Fixed points and complete lattices

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.