[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 50

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

| For instance, L is said to be "c-regular at j" if and only if
| the cardinality of the local flag L_x at j is c for all x in X_j,
| coded in symbols, if and only if |L_x at j| = c for all x in X_j.
|
| 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:
|
| L is  c-regular at j      iff   |L_x at j|   = c for all x in X_j.
|
| L is (<c)-regular at j    iff   |L_x at j|   < c for all x in X_j.
|
| L is (>c)-regular at j    iff   |L_x at j|   > c for all x in X_j.
|
| L is (=<c)-regular at j   iff   |L_x at j|  =< c for all x in X_j.
|
| L is (>=c)-regular at j   iff   |L_x at j|  >= c for all x in X_j.

Clearly, if any relation is (=<c)-regular on one
of its domains X_j and also (>=c)-regular on the
same domain, then it must be (=c)-regular on the
affected domain X_j, in effect, c-regular at j.

For example, let G = {r, s, t} and H = {1, ..., 9},
and consider the 2-adic relation F c G x H that is
bigraphed here:

    r           s           t
    o           o           o       G
   /|\         /|\         /|\
  / | \       / | \       / | \     F
 /  |  \     /  |  \     /  |  \
o   o   o   o   o   o   o   o   o   H
1   2   3   4   5   6   7   8   9

We observe that F is 3-regular at G and 1-regular at H.

Jon Awbrey

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