A lattice fixed-point theorem with constraints

This paper presents a lattice fixed-point theorem having applications in game theory and elsewhere. The results presented here form part of the author's Ph.D. thesis in Operations Research, conducted under the supervision of Robert Wilson. Let L be a complete lattice. Denote elements of L with small letters a, b, c, •••, and denote subsets of L with capital letters A, B, C •••. Consider a function U: L —• L with property P: for any ACL, U(\A) = /\U(A)9 where U(A)= {U(a)\aeA}. 2 Denote the composition of U with itself by U. Property P implies