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===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 <math>c\!</math>-regularity conditions where <math>c\!</math> is equal to 1.

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<p>Let ''P'' &sube; ''X'' &times; ''Y'' be an arbitrary 2-adic relation.  The following properties of P can be defined:</p>

<math>\begin{array}{lll}

 

P ~\text{is total at}~ X

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& \iff &

| ''P'' is "total" at ''X''

P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ X.

| iff

| ''P'' is (&ge;1)-regular at ''X''.

P ~\text{is total at}~ Y

& \iff &

| ''P'' is "total" at ''Y''

P ~\text{is}~ (\ge 1)\text{-regular}~ \text{at}~ Y.

| iff

| ''P'' is (&ge;1)-regular at ''Y''.

P ~\text{is tubular at}~ X

& \iff &

| ''P'' is "tubular" at ''X''

P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ X.

| iff

| ''P'' is (&le;1)-regular at ''X''.

P ~\text{is tubular at}~ Y

& \iff &

| ''P'' is "tubular" at ''Y''

P ~\text{is}~ (\le 1)\text{-regular}~ \text{at}~ Y.

| iff

\end{array}</math>

| ''P'' is (&le;1)-regular at ''Y''.

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