Let’s take a closer look at the numerical incidence properties of relations, concentrating on the assorted regularity conditions defined in the article on Relation Theory.
For example, 
has the property of being 
if and only if the cardinality of the local flag 
is equal to 
for all 
in 
coded in symbols, if and only if 
for all in 
In like fashion, one may define the numerical incidence properties 
and so on. For ease of reference, a few of these definitions are recorded below.
![\begin{array}{lll} L ~\text{is}~ c\text{-regular at}~ j & \iff & |L_{x @ j}| = c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (< c)\text{-regular at}~ j & \iff & |L_{x @ j}| < c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (> c)\text{-regular at}~ j & \iff & |L_{x @ j}| > c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\le c)\text{-regular at}~ j & \iff & |L_{x @ j}| \le c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\ge c)\text{-regular at}~ j & \iff & |L_{x @ j}| \ge c ~\text{for all}~ x \in X_j. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++L+%7E%5Ctext%7Bis%7D%7E+c%5Ctext%7B-regular+at%7D%7E+j+%26+%5Ciff+%26+%7CL_%7Bx+%40+j%7D%7C+%3D+c+%7E%5Ctext%7Bfor+all%7D%7E+x+%5Cin+X_j.++%5C%5C%5B6pt%5D++L+%7E%5Ctext%7Bis%7D%7E+%28%3C+c%29%5Ctext%7B-regular+at%7D%7E+j+%26+%5Ciff+%26+%7CL_%7Bx+%40+j%7D%7C+%3C+c+%7E%5Ctext%7Bfor+all%7D%7E+x+%5Cin+X_j.++%5C%5C%5B6pt%5D++L+%7E%5Ctext%7Bis%7D%7E+%28%3E+c%29%5Ctext%7B-regular+at%7D%7E+j+%26+%5Ciff+%26+%7CL_%7Bx+%40+j%7D%7C+%3E+c+%7E%5Ctext%7Bfor+all%7D%7E+x+%5Cin+X_j.++%5C%5C%5B6pt%5D++L+%7E%5Ctext%7Bis%7D%7E+%28%5Cle+c%29%5Ctext%7B-regular+at%7D%7E+j+%26+%5Ciff+%26+%7CL_%7Bx+%40+j%7D%7C+%5Cle+c+%7E%5Ctext%7Bfor+all%7D%7E+x+%5Cin+X_j.++%5C%5C%5B6pt%5D++L+%7E%5Ctext%7Bis%7D%7E+%28%5Cge+c%29%5Ctext%7B-regular+at%7D%7E+j+%26+%5Ciff+%26+%7CL_%7Bx+%40+j%7D%7C+%5Cge+c+%7E%5Ctext%7Bfor+all%7D%7E+x+%5Cin+X_j.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{lll} L ~\text{is}~ c\text{-regular at}~ j & \iff & |L_{x @ j}| = c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (< c)\text{-regular at}~ j & \iff & |L_{x @ j}| < c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (> c)\text{-regular at}~ j & \iff & |L_{x @ j}| > c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\le c)\text{-regular at}~ j & \iff & |L_{x @ j}| \le c ~\text{for all}~ x \in X_j. \\[6pt] L ~\text{is}~ (\ge c)\text{-regular at}~ j & \iff & |L_{x @ j}| \ge c ~\text{for all}~ x \in X_j. \end{array}")
Clearly, if any relation is 
on one of its domains 
and also 
on the same domain, then it must be 
on that domain, in short, 
at 
For example, let 
and 
and consider the dyadic relation 
that is bigraphed below:
 |
(38) |
We observe that 
is 3-regular at 
and 1-regular at 
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