Relation Theory • 4

Posted on October 30, 2021 by Jon Awbrey

Relation Theory • Local 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 Theory • Laws of Form • Peirce List

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