Relation Theory • 4

Relation TheoryLocal 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.

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.

6 Responses to Relation Theory • 4

  1. Pingback: Relation Theory • Discussion 2 | Inquiry Into Inquiry

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

  3. Pingback: Relation Theory • Discussion 3 | Inquiry Into Inquiry

  4. Pingback: Survey of Relation Theory • 5 | Inquiry Into Inquiry

  5. Pingback: Survey of Relation Theory • 2 | Inquiry Into Inquiry

  6. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.