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 -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 is a property which depends in turn on the properties of special subsets of known as its local flags. The local flags of a relation are defined in the following way.
Let be a -place relation
Select a relational domain and one of its elements
Then is a subset of called the flag of with at or the -flag of a mathematical object with the following definition.
Any property of the local flag is said to be a local incidence property of with respect to the locus
A -adic relation is said to be -regular at if and only if every flag of with at has the property where is taken to vary over the theme of the fixed domain
Expressed in symbols, is -regular at if and only if is true for all in