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

On Tuesday, December 22, 2015 at 1:49:12 AM UTC-8, Jan Hidders wrote:
> I don't follow you here. In your construction for Heath's theorem you are taking a set of polynomial equations and in them you substitute a variable with a function over the other variables. Are you now claiming that there is always an equivalent set of polynomial equations for this new system, no matter what that function is?
>

When we substitute a polynomial expression into polynomial expression we get yet another polynomial expression. Initially we had a system over x,y,z. We have eliminated y, and have gotten the system over x,z. It corresponds to projection of original relation

[x y z]
 1 1 1
 2 1 1
 3 2 1
 3 2 2

to x,z:

[x z]
 1 1
 2 1
 3 1
 3 2  

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

It is not obvious to me what to do if variety is not finite (generalized relation).  

> Btw. I find it a bit misleading to call this a proof of Heath's theorem, since in the relational model a functional dependency does not really imply there is a function to derive the dependent value.

For centuries function has been considered as a set of rules which describes a procedure how to transform an input to output. Mathematicians simply refused to believe in (or saw no purpose for) functions which can't be described via nice analytic formulas. Modern (20th century) treatment is that a function is a relation (set of ordered pairs). Going back to classic treatment (of functions), isn't it cute?

> Also here you are introducing additional restrictions that did not exist in the original relational model.

At the end of the day, theory's worth is determined by what predictions it gives. Cantor set theory would be unnoticed without discovery that transcendental numbers vastly outnumber algebraic ones. Tarski&Givant set theory without variables is still nowhere today because, despite much effort, it doesn't say anything new about sets. As for ideal-variety theory of relations, just two observations (Heath theorem analytic interpretation/proof and multirelations) are running thin on the application side yet. Received on Tue Dec 22 2015 - 18:50:02 CET