Re: Distributivity in Tropashko's Lattice Algebra

vc wrote:
> Mikito Harakiri wrote:
> > Leaving the lattice least element 10 and the greatest element 01 alone,
> > 00 has some interesting properties as well. Define
> >
> > header(A) = A join 00
> >
> > as a set of all attributes of A, or more rigorously, an empty relation
> > with the same header as A. Then,
> >
> > header(A join B) =
> > = (A join B) join 00 =
> > = (A join 00) join (B join 00) =
> > = header(A) join header(B)
> >
>
> I am not sure the formulas are very interesting. For the natural join,
> the attribute set is defined as the union of the attribute sets of the
> relations being joined whatever the relations cardinality might be.
> Your formulas show that the header of the join is the same whether we
> join empty or non-empty relations, literally the same as the
> definition does.

Yes, these formal identities are literally the same as the definitions.

> The same can be said about the union where the attribute set is defined
> by an intersection.
>
> > Likewise,
> >
> > header(A union B) =
> > = (A union B) join 00 =
> > = (A join 00) union (B join 00) =
> > = header(A) union header(B)
> >
> > In the second chain of equalities, we leveraged distributivity by
> > Marshall's criteria! Those are intuitively obvious identities proved
> > formally.
> >
> > I leave the exploration of dual definition
> >
> > rowid(A) = A union 00
>
>
> A union '00' = '00'.

This can't be.

A union '00' = '00'

implies 00 < A, and the partial order "<" being the same in upper and lower semilattice implies that

A join '00' = A

which we saw isn't.

> What's rowid(A) ?

A set of rowids? This is a dual concept to the relation header: table columns are identified by table attribute names, while the rows are identified by rowids. Unlike the set algebra of attribute names, the algebra of rowids doesn't seem to be boolean. Received on Tue Aug 16 2005 - 22:49:35 CEST