Applied Categorical Structures

, Volume 11, Issue 2, pp 117–124 | Cite as

Joins in the Frame of Nuclei

  • Martín H. Escardó


Joins in the frame of nuclei are hard to describe explicitly because a pointwise join of a set of closure operators on a complete lattice fails to be idempotent in general. We calculate joins of nuclei as least fixed points of inflationary operators on prenuclei. Using a recent fixed-point theorem due to Pataraia, we deduce an induction principle for joins of nuclei. As an illustration of the technique, we offer a simple (and also intuitionistic) proof of the localic Hofmann–Mislove Theorem.

frame locale compactness nuclei closure operator common fixed point Hofmann–Mislove Theorem 


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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Martín H. Escardó
    • 1
  1. 1.School of Computer ScienceUniversity of BirminghamBirminghamUK

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