## Relation Theory • 5

### Relation Theory

Two further classes of incidence properties will prove to be of great utility.

### Regional Incidence Properties

The definition of a local flag can be broadened from a point to a subset of a relational domain, arriving at the definition of a regional flag in the following way.

Let $L$ be a $k$-place relation $L \subseteq X_1 \times \ldots \times X_k.$

Choose a relational domain $X_j$ and a subset $M \subseteq X_j.$

Then $L_{M\,@\,j}$ is a subset of $L$ called the flag of $L$ with $M$ at $j,$ or the $(M\,@\,j)$-flag of $L,$ a mathematical object with the following definition.

$L_{M\,@\,j} ~ = ~ \{ (x_1, \ldots, x_j, \ldots, x_k) \in L ~ : ~ x_j \in M \}.$

### Numerical Incidence Properties

A numerical incidence property of a relation is a local incidence property predicated on the cardinalities of its local flags.

For example, $L$ is said to be $c$-regular at $j$ if and only if the cardinality of the local flag $L_{x\,@\,j}$ is $c$ for all $x$ in ${X_j}$ — to write it in symbols, if and only if $|L_{x\,@\,j}| = c$ for all ${x \in X_j}.$

In a similar fashion, one may define the numerical incidence properties, $(<\!c)$-regular at $j,$ $(>\!c)$-regular at $j,$ and so on.  For ease of reference, a few definitions are recorded below.

### Resources

cc: Category Theory • Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation TheoryLaws of FormPeirce List

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