A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems

Abstract

In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries. Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P , a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element ε greater than or not comparable to f(ε), W(a) , although constructed differently, is identical with the set A of Bourbaki (3) determined by a , f , and P 1 : { x|x ∈ P, x ≤ f(x)}. Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new.Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4).The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3).The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice.