PROC
Fixed points and complete lattices
Paula Kemp
Conference Publications 2007, 2007(Special): 568-572 doi: 10.3934/proc.2007.2007.568
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.
keywords: Increasing and Fixed points. Complete Lattices Decreasing
PROC
Characterizations of conditionally complete partially ordered sets
Paula Kemp
Conference Publications 2005, 2005(Special): 505-509 doi: 10.3934/proc.2005.2005.505
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.
keywords: and Fixed points. Increasing Decreasing Conditionally Complete

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