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Boolean games - classifying strategies and omitting cardinality assumptions. (English) Zbl 0541.90110
Summary: We deal with a transfinite game on Boolean algebras introduced by T. Jech. The game yields a fine method for handling $${\mathcal K}$$-closed dense subsets of Boolean algebras. We prove (without set-theoretical assumptions) the existence of a $$\gamma^+$$-closed dense subset for a certain type of Boolean algebras determined in the game of an uncountable length $$\gamma$$ which is a generalization of some results by M. Foreman [”Games played on Boolean algebras”, preprint]. We investigate relationships between certain cardinal characteristics of Boolean algebras, discuss the existence of positional strategies of trees, and give a couple of problems concerning the partially ordered set of all strategies.
##### MSC:
 91A99 Game theory 03G05 Logical aspects of Boolean algebras 06E05 Structure theory of Boolean algebras