I will describe certain distributive lattices that arise naturally in
the study of combinatorial games. The "birthday" of a game G is equal
to the height of its complete game tree (equivalently, the length of the
longest terminal line of play proceeding from G). In 2002, Calistrate,
Paulhus and Wolfe showed that the games of birthday at most n form a
distributive lattice L_n, and shortly thereafter Fraser, Hirshberg and
Wolfe characterized the join-irreducibles of L_n.
Although L_n has a natural model in combinatorial games, it can
also be described by an abstract induction. I will give this abstract
definition and show that L_n admits a unique nontrivial (order-preserving)
automorphism, which in turn reveals a fundamental companionship
among combinatorial games. I will also describe a distributive sublattice
L^0_n of L_n, in which the action of the automorphism is particularly
apparent. L^0_n also arises naturally in the study of games, by a
straightforward generalization of the Calistrate-Paulhus-Wolfe machinery.
This talk will be completely self-contained, and it is not
necessary to
attend the morning talk [part of Aaron's thesis defense at 10:30AM
in 939 Evans].
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