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

Published online by Cambridge University Press:  20 November 2018

Smbat Abian  and
Arthur B. Brown
Smbat Abian
Affiliation:
University of Pennsylvania
Arthur B. Brown
Affiliation:
Queens College
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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 P1: {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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 78 - 82
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Birkhoff, G., Lattice theory (A.M.S. Coll. PubL, 25, rev. éd.; New York, 1948).Google Scholar
2. Bourbaki, N., Théorie des ensembles (Paris, 1956) chapter in, p. 48, Example 6b.Google Scholar
3. Bourbaki, N., Sur le Théorème de Zorn, Archiv der Mathematik, 2 (1949-50), 434–437.Google Scholar