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.

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 TheoryLaws of FormPeirce List

This entry was posted in C.S. Peirce, Category Theory, Dyadic Relations, Logic, Logic of Relatives, Logical Graphs, Mathematics, Nominalism, Peirce, Pragmatism, Realism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization and tagged , , , , , , , , , , , , , , , . Bookmark the permalink.

5 Responses to Relation Theory • 5

  1. Pingback: Survey of Relation Theory • 4 | Inquiry Into Inquiry

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