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Unformatted text preview: A SHORT AND CONSTRUCTIVE PROOF OF TARSKI’S FIXEDPOINT THEOREM FEDERICO ECHENIQUE Abstract. I give short and constructive proofs of Tarski’s fixed point theorem, and of a muchused extension of Tarski’s fixedpoint theorem to setvalued maps. 1. Introduction I give short and constructive proofs of two related fixedpoint the orems. The first is Tarski’s fixedpoint theorem: If F is a monotone function on a nonempty complete lattice, the set of fixed points of F form a nonempty complete lattice. The second is a muchused extension of Tarski’s fixedpoint theorem to setvalued functions: If ϕ : X → 2 X is monotone—when 2 X is endowed with the induced set order—the set of fixedpoints of ϕ form a nonempty complete lattice. Tarski’s [4] original proof is beautiful and elegant, but nonconstructive and somewhat uninformative. Cousot and Cousot [1] give a construc tive proof of Tarski’s fixedpoint 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, [1] obtain certain subproducts from their approach that I do not obtain; I shall only be concerned with Tarski’s fixedpoint theorem, and its extension to setvalued functions. The extension to setvalued functions was developed by Smith [3] and Zhou [7]. I give a constructive proof of Zhou’s version of the result. Zhou’s version of the result is important in game theory ([5],[6]). Smith has a weaker monotonicity requirement than Zhou, but Smith does not obtain a lattice structure on the set of fixedpoints. 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.
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