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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the existence of fixed points in a totally ordered set


Authors: F. T. Metcalf and T. H. Payne
Journal: Proc. Amer. Math. Soc. 31 (1972), 441-444
DOI: https://doi.org/10.1090/S0002-9939-1972-0286722-7
MathSciNet review: 0286722
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Abstract | References | Additional Information

Abstract: The Tarski fixed point theorem, concerning an isotone mapping of a partially ordered set into itself, is extended to mappings which are not necessarily isotone, but which must map a totally ordered set into itself. The key requirement is that the function be continuous (in a certain sense) whenever it is ``decreasing". Then $ a \leqq f(a),f(b) \leqq b$, and $ a \leqq b \Rightarrow $ the existence of a fixed point of $ f$.


References [Enhancements On Off] (What's this?)

  • [1] Garrett Birkhoff, Lattice theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
  • [2] Alexander Abian, A fixed point theorem for nonincreasing mappings, Boll. Un. Mat. Ital. (4) 2 (1969), 200–201. MR 0244110


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0286722-7
Keywords: Fixed point, totally ordered set, conditionally complete
Article copyright: © Copyright 1972 American Mathematical Society

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