The weakly way-below relation
by
Joe Mashburn
University of Dayton

In 1970 Dana Scott defined a relation called the way-below relation on ordered sets. If X is an ordered set and a and b are elements of X then a << b if and only if for every directed subset D of X, if supD ≥ b then there is an element c of D such that a ≤ c. This relation, and the topology that it generates, has been useful in modeling information systems. More recently Keye Martin and Bob Coecke have revised this definition to create a relation that we will call the weakly way-below relation. They define a << _w b to mean that when D is a directed set and supD=b then there is an element c of D such that a ≤ c. This new relation arose in a model for classical and quantum states in physics when the traditional way-below relation failed to provide important information about the model. We will compare the two relations and the topologies that they generate. We will also consider a possible extension of the Coecke-Martin model to a system with an infinite number of possible states.

Date received: February 21, 2006