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

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.

> 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). Received on Fri Dec 11 2015 - 18:30:04 CET