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

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:

X={a,b,c}
Y={1,2,3}
R={(a,1),(a,3),(b,3),(c,2),(c,3)}

cl({a}) union cl({c}) = {a,c}
cl({a} union {c}) = {a,b,c} Received on Fri Dec 11 2015 - 20:43:29 CET