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'''Definition.'''  A <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is ''<math>P\!</math>-regular at <math>j\!</math>'' if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> is <math>P,\!</math> letting <math>x\!</math> range over the domain <math>X_j,\!</math> in symbols, if and only if <math>P(L_{x \,\text{at}\, j})\!</math> is true for all <math>x \in X_j.\!</math>

'''Definition.'''  A <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is ''<math>P\!</math>-regular at <math>j\!</math>'' if and only if every flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math> is <math>P,\!</math> letting <math>x\!</math> range over the domain <math>X_j,\!</math> in symbols, if and only if <math>P(L_{x \,\text{at}\, j})\!</math> is true for all <math>x \in X_j.\!</math>

Of particular interest are the local incidence properties of relations that can be calculated from the cardinalities of their local flags, and these are naturally called ''numerical incidence properties'' (NIPs).

Of particular interest are the local incidence properties of relations that can be calculated from the cardinalities of their local flags, and these are naturally called "numerical incidence properties" (NIPs).

 

For example, R is said to be "k regular at i" or "k regular at Xi" if and only if the cardinality |R&x@i| = k for all x C Xi.  In a similar fashion, one can define the NIPs "<k regular at i", ">k regular at i", and so on.  For ease of reference, I record a few of these definitions here:

In similar fashion, it is possible to define the numerical incidence properties <math>(< c)\text{-regular at}~ j,\!</math> <math>(> c)\text{-regular at}~ j,\!</math> and so on.  For ease of reference, a few of these definitions are recorded below.

R is k regular at i iff |R&x@i| = k for all x C Xi.

R is <k regular at i iff |R&x@i| < k for all x C Xi.

R is >k regular at i iff |R&x@i| > k for all x C Xi.

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

& \iff &

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

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

& \iff &

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

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

& \iff &

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

& \iff &

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

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

The definition of "local flags" can be broadened to give a definition of "regional flags".  Suppose R c X1x...xXn and choose a subset M c Xi.  Let "R&M@i" denote a subset of R called the "flag of R with M at i", or the "M@i flag of R", defined as:

The definition of "local flags" can be broadened to give a definition of "regional flags".  Suppose R c X1x...xXn and choose a subset M c Xi.  Let "R&M@i" denote a subset of R called the "flag of R with M at i", or the "M@i flag of R", defined as: