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Unformatted text preview: A SHORT AND CONSTRUCTIVE PROOF OF TARSKI’S FIXED-POINT THEOREM FEDERICO ECHENIQUE Abstract. I give short and constructive proofs of Tarski’s fixed- point theorem, and of a much-used extension of Tarski’s fixed-point theorem to set-valued maps. 1. Introduction I give short and constructive proofs of two related fixed-point the- orems. The first is Tarski’s fixed-point theorem: If F is a monotone function on a non-empty complete lattice, the set of fixed points of F form a non-empty complete lattice. The second is a much-used extension of Tarski’s fixed-point theorem to set-valued functions: If ϕ : X → 2 X is monotone—when 2 X is endowed with the induced set order—the set of fixed-points of ϕ form a non-empty complete lattice. Tarski’s  original proof is beautiful and elegant, but non-constructive and somewhat uninformative. Cousot and Cousot  give a construc- tive proof of Tarski’s fixed-point theorem. Their proof is long and quite involved, though. The proof I present is much simpler, and fits in a napkin. On the other hand,  obtain certain sub-products from their approach that I do not obtain; I shall only be concerned with Tarski’s fixed-point theorem, and its extension to set-valued functions. The extension to set-valued functions was developed by Smith  and Zhou . I give a constructive proof of Zhou’s version of the result. Zhou’s version of the result is important in game theory (,). Smith has a weaker monotonicity requirement than Zhou, but Smith does not obtain a lattice structure on the set of fixed-points. In addition, Smith needs a continuity assumption. 2. Definitions A set X endowed with a partial order ≤ is denoted h X, ≤i . h X, ≤i is a complete lattice if, for all nonempty B ⊆ X , V B and W B exist in Date : May 5, 2003....
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This note was uploaded on 04/18/2011 for the course COMPUTER S 1111 taught by Professor Name during the Spring '05 term at MIT.
- Spring '05