Re: x*x-1=0

In article <94rr74$l4j$1_at_news.tue.nl>,   hidders_at_win.tue.nl (Jan Hidders) wrote:
> Vadim Tropashko wrote:
> > In article <94mhpf$jdh$1_at_news.tue.nl>,
> > hidders_at_win.tue.nl (Jan Hidders) wrote:
> > > Note that if you have ordered tuples then the projection is also
> > > likely to be slightly different. If you say PROJ[#1,#2](R) then
> > > that will mean something else than PROJ[#2,#1](R).
> >
> > What are unordered tuples: sets or bags?
>
> Neither.

Do tuples have *internal* structure in terms of set/bags theory?

> The tuples are the elements of the sets and bags.
>Whether you
> allow an element to occur more than once inside a collection
 determines
> if it is a set or a bag.

Understand that -- this is why we have set operations upon them in the first place.

>
> Oh? What version are you reading? In mine he calls it the product.

Oops, actually, in mine too. It's 'TIMES' in formulas, however...

> > > But things get, from an algebraic perspective, a little more
> > > complicated because the cartesian product does not commute as the
> > > join does.
>

In exercise 6.1 Chris suggests to verify that product (along with some other operations) is commutative. On the other page he claims that join is commutative as well.

>
> > It might look less rigorous than CP defined on tuples, but it
> > captures one important concept that tuple theory is missing --
> > domains. In this context CP =/= MULTIPLY, right?
>
> Sorry, but no. Having domains or not has nothing to do with it; you
 can
> also have them with ordered tuples. The only difference is that the
> name of the fields (or columns, if you will) are replaced with
 numbers.
> This is often done in theoretical treatments of the algebra because
 you
> get basicaly the same algebra but without the renaming operation.

I agree that renaming could be viewed as algebraic operation. I wonder 1. if this is a productive definition
2. how would I define relational algebra without renaming (renaming, then, goes into metamodel). The fact that in classic algebras we never consider renaming as part of algebra confuses me 3. how algebra with renaming operation relates to the one without it

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