ONT Re: Reductions Among Relations
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RAR. Note 3
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Projective Reduction of Triadic Relations
We are ready to take up the question of whether
3-adic relations, in general, and in particular
cases, are "determined by", "reducible to", or
"reconstructible from" their 2-adic projections.
Suppose that L contained in XxYxZ is an arbitrary 3-adic relation,
and consider the three 2-adic relations that are gotten by taking
its projections, its "shadows", if you will, on each of the three
planes XY, XZ, YZ. Using the notation that I introduced before,
and compressing it just a bit or two in passing, one can write
these projections in each of the following ways, depending on
which turns out to be most convenient in a given context:
1. Proj_{X,Y} (L) = Proj_{1,2} (L) = Proj_XY L = Proj_12 L = L_XY = L_12.
2. Proj_{X,Z} (L) = Proj_{1,3} (L) = Proj_XZ L = Proj_13 L = L_XZ = L_13.
3. Proj_{Y,Z} (L) = Proj_{2,3} (L) = Proj_YZ L = Proj_23 L = L_YZ = L_23.
If you picture the relation L as a body in the 3-space XYZ, then
the issue of whether L is "reducible to" or "reconstuctible from"
its 2-adic projections is just the question of whether these three
projections, "shadows", or "2-faces" determine the body L uniquely.
Stating the matter the other way around, L is "not reducible to"
or "not reconstructible from" its 2-dim projections if & only if
there are two distinct relations L and L' which have exactly the
same projections on exactly the same planes.
The next series of Tables illustrates the projection operations
by means of their actions on the sign relations L(A) and L(B)
that I introduced earlier on, in the "Sign Relations" thread.
Recall that we had the following set-up:
| L(A) and L(B) are "contained in" or "subsets of" OxSxI:
|
| O = {A, B},
|
| S = {"A", "B", "i", "u"},
|
| I = {"A", "B", "i", "u"}.
| L(A) has the following eight triples
| of the form <o, s, i> in OxSxI:
|
| <A, "A", "A">
| <A, "A", "i">
| <A, "i", "A">
| <A, "i", "i">
| <B, "B", "B">
| <B, "B", "u">
| <B, "u", "B">
| <B, "u", "u">
| L(B) has the following eight triples
| of the form <o, s, i> in OxSxI:
|
| <A, "A", "A">
| <A, "A", "u">
| <A, "u", "A">
| <A, "u", "u">
| <B, "B", "B">
| <B, "B", "i">
| <B, "i", "B">
| <B, "i", "i">
Taking the 2-adic projections of L(A)
we obtain the following set of data:
| L(A)_OS has these four pairs
| of the form <o, s> in OxS:
|
| <A, "A">
| <A, "i">
| <B, "B">
| <B, "u">
| L(A)_OI has these four pairs
| of the form <o, i> in OxI:
|
| <A, "A">
| <A, "i">
| <B, "B">
| <B, "u">
| L(A)_SI has these eight pairs
| of the form <s, i> in SxI:
|
| <"A", "A">
| <"A", "i">
| <"i", "A">
| <"i", "i">
| <"B", "B">
| <"B", "u">
| <"u", "B">
| <"u", "u">
Taking the dyadic projections of L(B)
we obtain the following set of data:
| L(B)_OS has these four pairs
| of the form <o, s> in OxS:
|
| <A, "A">
| <A, "u">
| <B, "B">
| <B, "i">
| L(B)_OI has these four pairs
| of the form <o, i> in OxI:
|
| <A, "A">
| <A, "u">
| <B, "B">
| <B, "i">
| L(B)_SI has these eight pairs
| of the form <s, i> in SxI:
|
| <"A", "A">
| <"A", "u">
| <"u", "A">
| <"u", "u">
| <"B", "B">
| <"B", "i">
| <"i", "B">
| <"i", "i">
A comparison of the corresponding projections for L(A) and L(B)
reveals that the distinction between these two 3-adic relations
is preserved under the operation that takes the full collection
of 2-adic projections into consideration, and this circumstance
allows one to say that this much information, that is, enough to
tell L(A) and L(B) apart, can be derived from their 2-adic faces.
However, in order to say that a 3-adic relation L on OxSxI
is "reducible" or "reconstructible" in the 2-dim projective
sense, it is necessary to show that no distinct L' on OxSxI
exists such that L and L' have the same set of projections,
and this can take a rather more exhaustive or comprehensive
investigation of the space of possible relations on OxSxI.
As it happens, each of the relations L(A) and L(B) turns
out to be uniquely determined by its 2-dim projections.
This can be seen as follows. Consider any coordinate
position <s, i> in the plane SxI. If <s, i> is not
in L_SI then there can be no element <o, s, i> in L,
therefore we may restrict our attention to positions
<s, i> in L_SI, knowing that there exist at least
|L_SI| = Cardinality of L_SI = eight elements in L,
and seeking only to determine what objects o exist
such that <o, s, i> is an element in the objective
"fiber" of <s, i>. In other words, for what o in O
is <o, s, i> in ((Proj_SI)^(-1))(<s, i>)? Now, the
circumstance that L_OS has exactly one element <o, s>
for each coordinate s in S and that L_OI has exactly
one element <o, i> for each coordinate i in I, plus
the "coincidence" of it being the same o at any one
choice for <s, i>, tells us that L has just the one
element <o, s, i> over each point of SxI. Together,
this proves that both L(A) and L(B) are reducible in
an informative sense to 3-tuples of 2-adic relations,
that is, they are "projectively 2-adically reducible".
Next time I will give examples of 3-adic relations
that are not reducible to or reconstructible from
their 2-adic projections.
Jon Awbrey
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