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Re: completeness of the relational lattice

From: Vadim Tropashko <vadimtro_invalid_at_yahoo.com>
Date: Fri, 22 Jun 2007 16:05:22 -0700
Message-ID: <1182553522.724074.26350@q19g2000prn.googlegroups.com>


On Jun 22, 3:31 pm, Jan Hidders <hidd..._at_gmail.com> wrote:
> I can probably simplify using Marshall's rule. Assuming I
> have a function A(e) that gives the header of the result of e:
>
> r /\ (s \/ t) = (r /\ s) \/ (r /\ t) if (A(r) * (A(s)) - A(t) is
> empty and (A(r) * A(t)) - A(s) is empty

OK, to summarize we have 2-sorted algebra: 1. Relations (which is lattice)
2. Relation headers (which is standard BA) The function A maps relations to headers, and enjoys the following rule
A(R \/ B) = A(R) intersect A(B)
Note that function A is not a homomorhism, that is the dual identity A(R /\ B) = A(R) union A(B)
doesn't hold.

Please also note, that in my notation the algebra is not many sorted. I want to emphasize not this fact, however, but that the "crippled homomorphism" rule above becomes an identity 00 /\ (R \/ B) = (00 /\ A) \/ (00 /\ B)
which is a special case of distributivity. So you try to formulate the premise for the Spight criteria and distributivity in a limited context of the 00 element shows up -- I find it remarkable.

Let me also reiterate that some cases can't be cracked even with Spight criteria, and

(00 /\ R) \/ (11 /\ R)

is one of them. If Marhall casts doubts here on the existance of the 11 element, my reply would be take any relation "big enough" instead of 11.

BTW, if you have finction A, then we probably don't need element 00, right? (The element A(01) is just an empty set in the boolean algebra of headers, not the lattice element 00) Received on Fri Jun 22 2007 - 18:05:22 CDT

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