Jon Awbrey
Registered: Feb 2004
Posts: 551 |
Reductions Among Relations
RAR. Note 4
2. Projective Reduction of Triadic Relations (cont.)
It serves to develop a few concrete examples of 3-adic relations,
solid enough to reflect the lights of these various perspectives.
I will re-introduce here a couple of 3-adic relations
that I have often used as examples of "sign relations".
For now we are only interested in their properties as
3-adic relations, in the light of projective reduction.
The 3-adic relations L(A) and L(B) are defined as follows:
L(A) and L(B) are subsets of the cartesian product O x S x I, where
O = {A, B}, S = {"A", "B", "i", "u"}, and I = {"A", "B", "i", "u"}.
L(A) has the following eight triples
of the form <o, s, i> in O x S x I:
<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 O x S x I:
<A, "A", "A">
<A, "A", "u">
<A, "u", "A">
<A, "u", "u">
<B, "B", "B">
<B, "B", "i">
<B, "i", "B">
<B, "i", "i">
The next series of Tables illustrates the projection operations
by means of their actions on the sign relations L(A) and L(B).
Taking the 2-adic projections of L(A)
one obtains the following set of data:
L(A)_OS has these four pairs
of the form <o, s> in O x S:
<A, "A">
<A, "i">
<B, "B">
<B, "u">
L(A)_OI has these four pairs
of the form <o, i> in O x I:
<A, "A">
<A, "i">
<B, "B">
<B, "u">
L(A)_SI has these eight pairs
of the form <s, i> in S x I:
<"A", "A">
<"A", "i">
<"i", "A">
<"i", "i">
<"B", "B">
<"B", "u">
<"u", "B">
<"u", "u">
Taking the 2-adic projections of L(B)
one obtains the following set of data:
L(B)_OS has these four pairs
of the form <o, s> in O x S:
<A, "A">
<A, "u">
<B, "B">
<B, "i">
L(B)_OI has these four pairs
of the form <o, i> in O x I:
<A, "A">
<A, "u">
<B, "B">
<B, "i">
L(B)_SI has these eight pairs
of the form <s, i> in S x I:
<"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, at least enough to
tell L(A) and L(B) apart, can be derived from their 2-adic faces.
However, we are not done yet. In order to say that
a 3-adic relation L c O x S x I is "reducible to" or
"reconstructible from" 2-dimensional projective data,
we would need to show that no distinct L' c O x S x I
exists such that L and L' have identical projections,
and this takes a more comprehensive investigation of
the variety of all possible relations on O x S x I.
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 S x I. 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 S x I. In all,
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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04-27-2004 04:06 AM
