On the existence of fixed points in a totally ordered set
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 , and the existence of a fixed point of .
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Keywords: Fixed point, totally ordered set, conditionally complete
Article copyright: © Copyright 1972 American Mathematical Society