Changes

In a similar fashion, it is possible to define the numerical incidence properties <math>^{\backprime\backprime}(< c)\text{-regular at}~ j \, ^{\prime\prime},</math> <math>^{\backprime\backprime}(> c)\text{-regular at}~ j \, ^{\prime\prime},</math> and so on.  For ease of reference,  a few of these definitions are recorded below.

In a similar fashion, it is possible to define the numerical incidence properties <math>^{\backprime\backprime}(< c)\text{-regular at}~ j \, ^{\prime\prime},</math> <math>^{\backprime\backprime}(> c)\text{-regular at}~ j \, ^{\prime\prime},</math> and so on.  For ease of reference,  a few of these definitions are recorded below.

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="8" width="90%"

|

|

<math>\begin{array}{lll}

<math>\begin{array}{lll}

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.

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>L \subseteq X \times Y\!</math> be an arbitrary 2-adic relation.  The following properties of <math>L\!</math> can be defined:

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="8" width="90%"

|

|

<math>\begin{array}{lll}

<math>\begin{array}{lll}

P ~\text{is total at}~ X

L ~\text{is total at}~ X

& \iff &

& \iff &

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

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

\\[6pt]

\\[6pt]

P ~\text{is total at}~ Y

L ~\text{is total at}~ Y

& \iff &

& \iff &

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

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

\\[6pt]

\\[6pt]

P ~\text{is tubular at}~ X

L ~\text{is tubular at}~ X

& \iff &

& \iff &

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

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

\\[6pt]

\\[6pt]

P ~\text{is tubular at}~ Y

L ~\text{is tubular at}~ Y

& \iff &

& \iff &

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

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

\end{array}</math>

\end{array}</math>

|}

|}