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, 14:08, 13 April 2009→Commentary Note 11.9
===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 <math>c\!</math>-regularity conditions where <math>c\!</math> is equal to 1.
Among the variety of conceivable regularities affecting 2-adic relations, we pay special attention to the <math>c\!</math>-regularity conditions where <math>c\!</math> is equal to 1.
Let <math>P \subseteq X \times Y</math> be an arbitrary 2-adic relation. The following properties of <math>~P~</math> can be defined:
Let <math>P \subseteq X \times Y</math> be an arbitrary 2-adic relation. The following properties of <math>~P~</math> can be defined:
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We have already looked at 2-adic relations that separately exemplify each of these regularities.
We have already looked at 2-adic relations that separately exemplify each of these regularities. We also introduced a few bits of additional terminology and special-purpose notations for working with tubular relations:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
|
{| cellpadding="4"
<math>\begin{array}{lll}
P ~\text{is a pre-function}~ P : X \rightharpoonup Y
& \iff &
P ~\text{is tubular at}~ X.
\\[6pt]
P ~\text{is a pre-function}~ P : X \leftharpoonup Y
& \iff &
P ~\text{is tubular at}~ Y.
\end{array}</math>
|}
|}
For example, let ''X'' = ''Y'' = {0, …, 9} and let ''F'' ⊆ ''X'' × ''Y'' be the 2-adic relation that is depicted in the bigraph below:
For example, let ''X'' = ''Y'' = {0, …, 9} and let ''F'' ⊆ ''X'' × ''Y'' be the 2-adic relation that is depicted in the bigraph below:
{| align="center" cellspacing="6" width="90%"
|
<pre>
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
</pre>
|}
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''.
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''.
