Changes

In order to speak of generalized orders of 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 <math>n\!</math>-place relations that I wish to liberalize can be read off from their ''local incidence properties'' (LIPs).

In order to speak of generalized orders of 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 <math>n\!</math>-place relations that I wish to liberalize can be read off from their ''local incidence properties'' (LIPs).

'''Definition.''' A ''local incidence property'' of a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k\!</math> is one that is based on the following type of data.  Pick an element <math>x\!</math> in one of the domains <math>X_j\!</math> of <math>L.\!</math> Let <math>L_{x \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>x\!</math> at <math>j\!</math>'', or the ''<math>x \,\text{at}\, j\!</math> flag of <math>L.\!</math>'' The ''local flag'' <math>L_{x \,\text{at}\, j} \subseteq L\!</math> is defined as follows:

Definition.  A "local incidence property" of an n place relation R is one that is based on the following sorts of data.  Suppose R c X1x...xXn.  Pick an element x in one of the domains Xi of R.  Let "R&x@i" denote a subset of R called the "flag of R with x at i", or the "x@i flag of R".  The "local flag" R&x@i c R is defined as follows:

R&x@i  = {<x1, ... , xi, ... , xn> C R  : xi = x}.

| <math>L_{x \,\text{at}\, j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.\!</math>

Any property P of R&x@i constitutes a "local incidence property" of R with reference to the locus "x at i".

Any property P of R&x@i constitutes a "local incidence property" of R with reference to the locus "x at i".