ONT Re: Logic Of Relatives
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LOR. Note 51
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Among the vast variety of conceivable regularities affecting 2-adic relations,
we pay special attention to the c-regularity conditions where c is equal to 1.
| 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.
We have already looked at 2-adic relations that
separately exemplify each of these regularities.
Also, we introduced a few bits of additional terminology and
special-purpose notations for working with tubular relations:
| 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.
Thus, we arrive by way of this winding stair at the very special stamps
of 2-adic relations P c X x Y that are "total prefunctions" at X (or Y),
"total and tubular" at X (or Y), or "1-regular" at X (or Y), more often
celebrated as "functions" at X (or Y).
| If P is a pre-function P : X ~> Y that happens to be total at X, then P
| is known as a "function" from X to Y, typically indicated as P : X -> Y.
|
| To say that a relation P c X x Y is totally tubular at X is to say that
| it is 1-regular at X. Thus, we may formalize the following definitions:
|
| P is a "function" p : X -> Y iff P is 1-regular at X.
|
| P is a "function" p : X <- Y iff P is 1-regular at Y.
For example, let X = Y = {0, ..., 9} and let F c X x Y be
the 2-adic relation that is depicted in the bigraph below:
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
\ / /|\ \ | |\ \
\ / | \ \ | | \ \ F
/ \ / | \ \ | | \ \
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
We observe that F is a function at Y,
and we record this fact in either of
the manners F : X <- Y or F : Y -> X.
Jon Awbrey
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