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Some Fixed Point Theorems for Partially Ordered Sets

Published online by Cambridge University Press:  20 November 2018

Hartmut Höft  and
Margret Höft
Hartmut Höft
Affiliation:
Eastern Michigan University, Ypsilanti, Michigan
Margret Höft
Affiliation:
University of Michigan at Dearborn, Dearborn, Michigan
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Abstract

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A partially ordered set P has the fixed point property if every order-preserving map f : P → P has a fixed point, i.e. there exists x ∊ P such that f(x) = x. A. Tarski's classical result (see [4]), that every complete lattice has the fixed point property, is based on the following two properties of a complete lattice P:

  • (A) For every order-preserving map f : P → P there exists x ∊ P such that x ≦ f(x).

  • (B) Suprema of subsets of P exist; in particular, the supremum of the set {x|x ≦ f(x)} ⊂ P exists.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 5 , 01 October 1976 , pp. 992 - 997
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Abian, S. and Brown, A. B., A theorem on partially ordered sets, with applications to fixed point theorems, Can. J. Math. 13 (1961), 78–82.Google Scholar
2. Birkhoff, G., Lattice theory, 3rd edition (AMS Coll. Pub. XXV, 1973)Google Scholar
3. Rival, I., A fixed point theorem for finite partially ordered sets, Preprint Nr. 184, Januar 1975, Technische Hochschule Darmstadt.Google Scholar
4. Tarski, A., A lattice-theoreticalfixpoint theorem and its applications, Pacific J. Math. 5 (1955), 285–309.Google Scholar
5. Wong, J. S. W., Common fixed points of commuting monotone mappings, Can. J. Math. 19 (1967), 617–620.Google Scholar