Re: Comments on Norbert's topological extension of relational algebra

Op maandag 14 december 2015 20:36:52 UTC+1 schreef Tegiri Nenashi:
> On Saturday, December 12, 2015 at 4:29:41 AM UTC-8, Norbert_Paul wrote:
> > Tegiri Nenashi wrote:
> > > On Friday, December 11, 2015 at 10:33:48 AM UTC-8, Tegiri Nenashi wrote:
> > >> A [binary] relation induces Galois Connection and closure operator. It doesn't necessarily satisfy union axiom
> > >> http://mathoverflow.net/questions/35719/when-does-a-galois-connection-induce-a-topology
> > >> There are several ways to generate a topology from given binary relation
> > >> http://www.emis.de/journals/BMMSS/pdf/v31n1/v31n1p4.pdf
> > >
> > > Perhaps people are bewildered what Galois Connection has to do with it. One particular way by which Galois connections often arise is relation-generated. It involves posets -- powersets of some set X and Y, induced by a relation R between the elements of the original X and Y. Galois's original example
> > > was of this kind. Wille's Formal Concept Analysis is about relation-generated Galois Connections too.
> > >
> > > Now what about closure operator induced by relation-generated Galois Connection? It is still not necessarily Kuratovski closure. Counterexample:
> >
> > Topological closure is always Kuratovski closure.
> > So Galois-connection may be of theoretical (and maybe
> > even practical) interest but it is not part of my model.
>
> However it is interesting to know how much of topological structure can be taken away (generalized) and the result would still be compatible with relational algebra.
>
> True or false?
> Proposition. Let σ : 2^X → 2^X be a map (where 2^X is a powerset of X). Then, the following statements are equivalent.
> (1) σ is topological closure operator on X,
> (2) σ satisfies Kuratovski axiom, and there is a Galois connection (K, L) which satisfies σ = LK.

False, I think. The identity would satisfy always (2) but not necessarily (1).

What *I* am interested in is the connection with this work:

http://alpha.uhasselt.be/~lucp1080/queries_reals.pdf

:-)

  Jan Hidders Received on Tue Dec 15 2015 - 17:14:14 CET