Trading properties and Alexandrov kernels for Boolean functions

Simple games correspond to monotonic Boolean functions, but their study is associated with a distinct set of intuitions and questions. We show how "tr ading properties" from simple games can be adapted to the nonmonotonic cont ext, and how a number of these trading properties (some new and some old) a re tied to each other via two kernel operations of the Alexandrov topology. These results provide an answer to a question of Peter Hammer about the re lationship between the property referred to here as weak monotonicity and s ome better-known properties of Boolean functions. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 06E30; 94C10; 91A12.