Tarski's Fixpoint Lemma and combinatorial games

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.