Re: What to call this operator?

In article <1119976637.137573.61210_at_g47g2000cwa.googlegroups.com>, mikharakiri_nospaum_at_yahoo.com says...
> 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.

This is very interesting. Would you mind elaborating? Can you formalise the absorption law? How did you arrive at these partial order definitions?

> 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.

I quite liked the symmetry of the join and union of Tropashko's lattice algebra---that the attributes of a join result is the union of the attributes of its operands, and the attributes of a union result is the intersection of the attributes of its operands.

However, the symmetry/duality is lost in the semantics of the operations---the relation predicate of the result. If relvars A and B have predicates PA and PB, respectively, the expression A JOIN/<AND> B has the predicate PA AND PB (logical and) in both D&D and Tropashko. However, D&D's A <OR> B has the predicate PA OR PB (logical or), while Tropashko's A UNION B has a rather more complicated and less intuitive predicate (on first glance).

> 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.

Why is this absorption so important?

-- 
Jon