Construction of Homomorphisms of ^-continuous Lattices

We present a direct approach to constructing homomorphisms of A/-continuous lattices, a generalization of continuous lattices, into the unit interval, and show that an M -continuous lattice has sufficiently many homomorphisms into the unit interval to separate the points. In the past twenty years the concept of a continuous lattice and its generalizations have attracted more and more attention. It was the pioneering work of Dana Scott [15], [16] which led to the discovery that algebraic lattices and their generalization, continuous lattices, could be used to assign meanings to programs written in high-level programming languages. On the purely mathematical side, research into the structure theory of compact semilattices led Lawson [9] and others [7], [8] to consider the category of those compact semilattices which admit a basis of subsemilattice neighborhoods at each point. It was discovered in [8] that those objects are precisely the continuous lattices of Scott. One of the most important features of continuous lattices is that they admit sufficiently many homomorphisms (that is, mappings which preserve arbitrary infs and directed sups) into the unit interval to separate the points. The topological form of this result is due to Lawson [9]. For a complete lattice P and a family M of subsets of L, we define a corresponding relation <z^M on P by x <§ím y if and only if, for each S £ M, y < V S implies there exists s £ S such that x < s ; L is called M-continuous if the relation <^m satisfies the interpolation property (i.e. for every x, y £ L, x <^m y implies there is z e P such that x <sím z «a, y) and x = V{y £ L\y <Af x} for all x £ L. For a complete lattice T, a map /: P —► P is called an M-morphism, or, briefly, a homomorphism, if / preserves arbitrary infs and A/-sups. This paper is mainly devoted to the construction of homomorphisms of Mcontinuous lattices into the unit interval [0, 1]. We present a direct approach to the construction of such homomorphisms and show that A/-continuous lattices admit enough homomorphisms into [0, 1] to separate points. Received by the editors February 3, 1994 and, in revised form on July 21, 1994; originally communicated to the Proceedings oftheAMS by Andreas R. Blass. 1991 Mathematics Subject Classification. Primary 06B15, 06B35, 06D05.