Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Tarski's Fixpoint Lemma and combinatorial games

  • 45 Accesses

  • 2 Citations


Using Tarski's Fixpoint Lemma for order preserving maps of a complete lattice into itself, a new, lattice theoretic proof is given for the existence of persistent strategies for combinatorial games as well as for games with a topological tolerance and games on lattices. Further, the existence of winning strategies is obtained for games on superalgebraic lattices, which includes the case of ordinary combinatorial games. Finally, a basic representation theorem is presented for those lattices.

This is a preview of subscription content, log in to check access.


  1. 1.

    B.Banaschewski and G.Bruns (1988) The fundamental duality of partially ordered sets, Order 5, 61–74.

  2. 2.

    C.Berge (1953) Sur une théorie ensemblist des jeux alternatifs, J. Math. Pures Appl. 32, 129–184.

  3. 3.

    L.Kálmar (1928) Zur Theorie der abstrakten Spiele, Acta Sci. Math. Univ. Szeged 4, 65–85.

  4. 4.

    A.Pultr and J.Úlehla (1985) Remarks on strategies in combinatorial games, Discr. Appl. Math. 12, 165–173.

  5. 5.

    C. A. B.Smith (1966) Graphs and composite games, J. Comb. Theory 1, 51–81.

Download references

Author information

Additional information

Communicated by D. Duffus

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Banaschewski, B., Pultr, A. Tarski's Fixpoint Lemma and combinatorial games. Order 7, 375–386 (1990). https://doi.org/10.1007/BF00383202

Download citation

AMS subject classifications (1980)

  • 90D05
  • 90D42
  • 06A99

Key words

  • combinatorial games
  • Tarski's Fixpoint Lemma
  • persistent strategies
  • winning strategies
  • superalgebraic lattices