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

On Wednesday, December 23, 2015 at 2:49:21 AM UTC-8, Norbert_Paul wrote:
> To get your point it would be helpful to write down the polynomil expression
> and to /explicitly/ specify the transformation rules between polynomial expressions
> and their "corresponding" relations.
>
> > [x y z]
> > 1 1 1
> > 2 1 1
> > 3 2 1
> > 3 2 2
>
> What does that mean?
>
> 1x + 1y + 1z (1 1 1)
...
> or else
>
> x^1 + y^1 + z^1 (1 1 1)
> + x^2 + y^1 + z^1 (2 1 1)
> + x^3 + y^2 + z^1 (3 2 1)
> + x^3 + y^2 + z^2 (3 2 2)
>
> or, mwaybe,
>
> (x^1 + y^1 + z^1) (1 1 1)
...
>
> or something completetly different?

It means

x=1,y=1,z=1
x=2,y=1,z=1
x=3,y=2,z=1
x=3,y=2,z=2

Less sloppily it means

x=1 & y=1 & z=1
|
x=2 & y=1 & z=1
|
x=3 & y=2 & z=1
|
x=3 & y=2 & z=2

The polynomial system (of constraints) which roots we have just listed can be written in many ways. The basic idea how to construct it is start with one tuple, which is constrained with a system of three linear equations

x-1=0
y-1=0
z-1=0

and use union of varieties rule to add more tuples. The details:

https://vadimtropashko.wordpress.com/2014/01/02/relations-as-finite-varieties/

> > The second system consists of the two equations, the one constraining the domain of x, and
> > another is functional dependency itself (in explicit analytical form). It corresponds to
> > relation/finite variety
> >
> > x=1,y=1 x=2,y=1 x=3,y=2
>
> So merely a set of three discrete points?

Yes.

> What is so interesting about storing a finite point set
> S \subset R^n (or Q^n, or double^n) into an n-ary relation?

n-ary relation is a finite point set (of roots of multivariate polynomial system). Why is it interesting?

1. If we drop adjective "finite" as in "finite varieties" we'll get generalized relations
2. We can leverage analytic methods as just have been illustrated with proof of Heath's theorem
3. Instead of geometric objects (varieties) we can study dual algebraic objects (ideals). Ideals formalize multi-relations and this is one direction to go. Algebra of ideals is relational lattice (this has not been rigorously proven yet).