Re: Oids

Vadim Tropashko wrote:
> Summarising:
>
> header(A) = a0 = 00 join A
> rowids(A) = Ox = 00 union A

Hmmm.

Let's see if I've followed along correctly.

We have a terminology for some relation values, 00, 01, and the somewhat spooky 10. The two binary digits refer to the arity and the cardinality respectively.

00 join A = a0 = header(A) [ or just a in the earlier terminology ] 00 union A = 0x = isNonEmpty(A)

01 join A = A
01 union A = 01

10 join A = 10
10 union A = A

Is that the consensus?

[Just for my own amusement, I'm going to rewrite the above such that 10 is called "false" and 01 is called "true" and join is called "and" and inner union is called "or".]

false and A = false
false or A = A
true and A = A
true or A = true

So far, we've been describing 10 via universal quantification. The type of 10 we've been using is:
10 = forall A, reduce union over header(A)

but it would be less spooky, and more constructivist, if we defined it existentially:
10 = header(A)

I don't think I'm expressing this all that well, and I'm not convinced I'm using the terminology correctly, but do you see what I mean?

> (Change 'rowids' to 'foo' if you like. Although I find a little comfort
> with informal idea that header can be viewed is an additional hidden
> row in the table, while the list of rowids is an additional hidden
> column).

This doesn't seem right to me. Instead of "foo" I'm going to go with "isNonEmpty". I can see the attraction of the rowids idea, but it doesn't feel right, because duplicates are coalesced in sets, and when we project over zero columns, we're left with only 00 or 01, this being exactly the is-it-empty-or-not distinction.

> Now, it seems that you hit the nail, and the rowid maps Relational
> Lattice into a boolean algebra with just 2 elements 00 and 01. This is
> dual to header lattice homomorhism of Relational Lattice into a boolean
> algebra of sets. The symmetry is sligtly broken, but who would insist
> that rows and columns are completely symmetrical?

If we define 10 existentially, does that fix the symmetry? Instead of a single boolean algebra, we have one-per-type-of-A.

Marshall Received on Thu Aug 18 2005 - 04:39:40 CEST