SUO: Re: Relations And Their Divisitudes
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RATD. Note 23
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Compositional Analysis of Relations (cont.)
Let's recall the definition of the 2-adic projections:
The 2-adic "projections" p_12, p_13, p_23,
alternatively written as Proj_XY, Proj_XZ, Proj_YZ,
as applying to the 3-adic relation L c X x Y x Z,
together with the equivalent forms of application
p_12 (L) = L_XY, p_13 (L) = L_XZ, p_23(L) = L_YZ,
respectively, are defined as follows:
p_12 (L) = L_XY = {<x, y> in X x Y : <x, y, z> in L for some z in Z},
p_13 (L) = L_XZ = {<x, z> in X x Z : <x, y, z> in L for some y in Y},
p_23 (L) = L_YZ = {<y, z> in Y x Z : <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 X x Y x Z into
the space !L!_XY of 2-adic relations M c X x Y, likewise,
mutatis mutandis, for the projections Proj_XZ and Proj_YZ.
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 qualified phrase of "tacit extensions".
The "tacit extensions", te_12.3, te_13.2, te_23.1,
alternatively written TE_XY.Z, TE_XZ.Y, TE_YZ.X,
of the 2-adic relations U c X x Y, V c X x Z, W c Y x Z,
respectively, are defined by the following set of formulas:
te_12.3 (U) = TE_XY.Z (U) = {<x, y, z> : <x, y> in U},
te_13.2 (V) = TE_XZ.Y (V) = {<x, y, z> : <x, z> in V},
te_23.1 (W) = 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 the present set of considerations, we only have need of
the tacit extension of G c X x Y to te(G) c X x Y x Z and
the tacit extension of H c Y x Z to te(H) c X x Y x Z.
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 7. 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 8. 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 9. F as the Intersection of te(G) and te(H)
Jon Awbrey
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