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

Op dinsdag 15 december 2015 19:53:47 UTC+1 schreef Norbert_Paul:
> Jan Hidders wrote:
> > 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).
> I do not know much about Galois connections, but if identity can be constructed from a
> Galois conenction in the above manner then it is not a counter-example.
> Identity is the closure operator of the discrete topological space.

Ow, had not realised that. So on any discrete topology, the identity is a closure operator? And in fact the only closure operator?

  Jan Hidders Received on Wed Dec 16 2015 - 16:44:49 CET