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

On Friday, December 11, 2015 at 9:30:06 AM UTC-8, Norbert_Paul wrote:
> Tegiri Nenashi wrote:
> > On Friday, December 11, 2015 at 4:02:49 AM UTC-8, Nicola wrote:
> >>> Tegiri Nenashi wrote:
> >>>
> >>>> Assuming that algebra of binary relations is good fit for topology, then relations with named attributes (database relations) most likely aren't.
> >>>
> >>> I don't get that point. What objections are against the names "1" and "2", (or, maybe "one" and "two").
> >>
> >> I too don't fully understand it. You have commented in the past that the theory of multivariate relations does not generalize the theory of binary relations. While this is true in the sense that
> >> some operations, such as transitive closure, cannot be extended (at least, not straightforwardly), I do not see what prevents you from using relation algebra (not relational algebra!) in the
> >> context of multivariate relations, i.e., in databases, as a proper closed subsystem. Is it negation?
> >
> > This was a sentiment far from rigorous proposition. People who are aware of Tarski&Givant "A formalization of set theory without variables" would point out that certainly a parallel between binary
> > relations and predicate calculus (and, therefore, database theory) is well established. I will concede this point for now, and just agree that binary relations provide convenient middle ground to
> > meet.
>
> Just to clarify: A topological space (X,T) is composed of an /arbitrary/ set X
> with a topology T. X can also be the set of records in a relational databas table
> with /arbitrary/ arity. If T is represented by a binary relation R on X it should
> be modelled accordingly with all reedom relational database design gives you.

Are you saying that your contribution is introducing topological datatype? (This would answer Nicola's question why do you need special functions like INTERSECT as opposed to being able to reduce it to logical operator of relational algebra). Please note that in this case the term "topological databases" becomes a little misleading.  

> > Now, binary relations might provide a vehicle to get Norbert Paul's theory more accessible. For example what is topology in terms of adjacency relation? What queries topological method allows to
> > express which are not expressible with the help of adjacency relation?
>
> Again, given a set X with a binary relation R on X (hence R is a subset of X x X).
> Then
>
> T(R) := { A is subset of X | every x R a with a in A satisfies x in A }
>
> is a topology for X. This is the "topology in terms of R". Every topology T for a
> finite set X (say, of database records in a table X) has such a relation R such
> that T(R) = T (There are infinite cases, too).
>
> So for finite sets /everything/ that is "topological" can be expresse with such a
> binary relation. If it cannot it is not topological (in the mathematical meaning).

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

This is interesting stuff indeed. Received on Fri Dec 11 2015 - 19:33:43 CET