Stephen P. King (stephenk1@home.com)
Sat, 10 Apr 1999 20:50:11 -0400
Dear Hitoshi,
I understand now. :) Thanks... A question: Is it sufficient to consider
all interactions to be representable in terms of pairs? What if
associativity fails?
Later,
Stephen
Hitoshi Kitada wrote:
>
> Dear Stephen,
>
> Just on the following point:
>
> > > E.g., consider a set L={1,2,3}. (In the case of the universe, L may be an
> > > infinite set. At this point, to use the notion "cluster decomposition" b
> > > concerning the universe may be an abuse at least at the present stage of the
> > > theory. This point might be related with Obler's paradox as I mention below.)
> > >
> > > Then the set Q is
> > >
> > > Q={ {{1},{2},{3}},
> > > {{1},{2,3}}, {{1,2},{3}}, {{1,3},{2}},
> > > {{1,2,3}} },
> > >
> > > consisting of 5 elements. b varies over those elements.
> >
> > Forgive my mathematical naivete, but is Q here an example of a power
> > set?
>
> The power set P of L={1,2,3} is
>
> P={ {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
>
> that is different from Q. Q is the set of decomposition b of the set L. A
> decomposition b of L is a set defined as
>
> b={C1,C2,...,Ck},
>
> where Cj is a subset of L, which are mutually disjoint (i.e. Ci and Cj do not
> intersect when i is not equal to j), and the sum set of C1,...,Ck is equal to
> L. E.g.,
>
> b={{1,2}, {3}}, etc.
>
> Best wishes,
> Hitoshi
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