Acad. Sci. Hungar. 18 (1967), 337–339.
McKenzie, R.: Arithmetic of finite ordered sets: Cancellation of exponents, I, Order 16 (1999), 313–333.
McKenzie, R.: Arithmetic of finite ordered sets: Cancellation of exponents, II, Order 17 (2000),309–332.
Novotný, M.: Ñber gewisse Eigenschaften von Kardinaloperationen, Spisy Priírod. Fak. Univ. Brno (1960), 465–484.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
McKenzie, R. The Zig-Zag Property and Exponential Cancellation of Ordered Sets. Order 20, 185–221 (2003). https://doi.org/10.1023/B:ORDE.0000026529.04361.f8
Issue Date:
DOI: https://doi.org/10.1023/B:ORDE.0000026529.04361.f8