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| ===Commentary Note 11.9=== | | ===Commentary Note 11.9=== |
| | | |
− | <pre>
| + | 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. |
− | 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. | + | <blockquote> |
| + | <p>Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of P can be defined:</p> |
| | | |
− | | Let P c X x Y be an arbitrary 2-adic relation. | + | {| cellpadding="4" |
− | | The following properties of P can be defined:
| + | | ''P'' is "total" at ''X'' |
− | |
| + | | iff |
− | | P is "total" at X iff P is (>=1)-regular at X. | + | | ''P'' is (≥1)-regular at ''X''. |
− | | | + | |- |
− | | P is "total" at Y iff P is (>=1)-regular at Y. | + | | ''P'' is "total" at ''Y'' |
− | | | + | | iff |
− | | P is "tubular" at X iff P is (=<1)-regular at X. | + | | ''P'' is (≥1)-regular at ''Y''. |
− | | | + | |- |
− | | P is "tubular" 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''. |
| + | |} |
| + | </blockquote> |
| | | |
− | We have already looked at 2-adic relations that | + | We have already looked at 2-adic relations that separately exemplify each of these regularities. |
− | separately exemplify each of these regularities. | |
| | | |
− | Also, we introduced a few bits of additional terminology and | + | Also, we introduced a few bits of additional terminology and special-purpose notations for working with tubular relations: |
− | special-purpose notations for working with tubular relations: | |
| | | |
− | | P is a "pre-function" P : X ~> Y iff P is tubular at X. | + | :{| cellpadding="4" |
− | | | + | | ''P'' is a "pre-function" ''P'' : ''X'' ~> ''Y'' |
− | | P is a "pre-function" P : X <~ Y iff P is tubular at 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 | + | Thus, we arrive by way of this winding stair at the very special stamps of 2-adic relations ''P'' ⊆ ''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''). |
− | 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). | |
| | | |
| + | <pre> |
| | If P is a pre-function P : X ~> Y that happens to be total at X, then P | | | 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. | | | is known as a "function" from X to Y, typically indicated as P : X -> Y. |