Two further classes of incidence properties will prove to be of great utility.
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 be a -place relation
Choose a relational domain and a subset
Then is a subset of called the flag of with at or the -flag of a mathematical object with the following definition.
A numerical incidence property of a relation is a local incidence property predicated on the cardinalities of its local flags.
For example, is said to be -regular at if and only if the cardinality of the local flag is for all in — to write it in symbols, if and only if for all
In a similar fashion, one may define the numerical incidence properties, -regular at -regular at and so on. For ease of reference, a few definitions are recorded below.