# Example 410 x r n with x y iff x i y i i 1 n is a

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Chapter 6 / Exercise 124
College Algebra: Real Mathematics, Real People
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Example 4.10 . ( X, - ) = ( R n , ) with x y iff x i y i , i = 1 , . . . , n is a lattice with x y = (min( x 1 , y 1 ) , . . . , min( x n , y n )) and x y = (max( x 1 , y 1 ) , . . . , max( x n , y n )) . X is not a complete lattice, but ([0 , 1] n , ) is, with inf S = (inf { x 1 : x S } , inf { x 2 : x S } ) and sup S = (sup { x 1 : x S } , sup { x 2 : x S } ) . 35
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Chapter 6 / Exercise 124
College Algebra: Real Mathematics, Real People
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Chapter 4.5 Example 4.11 . Suppose that X 6 = , let P ( X ) denote the class of subsets of X , and for A, B ∈ P ( X ) , define A - B if A B . ( P ( X ) , - ) is a complete lattice, with inf S = T { E : E ∈ S} for S ⊂ P ( X ) and sup S = S { E : E ∈ S} . We can turn lattice orderings around and keep the lattice structure. Example 4.12 . Suppose that ( X, - ) is a lattice. Define x - - y if y - x . Show that ( X, - - ) is a lattice. [ Thus, ( R n , - ) , i.e. ( R n , ) , is a lattice, as is ( P ( X ) , ) . ] Example 4.13 . ( P ( N ) , ) is a lattice with many noncomparable pairs of elements, but the class of sets {{ 1 } , { 1 , 2 } , . . . , { 1 , 2 , . . . , n } , . . . } is a chain in P ( N ) . Problem 4.2 . Show that if A B and B is totally ordered, then A is totally ordered. As an application, show that, in ( R 2 , ) , any subset of the graph of a non-decreasing function from R to R is a chain. There are partially ordered sets that are not lattices. Example 4.14 . Let T 1 R 2 be the triangle with vertices at (0 , 0) , (1 , 0) and (0 , 1) . ( T 1 , ) is partially ordered, but is not a lattice because (0 , 1) (1 , 0) is not in T 1 . On the other hand, ( T 2 , ) is a lattice when T 2 is the square with vertices at (1 , 1) , (1 , 0) , (0 , 1) , and (0 , 0) . 5. Monotone comparative statics We now generalize in two directions: (1) we now allow for X and T to have more general properties, they are not just linearly ordered in what follows, and (2) we now also allow for the set of available points to vary with t , not just the utility function. 5.1. Product Lattices. Suppose that ( X, - X ) and ( T, - T ) are lattices. Define the order - X × T on X × T by ( x 0 , t 0 ) % X × T ( x, t ) iff x 0 % X x and t 0 % T t . (This is the unanimity order again.) Lemma 4.15 . ( X × T, - X × T ) is a lattice. Proof : ( x 0 , t 0 ) ( x, t ) = (max { x 0 , x } , max { t 0 , t } ) X × T . ( x 0 , t 0 ) ( x, t ) = (min { x 0 , x } , min { t 0 , t } ) X × T . 5.2. Supermodular Functions. Definition 4.16 . For a lattice ( L, - ) , f : L R is supermodular if for all ‘, ‘ 0 L , f ( 0 ) + f ( 0 ) f ( ) + f ( 0 ) , equivalently, f ( 0 ) - f ( 0 ) f ( ) - f ( 0 ) . Example 4.17 . Taking 0 = ( x 0 , t ) and = ( x, t 0 ) recovers Definition 4.2 . Problem 4.3 . Show that a monotonic convex transformation of a monotonic supermodular function is supermodular. 36
Chapter 4.5 Problem 4.4 . Let ( L, - ) = ( R n , ) . Show that f : L R is supermodular iff it has increasing differences in x i and x j for all i 6 = j . Show that a twice continuously differentiable f : L R is supermodular iff 2 f/∂x i ∂x j 0 for all i 6 = j . 5.3. Ordering Subsets of a Lattice. Definition 4.18 . For A, B L , L a lattice, the strong set order is defined by A - Strong B iff ( a, b ) A × B , a b A and a b B . Recall that interval subsets of R are sets of the form ( -∞ , r ), ( -∞ , r ], ( r, s ), ( r, s ], [ r, s ), [ r, s ], ( r, ), or [ r, ).
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