Chain Properties in P omega

In the past decade, semantical domains of programming languages have been modeled by c.p.o.‘s ([5,6]), i.e. partially ordered sets where every ascending chain has a least upper bound. As well important are chain-complete subsets of c.p.o.‘s which also have the “closed under chain” property. This paper is devoted to the study of chain-complete subsets of the lattice Pw, a universal domain for data types ([6,8]). A subset E of a c.p.0. is said to be chain-complete if it contains the least upper bound of every ascending chain lying in it (i.e. E). Let us first mention how chain-complete sets arise in mathematical semantics: (1) Suppose we model data types by c:ontinuousl lattices [S] and we want to know when one data type is “contained” or embedded in another. Mathematically, when does a subset E of a given continuous lattice D become a continuous lattice under the induced ordering? A sufUicient condition is that E is a retract of D ([5]). Retractions from D to D are continuous mappings satisfying I’ 0 I = r. A retract of D is given by the range of some retraction map t : D + D. Since r is idempotent, it can be verified that the range r(D) consists of all the fixed points of r, i.e.