On lattices from combinatorial game theory modularity and a representation theorem: finite case. (English) Zbl 1293.91033
Authors’ abstract: We show that a self-generated set of combinatorial games \(S\) may not be hereditarily closed but, strong self-generation and hereditary closure are equivalent in the universe of short games. In [R. J. Nowakowski, “Unsolved problems in combinatorial games”, Preprint], the question ‘Is there a set which will give a non-distributive but modular lattice?’ appears. A useful necessary condition for the existence of a finite non-distributive modular \(\mathcal{L}(S)\) is proved. We show the existence of \(S\) such that \(\mathcal{L}(S)\) is modular and not distributive, exhibiting the first known example. More, we prove a representation theorem with games that allows the generation of all finite lattices in game context. Finally, a computational tool for drawing lattices of games is presented.
Reviewer: Michel Rigo (Liège)
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