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===Commentary Note 11.13===

===Commentary Note 11.13===

As we make our way toward the foothills of Peirce's 1870 LOR, there are several pieces of equipment that we must not leave the plains without, namely, the utilities that are variously referred to as ''arrows'', ''morphisms'', ''homomorphisms'', ''structure-preserving maps'', and several other names, depending on the altitude of abstraction that one happens to be traversing at the moment in question.  As a moderate to middling but not too beaten track, I will lay out the definition of a morphism in a number of the forms that we will need right away.

As we make our way toward the foothills of Peirce's 1870 LOR, there are several pieces of equipment that we must not leave the plains without, namely, the utilities that are variously referred to as ''arrows'', ''morphisms'', ''homomorphisms'', ''structure-preserving maps'', and several other names, depending on the altitude of abstraction that one happens to be traversing at the moment in question.  As a moderate to middling but not too beaten track, let us take up a few ways of defining a morphism that will serve us in the present discussion.

Let's say that we have three functions ''J'', ''K'', ''L'' that have the following types and that satisfy the equation that follows:

Suppose we have three functions <math>J, K, L\!</math> given by the type descriptors and the equation that follows:

: ''J'' : ''X'' &larr; ''Y''

 

: ''K'' : ''X'' &larr; ''X'' &times; ''X''

 

J & : & X & \leftarrow & Y

: ''L'' : ''Y'' &larr; ''Y'' &times; ''Y''

 

K & : & X & \leftarrow & X \times X

: ''J''(''L''(''u'', ''v'')) = ''K''(''Ju'', ''Jv'')

L & : & Y & \leftarrow & Y \times Y

J(L(u, v)) & = & K(Ju, Jv)

Our sagittarian leitmotif can be rubricized in the following slogan:

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".

<p>The image of the ligature is the compound of the images.</p>

<p>Where <math>J\!</math> is the ''image'', <math>K\!</math> is the ''compound'', and <math>L\!</math> is the ''ligature''.</p>

Figure 19 presents us with a picture of the situation in question.

Figure&nbsp;19 presents us with a picture of the situation in question.

<pre>

<pre>

o-----------------------------------------------------------o

o-----------------------------------------------------------o

Figure 19.  Structure Preserving Transformation J : K <- L

Figure 19.  Structure Preserving Transformation J : K <- L

</pre>

</pre>

Here, I have used arrowheads to indicate the relational domains at which each of the relations ''J'',&nbsp;''K'',&nbsp;''L'' happens to be functional.

Here, I have used arrowheads to indicate the relational domains at which each of the relations ''J'',&nbsp;''K'',&nbsp;''L'' happens to be functional.

Table 20 gives the constraint matrix version of the same thing.

Table&nbsp;20 gives the constraint matrix version of the same thing.

<pre>

<pre>

Table 20.  Arrow:  J(L(u, v)) = K(Ju, Jv)

Table 20.  Arrow:  J(L(u, v)) = K(Ju, Jv)

o---------o---------o---------o---------o

o---------o---------o---------o---------o

</pre>

</pre>

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'',&nbsp;''v'')) = ''K''(''Ju'',&nbsp;''Jv'').

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'',&nbsp;''v'')) = ''K''(''Ju'',&nbsp;''Jv'').