On lattices from combinatorial game theory modularity and a representation theorem: finite case

• Autor(a) principal: Carvalho, Alda
• Data de Publicação: 2014
• Outros Autores: Santos, Carlos Pereira dos, Dias, Catia, Coelho, Francisco, Neto, João Pedro, Nowakowski, Richard, Vinagre, Sandra
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.21/5012
• topic: Combinatorial game theory; Lattices; Modularity; Representation theorems

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 [13], 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 L(S) is proved. We show the existence of S such that 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. (C) 2014 Elsevier B.V. All rights reserved.