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completely distributive lattices

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  • Janis Buls
    Would you recommend me literature about completely distributive lattices? Thank you. J. Buls buls@fmf.lu.lv
    Message 1 of 2 , Nov 23, 2003
      Would you recommend me
      literature about
      completely distributive lattices?

      Thank you.

      J. Buls
      buls@...
    • Steve Vickers
      There is a decent introduction in Peter Johnstone s Stone Spaces (Cambridge University Press), written from the point of view that completely distributive
      Message 2 of 2 , Nov 24, 2003
        There is a decent introduction in Peter Johnstone's "Stone Spaces"
        (Cambridge University Press), written from the point of view that
        completely distributive lattices are (up to isomorphism) exactly the
        Scott topologies on continuous posets (a.k.a. continuous dcpos, i.e.
        continuous directed complete partial orders).

        Johnstone uses the standard definition. However, there is a problem with
        it in that you end up using the axiom of choice a lot.

        There is also a good definition of "constructively completely
        distributive lattice", classically equivalent to the standard one, that
        has been studied (first) by Fawcett and Wood and (subsequently) by
        Rosebrugh and Wood in a series of papers all entitled "Constructive
        complete distributivity ..."

        Fawcett and Wood "Constructive complete distributivity I" Math. Proc.
        Cam. Phil. Soc. 107 (1990) 81-90.
        Rosebrugh and Wood "Constructive complete distributivity II" Math.
        Proc. Cam. Phil. Soc. 110 (1991) 245-249.

        Another reference: Raney, "A subdirect-union representation for
        completely distributive complete lattices", Proc. Amer. Math. Soc. 4
        (1953) 518-522, showed how each completely distributive lattice could be
        got (up to isomorphism) as the lattice of rounded upper sets in a set
        with a dense transitive relation.

        I developed these in my own paper "Information systems for continuous
        posets", Theoretical Computer Science 114 (1993) 201-229..

        Steve Vickers.


        Janis Buls wrote:

        >Would you recommend me
        >literature about
        >completely distributive lattices?
        >
        >Thank you.
        >
        >J. Buls
        >buls@...
        >
        >
        >
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