ONT Re: Relations In General
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RIG. Note 2
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Local Incidence Properties of Relations
In order to speak of generalized relations I need to outline the dimensions
of variation along which I intend the characters of already familiar orders
of relations to be broadened. Generally speaking, the taxonomic features
of k-place relations that I wish to liberalize can be read off from their
"local incidence properties" (LIP's).
A "local incidence property" (LIP) of a relation L
is one that depends on the properties of specified
subsets of L that are known as its "local flags".
Suppose that L is a k-place relation L c X_1 x ... x X_k.
Choose a relational domain X_j and one of its elements x.
The notation "L_x@i" denotes a subset of L that we shall
call "the flag of L with x at j", or "the x@j-flag of L".
Given all this, the flag L_x@j c L is defined as follows:
L_x@j = {<x_1, ..., x_j, ..., x_k> in L : x_j = x}.
Any property C of the local flag L_x@j c L may then be classified as
a "local incidence property" of L with respect to the locus "x at j".
A k-adic relation L c X_1 x ... x X_k is "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. Coded up more symbolically,
L is C-regular at j if and only if C(L_x@j) is true for all x in X_j.
Jon Awbrey
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