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, 15:52, 27 November 2007→Commentary Note 11.8: markup
===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.
Now let's re-examine the "numerical incidence properties" of relations,
concentrating on the definitions of the assorted regularity conditions.
<blockquote>
<p>For instance, L is said to be "''c''-regular at ''j''" if and only if the cardinality of the local flag ''L''<sub>''x''.''j''</sub> is ''c'' for all ''x'' in ''X'<sub>''j''</sub>, coded in symbols, if and only if |''L''<sub>''x''.''j''</sub>| = ''c'' for all ''x'' in ''X<sub>''j''</sub>.</p>
<p>In a similar fashion, one can define the NIP's "<''c''-regular at ''j''", ">''c''-regular at ''j''", and so on. For ease of reference, I record a few of these definitions here:</p>
| For instance, L is said to be "c-regular at j" if and only if
:{| cellpadding="6"
| the cardinality of the local flag L_x@j is c for all x in X_j,
| ''L'' is ''c''-regular at ''j''
| coded in symbols, if and only if |L_x@j| = c for all x in X_j.
| iff
| |''L''<sub>''x''.''j''</sub>| = ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
|-
| ''L'' is (<''c'')-regular at ''j
| iff
|
| |''L''<sub>''x''.''j''</sub>| < ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| L is c-regular at j iff |L_x@j| = c for all x in X_j.
|-
|
| L is (>c)-regular at j
| L is (<c)-regular at j iff |L_x@j| < c for all x in X_j.
| iff
|
| |''L''<sub>''x''.''j''</sub>| > ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| L is (>c)-regular at j iff |L_x@j| > c for all x in X_j.
|-
|
| L is (=<c)-regular at j
| L is (=<c)-regular at j iff |L_x@j| =< c for all x in X_j.
| iff
|
| |''L''<sub>''x''.''j''</sub>| ≤ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| L is (>=c)-regular at j iff |L_x@j| >= c for all x in X_j.
|-
| L is (>=c)-regular at j
| iff
| |''L''<sub>''x''.''j''</sub>| ≥ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
|}
</blockquote>
<pre>
Clearly, if any relation is (=<c)-regular on one
Clearly, if any relation is (=<c)-regular on one
of its domains X_j and also (>=c)-regular on the
of its domains X_j and also (>=c)-regular on the
