The Least Fixed Point Property for ω-chain Continuous Functions

A partially ordered set P is ω-chain complete if every countable chain (including the empty set) in P has a supremum. … Notice that an ω-chain continuous function must preserve order. P has the (least) fixed point property for ω-chain continuous functions if every ω-chain continuous function from P to itself has (least) fixed point. It has been shown that a partially ordered set does not have to be ω-chain complete to have the least fixed point property for ω-chain continuous functions. This answers a question posed by G. Plotkin in 1978. I.I. Kolodner has shown that an ω-chain complete partially ordered set has the least fixed point property for ω-chain continuous functions. Plotkin and Smythe and others have used ω-chain complete partially ordered sets in their study of models for theoretical computer science in order to have fixed or least fixed point properties. The result should also be compared with G. Markowsky’s result that to have the least fixed point property (every order preserving function has a least fixed point), a partially ordered set must be chain complete. It is the purpose of this paper to look at some cases in which ω-chain completeness and the least fixed point property for ω-chain continuous functions are equivalent