[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 49

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We come now to the very special cases of 2-adic relations that are
known as functions.  It will serve a dual purpose on behalf of the
exposition if we take the class of functions as a source of object
examples to clarify the more abstruse concepts in the RIG material.

To begin, let's recall the definition of a local flag:

L_x at j  =  {<x_1, ..., x_j, ..., x_k> in L : x_j = x}.

In the case of a 2-adic relation L c X_1 x X_2 = X x Y,
we can reap the benefits of a radical simplification in
the definitions of the local flags.  Also in this case,
we tend to denote L_u at 1 by "L_u at X" and L_v at 2 by "L_v at Y".

In the light of these considerations, the local flags of
a 2-adic relation L c X x Y may be formulated as follows:

L_u at X  =  {<x, y> in L : x = u}

       =  the set of all ordered pairs in L incident with u in X.

L_v at Y  =  {<x, y> in L : y = v}

       =  the set of all ordered pairs in L incident with v in Y.

A sufficient illustration is supplied by the earlier example E.

0   1   2   3   4   5   6   7   8   9
o   o   o   o   o   o   o   o   o   o   X
     \  |\ /|\   \   \  |   |\
      \ | / | \   \   \ |   | \         E
       \|/ \|  \   \   \|   |  \
o   o   o   o   o   o   o   o   o   o   Y
0   1   2   3   4   5   6   7   8   9

The local flag E_3 at X is displayed here:

0   1   2   3   4   5   6   7   8   9
o   o   o   o   o   o   o   o   o   o   X
           /|\
          / | \                         E_3 at X
         /  |  \
o   o   o   o   o   o   o   o   o   o   Y
0   1   2   3   4   5   6   7   8   9

The local flag E_2 at Y is displayed here:

0   1   2   3   4   5   6   7   8   9
o   o   o   o   o   o   o   o   o   o   X
     \  |  /
      \ | /                             E_2 at Y
       \|/
o   o   o   o   o   o   o   o   o   o   Y
0   1   2   3   4   5   6   7   8   9

Jon Awbrey

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