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, 21:10, 3 December 2007→Commentary Note 11.9: markup
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<pre>+
−| The following properties of P can be defined:+|+
===Commentary Note 11.9===
===Commentary Note 11.9===
−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"
−| ''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.
