Compact Posets and Semilattices

Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove and Dana S. Scott
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Gerhard Gierz: Technische Hochschule Darmstadt, Fachbereich Mathematik
Karl Heinrich Hofmann: Tulane University, Department of Mathematics
Klaus Keimel: Technische Hochschule Darmstadt, Fachbereich Mathematik
Jimmie D. Lawson: Louisiana State University, Department of Mathematics
Michael W. Mislove: Tulane University, Department of Mathematics
Dana S. Scott: Merton College

Chapter Chapter VI in A Compendium of Continuous Lattices, 1980, pp 271-303 from Springer

Abstract: 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.

Keywords: Maximal Chain; Compact Hausdorff Space; Continuous Lattice; Compact Element; Algebraic Lattice (search for similar items in EconPapers)
Date: 1980
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DOI: 10.1007/978-3-642-67678-9_7

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