Re: Relational Lattice, what is it good for?

Marshall Spight wrote:
> Sets of attributes are given with lower case letters, according
> to which relations these attributes appear in. Thus, (ab) is the
> set of attributes that are common to A and B. This set may be empty.
> The letters of the name are normalized to be in alphabetical order.

This is fine, as long as (ab) is the set of attributes. Below, however, you use it in tuple set notation.

> Definitions
> The Inner Union, A || B is:
>
> { (ab) | (ab) in A or (ab) in B }

I understand this as abbreviation of

 { (ab) | (ab) in pi_a_b A or (ab) in pi_a_b B }

Alternatively, you have to use existential qualifier, as Jon written in the old exchange.

> 4) = { (a,ab,ac,abc,bc) |
> ( (a,ab,ac,abc) in A and (ab,bc,abc) in B )
> or
> ( (a,ab,ac,abc) in A and (ac,bc,abc) in C )
> }
>
> 5) = { (a,ab,ac,abc,bc) |
> (a,ab,ac,abc) in A and
> ((ab,bc,abc) in B or (ac,bc,abc) in C) }

This step is not obvious. Again, careful manipulation would involve projection.

> Suggestions, comments, and particularly
> corrections/refutations/confirmations
> welcome.

You aren't reading Imre Lakatos, by any chance? Anyhow, that's an outstanding effort (by this group standards) that has a chance to be repaired.

A similar manipulation for dual law (with jon and union swapped) reveals a that it has to meet the same criteria. Although I seem to have (a pathological?) counterexample.

Let

`1` := {x=1}
`2b` := {x=2, y=b}

00,01,10, and 11 are defined as before. Then

00 || (`1` && `2b`) = 00
(00 || `1`) && (00 || `2b`) = 01 && 01 = 01 Received on Sat Feb 18 2006 - 01:56:56 CET