ONT Re: Relations And Their Divisitudes
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RATD. Note 13
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Viewing k-adic relations as k-dimensional bodies in k-dimensional spaces,
it should be clear now what is meant by the m-projective reducibility of
a k-adic relation L c X = X_1 x ... x X_k, for m in the interval [1, k].
L is "m-projectively reducible" if and only if L is uniquely determined by
its m-adic projection data, which is equivalent to the following statement:
If there exists a relation L' c X that has the same m-adic projection data
as L, then L' = L.
Said the other way around, L is "m-projectively irreducible" if and only if
there exists a relation L' c X that has the same m-adic projection data as L,
and yet L' =/= L. In this case, the pair of relations L, L' constitute what
is known as an "m-projectively indiscernible couplet" of k-adic relations.
It should be visibly clear at this point that there are likely to be
lots and lots of m-projectively irreducible k-adic relations, for m
in the half-open interval [1, k).
For example, staying within the bounds of our plane and solid
geometric intuitions, a sphere and a solid ball that have the
same center and the same radius cannot be told apart by means
of their plane projections. Thus, they form a 2-projectively
indiscernible pair of 3-adic relations in R^3, making both of
them count as 2-projectively irreducible 3-adic relations.
Jon Awbrey
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