Re: What to call this operator?

Jon Heggland wrote:
> In article <IpVve.12426$pa3.12390_at_newsread2.news.atl.earthlink.net>,
> david.cressey_at_earthlink.net says...
> > But seriously, I'm unable to figure out from the formal description what it
> > is. more importantly, I'm unable to figure out what it is FOR. Not that
> > there's anything wrong with your formal description. It's just my own
> > unfamiliarity with it that causes problems.
>
> It is part of a minimalistic approach to relational algebra that is more
> geared towards logic instead of set theory. <OR> is a generalisation of
> union. If the relations have the same heading, the result is a union.
>
> I think the point is to have a counterpart to <AND> (which covers join,
> product, intersection, selection and extension) that covers union, but
> places no restrictions on the types of the operands, and has simple
> logic-based semantics.

I wonder how far this algebra is developed. Both binary operations <AND> and <OR> are idempotent, commutative and accociative. This three properties are enough to define a semilattice. That is we have 2 semilattices:
1. the upper semilattice for <OR>
2. the lowe semilattice for <AND>
Each semilattice defines a partial order relation, one for upper lattice
x<=y iff y=x<OR>y
and one for lower one
x<=y iff x=x<AND>y
Even though I slopily used the same symbol "<=", these two partial orders are incompatible unless the algebra meet the *Absorption Law*. Absorption law merges the two semilattices into a single lattice.

Unfortunately, D&D algebra doesn't meet the absorption law. The Lattice algebra that I mentioned in the other thread does. The partial order there is a generalization of the "is subset of" relation applied to the any pair of database relations, even those with different headings.

Neither of those algebras (D&D,nor Lattice) is boolean. D&D has nice distributivity property, although without absorption it doesn't buy us much. Lattice algebra doesn't have distributivity, which looks like serious obstacle when transforming expressions.     

operations <AND> and <OR> semilatt Received on Tue Jun 28 2005 - 18:37:17 CEST