[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 48

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Let's continue to work our way through the rest of the first
set of definitions, making up appropriate examples as we go.

| Let P c X x Y be an arbitrary 2-adic relation.
| The following properties of P can be defined:
|
| P is "total" at X     iff   P is (>=1)-regular at X.
|
| P is "total" at Y     iff   P is (>=1)-regular at Y.
|
| P is "tubular" at X   iff   P is (=<1)-regular at X.
|
| P is "tubular" at Y   iff   P is (=<1)-regular at Y.

E_1 exemplifies the quality of "totality at X".

0   1   2   3   4   5   6   7   8   9
o   o   o   o   o   o   o   o   o   o   X
 \   \  |\ /|\   \   \  |   |\   \  |
  \   \ | / | \   \   \ |   | \   \ |   E_1
   \   \|/ \|  \   \   \|   |  \   \|
o   o   o   o   o   o   o   o   o   o   Y
0   1   2   3   4   5   6   7   8   9

E_2 exemplifies the quality of "totality at Y".

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

E_3 exemplifies the quality of "tubularity at X".

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

E_4 exemplifies the quality of "tubularity at Y".

0   1   2   3   4   5   6   7   8   9
o   o   o   o   o   o   o   o   o   o   X
           /|\   \   \      |\
          / | \   \   \     | \         E_4
         /  |  \   \   \    |  \
o   o   o   o   o   o   o   o   o   o   Y
0   1   2   3   4   5   6   7   8   9

| If P c X x Y is tubular at X, then P is known as a "partial function"
| or a "pre-function" from X to Y, frequently signalized by renaming P
| with an alternative lower case name, say "p", and writing p : X ~> Y.
|
| Just by way of formalizing the definition:
| 
| P is a "pre-function" P : X ~> Y   iff   P is tubular at X.
|
| P is a "pre-function" P : X <~ Y   iff   P is tubular at Y.

So, E_3 is a pre-function e_3 : X ~> Y,
and E_4 is a pre-function e_4 : X <~ Y.

Jon Awbrey

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