Pacific Journal of Mathematics 

About: Pacific Journal of Mathematics is an academic journal published by Mathematical Sciences Publishers. The journal publishes majorly in the area(s): Bounded function & Invariant (mathematics). It has an ISSN identifier of 0030-8730. Over the lifetime, 10566 publications have been published receiving 213821 citations. The journal is also known as: Journal of mathematics.

A lattice-theoretical fixpoint theorem and its applications

Abstract: 1. A lattice-theoretical fixpoint theorem. In this section we formulate and prove an elementary fixpoint theorem which holds in arbitrary complete lattices. In the following sections we give various applications (and extensions) of this result in the theories of simply ordered sets, real functions, Boolean algebras, as well as in general set theory and topology. * By a lattice we understand as usual a system 21 = (A 9 <) formed by a non-empty set A and a binary relation <; it is assumed that < establishes a partial order in A and that for any two elements a f b E A there is a least upper bound (join) a u b and a greatest lower bound (meet) an b. The relations >L, <, and > are defined in the usual way in terms of <. The lattice 21 = (A, <) is called complete if every subset B of A has a least upper bound ΌB and a greatest lower bound Πβ. Such a lattice has in particular two elements 0 and 1 defined by the formulas 0 = ΓU and 1 = 11,4. Given any two elements a 9 b E A with a < b, we denote by [a 9 b] the interval with the endpoints a and b, that is, the set of all elements x E A for which a < x < b; in symbols, [ a,b] = E x [x E A and a .< x .< b ]. The system \[α,6], <) is clearly a lattice; it is a complete if 21 is complete. We shall consider functions on A to A and, more generally, on a subset B of A to another subset C of A. Such a function / is called increasing if, for any 1 For notions and facts concerning lattices, simply ordered systems, and Boolean algebras consult [l].

On general minimax theorems

Abstract: There have been several generalizations of this theorem. J. Ville [9], A. Wald [11], and others [1] variously extended von Neumann's result to cases where M and N were allowed to be subsets of certain infinite dimensional linear spaces. The functions / they considered, however, were still linear. M. Shiffman [8] seems to have been the first to have considered concave-convex functions in a minimax theorem. H. Kneser [6], K. Fan [3], and C. Berge [2] (using induction and the method of separating two disjoint convex sets in Euclidean space by a hyperplane) got minimax theorems for concave-convex functions that are appropriately semi-continuous in one of the two variables. Although these theorems include the previous results as special cases, they can also be shown to be rather direct consequences of von Neumann's theorem. H. Nikaidό [7], on the other hand, using Brouwer's fixed point theorem, proved the existence of a saddle point for functions satisfying the weaker algebraic condition of being quasi-concave-convex, but the stronger topological condition of being continuous in each variable. Thus, there seem to be essentially two types of argument: one uses some form of separation of disjoint convex sets by a hyperplane and yields the theorem of Kneser-Fan (see 4.2), and the other uses a fixed point theorem and yields Nikaidό's result. ΐn this paper, we unify the two streams of thought by proving a minimax theorem for a function that is quasi-concave-convex and appropriately semi-continuous in each variable. The method of proof differs radically from any used previously. The difficulty lies in the fact that we cannot use a fixed point theorem (due to lack of continuity) nor the separation of disjoint convex sets by a hyperplane (due to lack of convexity). The key tool used is a theorem due to Knaster, Kuratowski, Mazurkiewicz based on Sperner's lemma. It may be of some interest to point out that, in all the minimax theorems, the crucial argument is carried out on spaces M and N that