Changes

===Commentary Note 11.14===

===Commentary Note 11.14===

First, a correction.  Ignore for now the gloss that I gave in regard to Figure 19:

First, a correction.  Ignore for now the gloss that I gave in regard to Figure 19:

{| align="center" cellspacing="6" width="90%" <!--QUOTE-->

{| align="center" cellspacing="6" width="90%"

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

<p>Here, I have used arrowheads to indicate the relational domains at which each of the relations <math>J, K, L\!</math> happens to be functional.</p>

|}

|}

It is more like the feathers of the arrows that serve to mark the relational domains at which the relations ''J'', ''K'', ''L'' are functional, but it would take yet another construction to make this precise, as the feathers are not uniquely appointed but many splintered.

It is more like the feathers of the arrows that serve to mark the relational domains at which the relations <math>J, K, L\!</math> are functional, but it would take yet another construction to make this precise, as the feathers are not uniquely appointed but many splintered.

Now, as promised, let's look at a more homely example of a morphism, say, any one of the mappings ''J''&nbsp;:&nbsp;'''R'''&nbsp;&rarr;&nbsp;'''R''' (roughly speaking) that are commonly known as ''logarithm functions'', where you get to pick your favorite base.  In this case, ''K''(''r'',&nbsp;''s'') = ''r''&nbsp;+&nbsp;''s'' and ''L''(''u'',&nbsp;''v'') = ''u''&nbsp;<math>\cdot</math>&nbsp;''v'', and the defining formula ''J''(''L''(''u'',&nbsp;''v'')) = ''K''(''Ju'',&nbsp;''Jv'') comes out looking like ''J''(''u''&nbsp;<math>\cdot</math>&nbsp;''v'') = ''J''(''u'')&nbsp;+&nbsp;''J''(''v''), writing a dot (<math>\cdot</math>) and a plus sign (+) for the ordinary 2-ary operations of arithmetical multiplication and arithmetical summation, respectively.

Now, as promised, let's look at a more homely example of a morphism, say, any one of the mappings ''J''&nbsp;:&nbsp;'''R'''&nbsp;&rarr;&nbsp;'''R''' (roughly speaking) that are commonly known as ''logarithm functions'', where you get to pick your favorite base.  In this case, ''K''(''r'',&nbsp;''s'') = ''r''&nbsp;+&nbsp;''s'' and ''L''(''u'',&nbsp;''v'') = ''u''&nbsp;<math>\cdot</math>&nbsp;''v'', and the defining formula ''J''(''L''(''u'',&nbsp;''v'')) = ''K''(''Ju'',&nbsp;''Jv'') comes out looking like ''J''(''u''&nbsp;<math>\cdot</math>&nbsp;''v'') = ''J''(''u'')&nbsp;+&nbsp;''J''(''v''), writing a dot (<math>\cdot</math>) and a plus sign (+) for the ordinary 2-ary operations of arithmetical multiplication and arithmetical summation, respectively.

<pre>

<pre>

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

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

|                                                          |

|                                                          |

|                      {+}        {.}                      |

|                      {+}        {.}                      |

|                      @           @                       |

|                      O           O                       |

|                      /|\        /|\                      |

|                      /|\        /|\                      |

|                    / | \      / | \                    |

|                    / | \      / | \                    |

|                      \  \  \  /                      |

|                      \  \  \  /                      |

|                        \ / \ / \ /                        |

|                        \ / \ / \ /                        |

|                        @   @   @                         |

|                        O   O   O                         |

|                        J  J  J                        |

|                        J  J  J                        |

|                                                          |

|                                                          |

Figure 21.  Logarithm Arrow J : {+} <- {.}

Figure 21.  Logarithm Arrow J : {+} <- {.}

</pre>

</pre>

Thus, where the "image" ''J'' is the logarithm map, the "compound" ''K'' is the numerical sum, and the the "ligature" ''L'' is the numerical product, one obtains the immemorial mnemonic motto:

Thus, where the "image" ''J'' is the logarithm map, the "compound" ''K'' is the numerical sum, and the the "ligature" ''L'' is the numerical product, one obtains the immemorial mnemonic motto: