, Volume 29, Issue 1, pp 75–84 | Cite as

Lattices of Games

  • Michael Henry AlbertEmail author
  • Richard J. Nowakowski


We show that, for any set S of combinatorial games, the set of games all of whose immediate options belong to S forms a complete lattice. If every option of a game in S also lies in S, then this lattice is completely distributive.


Combinatorial game Distributive lattices 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, M., Nowakowski, R.J., Wolfe, D.: Lessons in Play. A.K. Peters, Wellesley (2007)zbMATHGoogle Scholar
  2. 2.
    Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for your Mathematical Plays, 2nd edn., vols. 1–4. A.K. Peters, Wellesley (2001–2004)Google Scholar
  3. 3.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence (1967)Google Scholar
  4. 4.
    Calistrate, D., Paulhus, M., Wolfe, D.: On the lattice structure of finite games. In: Nowakowski, R. (ed.) More Games of No Chance, pp. 25–30. Cambridge University Press, Mathematical Sciences Research Institute Publications 42 (2002)Google Scholar
  5. 5.
    Conway, J.H.: On Numbers and Games, 2nd edn. A.K. Peters, Wellesley (2001)zbMATHGoogle Scholar
  6. 6.
    Fraser, W., Hirshberg, S., Wolfe, D.: The structure of the distributive lattice of games born by day n. Integers 5(2), A6, 11 pp. (2005, electronic)Google Scholar
  7. 7.
    Raney, G.N.: Completely distributive complete lattices. Proc. Am. Math. Soc. 3(5), 677–680 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Siegel, A.N.: Combinatorial Game Suite. (2000)
  9. 9.
    Tunnicliffe, W.R.: On defining “completely distributive”. Algebra Univers. 19(3), 397–398 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Wolfe, D., Fraser, W.: Counting the number of games. Theor. Comput. Sci. 313, 527–532 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OtagoDunedinNew Zealand
  2. 2.Department of Mathematics & StatisticsDalhousie UniversityHalifaxCanada

Personalised recommendations