We extend, using an elementary method, results of Banach (1924), Fan (1952), and Sanders (1961), which concern a finite collection {f (i) : A(i) -> A(i)(+1)}(i=1)(n) of mappings with A(n)(+1) = A(1) which is decomposable as f(i)(B-i) = A(i+1)Bi+1, where B-i subset of A(i) for all i and B-n(+1) = B-1. Our theorem determines when such a collection is decomposable. We also show that such a set B-1 is unique up to an addition of a certain set, which was conjectured by Sanders.
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