Dissertation: The Least Fixed Point Property for ω-chain Continuous Functions

The basic definitions are given in the first section, including those for ω-chain continuity, ω-chain completeness, and the least fixed point property for ω-chain continuous functions. Some of the relations between completeness and fixed point properties in partially ordered sets are stated and it is briefly shown how the question basic to the dissertation arises. In the second section, two examples are given showing that a partially ordered set need not be ω-chain complete to have the least fixed point property for ω-chain continuous functions. Retracts are discussed in section 3, where it is seen that they are not sufficient to characterize those partially ordered sets having the least fixed point property for ω-chain continuous functions. In section 4, the relation between finite width and the least fixed point property for ω-chain continuous functions is explored. Section 5 introduces the notion of a layered partially ordered set and discusses some of its problems