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

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.

On the v31n1p4.pdf paper:

Def. 2.2: /Every/ set of subsets of a set X is a subbase of a topology for X.

My method of generating a topology from an arbitrary relation R is similar to case (ii), but instead of R I take its preorder R^*. Received on Sat Dec 12 2015 - 13:29:39 CET