On the existence of fixed points in a totally ordered set
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- by F. T. Metcalf and T. H. Payne PDF
- Proc. Amer. Math. Soc. 31 (1972), 441-444 Request permission
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
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- Alexander Abian, A fixed point theorem for nonincreasing mappings, Boll. Un. Mat. Ital. (4) 2 (1969), 200–201. MR 0244110
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 31 (1972), 441-444
- DOI: https://doi.org/10.1090/S0002-9939-1972-0286722-7
- MathSciNet review: 0286722