Re: Distributivity in Tropashko's Lattice Algebra

Vadim Tropashko wrote:
> > Note however,
> > that in the article there is a certain controversy about TABLE_DUM and
> > TABLE_DEE. Apparently, in the attempt to make them symmetrical
> >
> > TABLE_DUM is defined with arity = 0, cardinality = *1*, and
> > TABLE_DEE is defined with arity = 1, cardinality = 0
>
> There is no controversy. The article introduced the two lattice units
> that I can't reproduce pictirially here due to ascii constraint. I have
> to introduce ascii-frendly symbols:
>
> "01" is defined as a neutral element that satisfies the following
> identities:
>
> 01 join A = A
> 01 union A = 0

Let 'join' be the natural join from your aticle, 'union' the generalized union from your article, '01' the relation with arity 0 and card 1, '10' the relation with the header of probably countably infinite cardinality (since it's a union of all the headers in some implementation of your algebra), '00' the relation with arity zero and cardinality zero.

Then '01' join A = A
but
'01' union A = '00' (unless '0' was typo)

>
> "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.

It's unclear what you mean. See below.

>
> 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

Nope, '00' join A = A. It's no different from '01' join A Also, '00' union A = '00'. Besides, since DUM='00' (and DEE='10'), '00' is no novelty, '10', on the other hand, is.

N.B. You may define of course 'a' as '00' join A, but then the question is what is '00' ? Received on Tue Aug 16 2005 - 21:19:37 CEST