Compact Posets and Semilattices

Abstract

As the title of the chapter indicates, we now turn our attention from the principally algebraic properties of continuous lattices to the position these lattices hold in topological algebra as certain compact semilattices. Indeed, as the Fundamental Theorem 3.4 shows, continuous lattices are exactly the compact semilattices with small semilattices in the Lawson topology. Thus, continuous lattices not only comprise an intrinsically important subcategory of the category of compact semilattices but also form the most well-understood category of compact semilattices. In fact there are only two known examples of compact semilattices which are not continuous lattices; these are presented in Section 4. The paucity of such examples attests to the unknown nature of compact semilattices in general.