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

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. Received on Mon Dec 14 2015 - 20:36:34 CET