A note on the Banach mapping theorem and its related conjecture

We extend, through an elementary way, the results of Banach [1], Fan [3], and Sanders [7], which concern a finite collection {fi : Ai → Ai+1}i=1 of mappings with An+1 = A1 is decomposable as follows: fi(Bi) = Ai+1\Bi+1, where Bi ⊆ Ai for all i and Bn+1 = B1. Our theorem determines when such a collection is decomposable. We also show that such a set B1 is unique up to an addition of a certain set, which was conjectured by Sanders in [7]. Let A1 and A2 be two sets and let f1 : A1 → A2 and f2 : A2 → A1 be two mappings. If both f1 and f2 are injections, then the Banach mapping theorem [1] asserts that there exist B1 ⊆ A1 and B2 ⊆ A2 such that f1(B1) = A2\B2 and f2(B2) = A1\B1, and the Cantor-Schroeder-Bernstein theorem [5, Theorem 4.5.5] asserts that there exists a bijection from A1 to A2. It is well known that the former theorem implies the latter one, by defining a desired bijection to be f1 if restricted to B1, and the inverse of f2 otherwise. In fact, the Banach mapping theorem can be generalized by removing the condition of injections; refer to [5, p.102, Exercise 4.21], and our previous work [6] for an elementary proof. In order to extend the Banach mapping theorem, let n be a positive integer, {Ai} n+1 i=1 be a collection of sets with An+1 = A1, and {fi : Ai → Ai+1} n i=1 be a collection of mappings. We say that {fi}i=1 is decomposable if there exists a subset B1 of A1, possibly empty, such that the recursive definition Bi+1 = Ai+1\fi(Bi) for 1 ≤ i ≤ n implies Bn+1 = B1; in this case, we also say that {fi}i=1 is decomposed by B1. Let F = fn ◦ · · · ◦ f1 and let Q be the union of all subsets P of A1 such that P is invariant for F , i.e., F (P ) = P . The Banach mapping theorem says exactly that if n = 2 then {fi}i=1 is always decomposable. We briefly summarize its generalizations in literature as follows. Assume that each fi is injective and each Ai is infinite. Fan [3] extended this result for all even n. Froda [4], in a review, pointed out that this is not true for odd n. Sanders [7] showed that when n ≥ 3 is odd, in order that {fi}i=1 can be decomposable, it is sufficient and necessary that there exists a subset Q′ of Q such ∗Department of Applied Mathematics, National Yang Ming Chiao Tung University, 1001 University Road, Hsinchu 30010, TAIWAN, Tel: +866-3-5712121 ext. 56463, Fax: +866-3-5131223, E-mail: mcli@math.nctu.edu.tw