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PROC

Tarski proved in 1955 that every complete lattice has the fixed
point property. Later, Davis proved the converse that every lattice with the fixed point property is complete. For a chain complete ordered set, there is the well known Abian-Brown fixed point result. As a consequence of the Abian-Brown result, every chain complete ordered set with a smallest element has the fixed point property. In this paper, a new characterization of a complete lattice is given. Also, fixed point theorems are given for decreasing functions where the partially ordered set need not be dense as is the usual case for fixed point results for decreasing functions.

PROC

In the mid 1950's Tarski showed that a
complete lattice P has the property that every increasing
increasing function from P into itself has a fixed point. Anne
Davis proved the converse of this result that every lattice with
the fixed point property is complete. In this paper, the author
proves new equivalences for conditionally complete partially
ordered sets with a type of fixed point property. Some comments
about these theorems are also given in the paper.

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