Propaedeutic to chain-complete posets with basis

Abstract

In this paper, several relations between members of a chain-complete poset (weaker than the relations relatively compact and IN introduced by Scott in his study of bases) are considered, with an eye toward examining whether all the properties of bases can be obtained from weaker axioms. It turns out that for all but one relation (relatively chain compact) one gets a weaker notion of bases than using relatively compact or IN. For relatively chain compact we conjecture that one gets a weaker notion of basis. However, we show that in the presence of bounded joins the existence of a basis derived from the relation of relatively chain compact implies the existence of a Scott basis if the cardinality of the chain-complete poset is less than the ω-th infinite cardinal. If one accepts the Continuum Hypothesis this means that the weaker relation is adequate in many posets likely to be of interest to computer science. Finally, even in the presence of the "weaker" type of basis in a complete lattice, the meet operation is continuous.