ONT Re: Reductions Among Relations
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RAR. Note 14
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Compositional Analysis of Relations (cont.)
The 2-adic "projections" Proj_XY, Proj_XZ, Proj_YZ, for
any 3-adic relation L c XxYxZ, along with the equivalent
forms of application L_XY, L_XZ, L_YZ, respectively, are
defined as follows:
Proj_XY (L) = L_XY = {<x, y> in XxY : <x, y, z> in L for some z in Z},
Proj_XZ (L) = L_XZ = {<x, z> in XxZ : <x, y, z> in L for some y in Y},
Proj_YZ (L) = L_YZ = {<y, z> in YxZ : <x, y, z> in L for some x in X}.
In light of these definitions, Proj_XY is a mapping
from the space !L!_XYZ of 3-adic relations L c XxYxZ
into the space !L!_XY of 2-adic relations M c XxY, and
similarly, mutatis mutandis, for the other projections.
In mathematics, the inverse relation of a projection is
usually called an "extension", but in view of the ample
confusion that we already have in logic over extensions
and intensions and comprehensions and so on, I will try
to guard against the chance of chaos in this context by
always using the adjective form of "tacit extensions".
The "tacit extensions" TE_XY_Z, TE_XZ_Y, TE_YZ_X,
of the 2-adic relations U c XxY, V c XxZ, W c YxZ,
respectively, can be defined in the following way:
TE_XY_Z (U) = {<x, y, z> : <x, y> in U},
TE_XZ_Y (V) = {<x, y, z> : <x, z> in V},
TE_YZ_X (W) = {<x, y, z> : <y, z> in W}.
It will be clear enough to write TE(U), TE(V), TE(W),
respectively, so long as the contexts are understood.
In our present application, we are making use of
the tacit extension of G c XxY to TE(G) c XxYxZ and
the tacit extension of H c YxZ to TE(H) c XxYxZ, only.
Here are the snapshots:
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| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | * \ |
| o o ** o |
| |\ / \*** /| |
| | \ / *** / | |
| | \ / ***\ / | |
| | \ *** / | |
| | / \*** / \ | |
| | / *** / \ | |
| |/ ***\ / \| |
| o X /** Y Z o |
| |\ \//* | / /| |
| | \ /// | / / | |
| | \ ///\ | / / | |
| | \ /// \ | / / | |
| | \/// \ | / / | |
| | /\/ \ | / / | |
| | *//\ \|/ / * | |
| X */ Y o Y * Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
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Figure 2. Tacit Extension of G to X x Y x Z
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| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / * | \ |
| o ** o o |
| |\ ***/ \ /| |
| | \ *** \ / | |
| | \ /*** \ / | |
| | \ *** / | |
| | / \ ***/ \ | |
| | / \ *** \ | |
| |/ \ /*** \| |
| o X Y **\ Z o |
| |\ \ | *\\/ /| |
| | \ \ | \\\ / | |
| | \ \ | /\\\ / | |
| | \ \ | / \\\ / | |
| | \ \ | / \\\/ | |
| | \ \ | / \/\ | |
| | * \ \|/ /\\* | |
| X * Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
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Figure 3. Tacit Extension of H to X x Y x Z
Finally, we can now supply a visual interpretation
that helps us to see the meaning of a formula like:
F(x, y, z) = G(x, y) & H(y, z).
The conjunction that is indicated by "&" corresponds as usual
to an intersection of two sets, however, in this case it is
the intersection of the tacit extensions TE(G) and TE(H).
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| |
| o |
| /|\ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| / | \ |
| o o o |
| |\ / \ /| |
| | \ / F \ / | |
| | \ / * \ / | |
| | \ *** / | |
| | / \//*\\/ \ | |
| | / /\/ \/\ \ | |
| |/ ///\ /\\\ \| |
| o X /// Y \\\ Z o |
| |\ \/// | \\\/ /| |
| | \ /// | \\\ / | |
| | \ ///\ | /\\\ / | |
| | \ /// \ | / \\\ / | |
| | \/// \ | / \\\/ | |
| | /\/ \ | / \/\ | |
| | *//\ \|/ /\\* | |
| X */ Y o Y \* Z |
| \ * | | * / |
| \ G | | H / |
| \ | | / |
| \ | | / |
| \ | | / |
| \ | | / |
| \| |/ |
| o o |
| |
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Figure 4. F as the Intersection of TE(G) and TE(H)
Jon Awbrey
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