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Compact Posets and Semilattices

In: A Compendium of Continuous Lattices

Author

Listed:
  • 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)

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.

Suggested Citation

  • Gerhard Gierz & Karl Heinrich Hofmann & Klaus Keimel & Jimmie D. Lawson & Michael W. Mislove & Dana S. Scott, 1980. "Compact Posets and Semilattices," Springer Books, in: A Compendium of Continuous Lattices, chapter 0, pages 271-303, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-67678-9_7
    DOI: 10.1007/978-3-642-67678-9_7
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