Lattices and Tarski's Theorem
نویسنده
چکیده
In this lecture, we develop the theory of supermodular games; key references are the papers of Topkis [7], Vives [8], and Milgrom and Roberts [3]. Our development closely follows that of Milgrom and Roberts, though we will also note other references where necessary. We start with some basic definitions and facts about lattices. Given a set X, a binary relation is a partial ordering on X if it is reflexive (i.e., x x for all x ∈ X); transitive (i.e., x y and y z implies x z); and antisymmetric (i.e., x y and y x implies x = y). The relation is a total ordering if x y or y x for all x, y. Given any set S ⊂ X, an element x is called an upper bound of S if x y for all y ∈ S; similarly, x is called a lower bound of S if y x for all y ∈ S. We say that x is a supremum or least upper bound of S in X if x is an upper bound of S, and for any other upper bound x ′ of S, we have x ′ x; note that the supremum is unique if it exists. In this case we write x = sup S. We similarly define infimum (or greatest lower bound), and denote it by inf S. We will occasionally need to be explicit about the underlying set in which we are computing the supremum or infimum; in such situations, we will write sup X S or inf X S for the supremum of S in X, and the infimum of S in X, respectively. The partially ordered set (X,) is a lattice if for all pairs x, y ∈ X, the elements sup{x, y} and inf{x, y} exist in X. The lattice (X,) is a complete lattice if in addition, for all nonempty subsets S ⊂ X, the elements sup S and inf S exist in X. A set S is a sublattice of (X,) if for any two x, y ∈ S, the elements sup X {x, y} and inf X {x, y} lie in S. Note that (S,) can be a lattice without being a sublattice; i. The following theorem is a basic result in theory of lattices. Note that a function f : X → X is increasing if x y implies f (x) f (y). …
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