Re: Distributivity in Tropashko's Lattice Algebra

Vadim Tropashko wrote:
> "01" is defined as a neutral element that satisfies the following
> identities:
>
> 01 join A = A
> 01 union A = 0
>
> "10" is defined as a neutral element that satisfies the following
> identities:
>
> 10 join A = A
> 10 union A = A
>
> Note that DEE and DUM in D&D algebra satisfy only a single identity
> each.
>
> Now, the novelty is that there is a *third* special element 00 the
> relation with with arity = 0 and cardinality = 0, which is not a unit
> in Relational Lattice. It is a mapper of a relation into a header
> relation!
>
> 00 join A = a

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)

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

for now. Received on Tue Aug 16 2005 - 20:39:43 CEST