ONT Re: Reductions Among Relations
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RAR. Note 2
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Projective Reduction of Relations in General
I will start out with the notion of "projective reduction"
of relations, in part because it is easier and much more
intuitive (in the visual sense), but also because there
are a number of tools that we need for the other brand
of reduction that arise quite naturally as a part of
the projective setting.
Before we get into the operational machinery and the
vocational vocabulary of it all, let me just suggest
that you keep in mind the following style of picture,
which pretty much says it all, in that reducing to a
unity the "motley of the ten thousand terms" (MOT^4)
manner that the aptest genres and the fittest motifs
of representations are genreally found to immanifest.
Picture a k-adic relation L as a body
that resides in k-dimensional space X.
If the dimensions are X_1, ..., X_k,
then the "extension" of L, an object
that I will, for the whole time that
I am working this "extensional" vein,
regard as tantamount to the relation
itself, is a subset of the cartesian
product space X = X_1 x ... x X_k.
If you pick out your favorite family F of domains among these
dimensions, say, X_F = {X_j : j in F}, then the "projection" of
a point x of X on the subspace that gets "generated" along these
dimensions of X_F can be denoted by the notation "Proj_F (x)".
By extension, the projection of any relation L on that subspace
is denoted by "Proj_F (L)", or still more simply, by "Proj_F L".
The question of "projective reduction" for k-adic relations
can be stated with moderate generality in the following way:
| Given a set of k-place relations in the same space X and
| a set of projections from X to the associated subspaces,
| do the projections afford sufficient data to tell the
| different relations apart?
Next time, in order to make this discussion more concrete,
I will focus on some examples of triadic relations. In fact,
to start within the bounds of no doubt familiar examples by now,
I will begin with the examples of sign relations that I used before.
http://suo.ieee.org/email/msg00729.html
http://suo.ieee.org/email/msg01224.html
Jon Awbrey
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