Decomposable collections of sets.
Barry Burd · Notre Dame Journal of Formal Logic · 1984
The notion of a weave was first defined by Gaisί Takeuti as an approach to the problem of Borel Determinateness.In a paper co-authored by Burd and Takeuti [1] some of the game-like properties of weaves were explored.A weave is a set-theoretic object that corresponds to a two-person game in which each player presents a choice of moves from a set of possibilities rather than a single move.In another paper [2] Green and Takeuti used the weave idea to prove a theorem about Boolean polynomials.The Green-Takeuti paper gives sufficient conditions enabling a Boolean polynomial to be factored into a Boolean statement in which no atom appears more than once.Consider a Boolean polynomial to be a collection of sets, each atom being an element, each term being a set of elements.In this context the Green-Takeuti theorem is a theorem about the decomposition of collections of sets.In this paper we present a new proof of the Green-Takeuti theorem, and extend the proof to cover the case where the collection of sets is infinite (i.e., the Boolean polynomial is infinitary).We do this using the notion of a weave. Definition 1Let P be a set.Let W and W' be nonempty subsets of iP(P) -I φ ί.The pair is a Weave of P iff both:(1) for each set W in W, and each set W' in W, the intersection W Π W f is a singleton set (2) for each element p in P, there is a set W in W and a set W' in W such that wnw'=\p\.The set P is called the set of Points of the weave.We denote this by writing P = Points((W,W>).Notice that Clause 2 in the definition of a weave *The author wishes to thank Gaisi Takeuti, Harriet Ritter, and Alan Candiotti for their aid and support.