Three counterexamples concerning ω-chain completeness and fixed point properties

Abstract

A partially ordered set, is <o -chain complete if, for every countable chain, or o> -chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an o> -chain complete partially ordered set has a least element, denoted by 0.A function / mapping a partially ordered set P into a partially ordered set Q is chain continuous if for any nonempty chain C in P, which has a supremum, /(sup P C) = sup Q /(C). It is o>-chain continuous if for any nonempty countable chain C in P, which has a supremum, /(sup P C) = sup Q /(C). A partially ordered set P has the least fixed point property if every order-preserving function from P to itself has a least fixed point. It has the fixed point property if every order-preserving function from P to itself has a fixed point. It has the least fixed point property for o>-chain continuous functions if every <o -chain continuous function from P to itself has a least fixed point.In 1955 A. Tarski (13, Thm. 1) and A. C. Davis (2, Thm. 2) characterised complete lattices as those lattices having the fixed point property. In 1976 G. Markowsky (6, Thm. 11) characterised the partially ordered sets having the least fixed point property as those which are chain complete. With the development by Dana Scott (8, 9) of lattices as models in theoretical computer science and the generalisation of these ideas by Smyth and Plotkin (12) and M. Wand (14) it has become important to know what kinds of partially ordered sets have the least fixed point property for a -chain continuous functions. In view of Markowsky's characterisation it was natural for Plotkin to ask in Pisa in 1978 if o> -chain completeness follows from the least fixed point property for a) -chain continuous functions. Kolodner has shown in (5) that the converse is true. The following example shows that the answer to Plotkin's question is no. Example 1. Let P l be the partially ordered set defined by the diagram on the next page. Let C = {c n |neN}, X = { x J n e N } and Y = {y n |neN}. Then x = s u p X and C has no supremum in P. Proposition 1. Every <o-chain continuous function from P x to itself has a least fixed point.