Fixed points and complete lattices

Pages: 568 - 572, Issue Special, September 2007

 Abstract        Full Text (140.3K)              

Paula Kemp - Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, United States (email)

Abstract: 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:  Complete Lattices, Decreasing, Increasing, and Fixed points.
Mathematics Subject Classification:  Primary 03E25.

Received: September 2006;      Revised: April 2007;      Published: September 2007.