A motivation and generalization of scott's notion of a continuous lattice

Abstract

In [6], Scott introduced continuous lattices as the correct setting for an abstract theory of computation. The motivation and definition of continuous lattices was primarily in topological terms. In [7], Scott discussed continuous lattices primarily from a topological point of view. However, buried in [7] is an indication (without proof) of how to approach continuous lattices from a purely order-theoretic perspective. This order-theoretic approach seems to have escaped the notice of most computer scientists. We develop this approach in the more general setting of chain-complete posets and offer some arguments in support of the thesis that "continuous posets" (chain-complete posets with a basis) are the proper setting for an abstract theory of computation. Our definition of basis also generalizes that used by Markowsky and Rosen [5]. Finally, we discuss a number of constructions which construct posets with a basis from posets with bases.