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Revision as of 17:58, 27 November 2007
, 17:58, 27 November 2007→Commentary Note 11.8: markup
| |''L''<sub>''x''.''j''</sub>| = ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| |''L''<sub>''x''.''j''</sub>| = ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
|-
|-
| ''L'' is (<''c'')-regular at ''j
| ''L'' is (<''c'')-regular at ''j''
| iff
| iff
| |''L''<sub>''x''.''j''</sub>| < ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| |''L''<sub>''x''.''j''</sub>| < ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
|-
|-
| L is (>c)-regular at j
| L is (>''c'')-regular at ''j''
| iff
| iff
| |''L''<sub>''x''.''j''</sub>| > ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| |''L''<sub>''x''.''j''</sub>| > ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
|-
|-
| L is (=<c)-regular at j
| L is (≤''c'')-regular at ''j''
| iff
| iff
| |''L''<sub>''x''.''j''</sub>| ≤ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| |''L''<sub>''x''.''j''</sub>| ≤ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
|-
|-
| L is (>=c)-regular at j
| L is (≥''c'')-regular at ''j''
| iff
| iff
| |''L''<sub>''x''.''j''</sub>| ≥ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
| |''L''<sub>''x''.''j''</sub>| ≥ ''c'' for all ''x'' in ''X''<sub>''j''</sub>.
Clearly, if any relation is (≤''c'')-regular on one of its domains ''X''<sub>''j''</sub> and also (≥''c'')-regular on the same domain, then it must be (=''c'')-regular on the affected domain ''X''<sub>''j''</sub>, in effect, ''c''-regular at ''j''.
Clearly, if any relation is (≤''c'')-regular on one of its domains ''X''<sub>''j''</sub> and also (≥''c'')-regular on the same domain, then it must be (=''c'')-regular on the affected domain ''X''<sub>''j''</sub>, in effect, ''c''-regular at ''j''.
For example, let ''G'' = {''r'', ''s'', ''t'} and ''H'' = {1, …, 9}, and consider the 2-adic relation ''F'' ⊆ ''G'' × ''H'' that is bigraphed here:
For example, let ''G'' = {''r'', ''s'', ''t''} and ''H'' = {1, …, 9}, and consider the 2-adic relation ''F'' ⊆ ''G'' × ''H'' that is bigraphed here:
<pre>
<pre>
