Relation Theory • 5

Posted on October 31, 2021 by Jon Awbrey

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.

Numerical Incidence Properties

Resources

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

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