Re: Distributivity in Tropashko's Lattice Algebra

"Vadim Tropashko" <vadimtro_invalid_at_yahoo.com> wrote in message news:1124224345.270872.245610_at_g49g2000cwa.googlegroups.com...
>
> vc wrote:
>> Vadim Tropashko wrote:
>> >
>> > "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)
>
> I see that you've already addressed this point. I'll add that A is
> bounded between the tattice least and greatest element
>
> 01 < A < 10
>
> while
>
> A and 00 is not comparable
>
>> >
>> > "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.
>
> In D&D algebra
>
> A union DEE != A
> A union DEE != DEE
> A join DEE = A
>
> A union DUM = A
> A join DEE != DEE
> A join DEE != A

But in your algebra:

A union DEE = DUM
A join DEE = A

A union DUM = DUM
A join DUM= 'a0'

>
>
>> > 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.
>
> It's a novelty for me;-)

What's 'a novelty' for you ? That '00'= DUM and DEE='10' ? Or that '10' is a novelty ?

>
Received on Wed Aug 17 2005 - 13:05:30 CEST