## Relation Theory • 4

### Relation Theory • Local Incidence Properties

The next few definitions of local incidence properties of relations are given at a moderate level of generality in order to show how they apply to $k$-place relations.  In the sequel we’ll see what light they throw on a number of more familiar two-place relations and functions.

A local incidence property of a relation $L$ is a property which depends in turn on the properties of special subsets of $L$ known as its local flags.  The local flags of a relation are defined in the following way.

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

Select a relational domain ${X_j}$ and one of its elements $x.$

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

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

Any property $C$ of the local flag $L_{x\,@\,j}$ is said to be a local incidence property of $L$ with respect to the locus $x\,@\,j.$

A $k$-adic relation $L \subseteq X_1 \times \ldots \times X_k$ is said to be $C$-regular at $j$ if and only if every flag of $L$ with $x$ at $j$ has the property $C,$ where $x$ is taken to vary over the theme of the fixed domain $X_j.$

Expressed in symbols, $L$ is $C$-regular at $j$ if and only if $C(L_{x\,@\,j})$ is true for all $x$ in $X_j.$

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