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===Commentary Note 11.8===

===Commentary Note 11.8===

Now let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.

Next let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.

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For instance, <math>L\!</math> is said to be <math>^{\backprime\backprime} c\text{-regular at}~ j \, ^{\prime\prime}</math> if and only if the cardinality of the local flag <math>L_{x \star j}</math> is equal to <math>c\!</math> for all <math>x \in X_j,</math> coded in symbols, if and only if <math>|L_{x \star j}| = c</math> for all <math>x \in X_j.</math>

<p>For instance, L is said to be "''c''-regular at ''j''" if and only if the cardinality of the local flag ''L''_''x''.''j'' is ''c'' for all ''x'' in ''X'_''j'', coded in symbols, if and only if |''L''_''x''.''j''| = ''c'' for all ''x'' in ''X_''j''.</p>

<p>In a similar fashion, one can define the NIP's "&lt;''c''-regular at ''j''", "&gt;''c''-regular at ''j''", and so on.  For ease of reference, I record a few of these definitions here:</p>

In a similar fashion, one can 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, I record a few of these definitions here:

:{| cellpadding="6"

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

| ''L'' is ''c''-regular at ''j''

|

| iff

| &#124;''L''_''x''.''j''&#124; = ''c'' for all ''x'' in ''X''_''j''.

L ~\text{is}~ c\text{-regular at}~ j

& \iff &

| ''L'' is (&lt;''c'')-regular at ''j''

|L_{x \star j}| = c ~\text{for all}~ x \in X_j.

| iff

| &#124;''L''_''x''.''j''&#124; &lt; ''c'' for all ''x'' in ''X''_''j''.

L ~\text{is}~ (< c)\text{-regular at}~ j

& \iff &

| L is (&gt;''c'')-regular at ''j''

|L_{x \star j}| < c ~\text{for all}~ x \in X_j.

| iff

| &#124;''L''_''x''.''j''&#124; &gt; ''c'' for all ''x'' in ''X''_''j''.

L ~\text{is}~ (> c)\text{-regular at}~ j

& \iff &

| L is (&le;''c'')-regular at ''j''

|L_{x \star j}| > c ~\text{for all}~ x \in X_j.

| iff

| &#124;''L''_''x''.''j''&#124; &le; ''c'' for all ''x'' in ''X''_''j''.

L ~\text{is}~ (\le c)\text{-regular at}~ j

& \iff &

| L is (&ge;''c'')-regular at ''j''

|L_{x \star j}| \le c ~\text{for all}~ x \in X_j.

| iff

| &#124;''L''_''x''.''j''&#124; &ge; ''c'' for all ''x'' in ''X''_''j''.

L ~\text{is}~ (\ge c)\text{-regular at}~ j

& \iff &

|L_{x \star j}| \ge c ~\text{for all}~ x \in X_j.

\end{array}</math>

|}

|}