ONT Re: Logic Of Relatives
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LOR. Note 55
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As we make our way toward the foothills of Peirce's 1870 LOR, there
is one piece of equipment that we dare not leave the plains without --
for there is little hope that "l'or dans les montagnes là" will lie
among our prospects without the ready use of its leverage and lifts --
and that is a facility with the utilities that are variously called
"arrows", "morphisms", "homomorphisms", "structure-preserving maps",
and several other names, in accord with the altitude of abstraction
at which one happens to be working, at the given moment in question.
As a middle but not too beaten track, I will lay out the definition
of a morphism in the forms that we will need right off, in a slight
excess of formality at first, but quickly bringing the bird home to
roost on more familiar perches.
Let's say that we have three functions J, K, L
that have the following types and that satisfy
the equation that follows:
| J : X <- Y
|
| K : X <- X x X
|
| L : Y <- Y x Y
|
| J(L(u, v)) = K(Ju, Jv)
Our sagittarian leitmotif can be rubricized in the following slogan:
>-> The image of the ligature is the compound of the images. <-<
Where J is the "image", K is the "compound", and L is the "ligature".
Figure 19 presents us with a picture of the situation in question.
o-----------------------------------------------------------o
| |
| K L |
| @ @ |
| /|\ /|\ |
| / | \ / | \ |
| v | \ v | \ |
| o o o o o o |
| X X X Y Y Y |
| o o o o o o |
| ^ ^ ^ / / / |
| \ \ \ / / |
| \ \ / \ / / |
| \ \ \ / |
| \ / \ / \ / |
| @ @ @ |
| J J J |
| |
o-----------------------------------------------------------o
Figure 19. Structure Preserving Transformation J : K <- L
Here, I have used arrowheads to indicate the relational domains
at which each of the relations J, K, L happens to be functional.
Table 20 gives the constraint matrix version of the same thing.
Table 20. Arrow: J(L(u, v)) = K(Ju, Jv)
o---------o---------o---------o---------o
| # J | J | J |
o=========o=========o=========o=========o
| K # X | X | X |
o---------o---------o---------o---------o
| L # Y | Y | Y |
o---------o---------o---------o---------o
One way to read this Table is in terms of the informational redundancies
that it schematizes. In particular, it can be read to say that when one
satisfies the constraint in the L row, along with all of the constraints
in the J columns, then the constraint in the K row is automatically true.
That is one way of understanding the equation: J(L(u, v)) = K(Ju, Jv).
Jon Awbrey
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