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 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.

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For example, <math>L\!</math> is <math>c\text{-regular at}~ j\!</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}\!</math> is equal to <math>c\!</math> for all <math>x \in X_j,\!</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c\!</math> for all <math>x \in X_j.\!</math>

For example, <math>L\!</math> is <math>c\text{-regular at}~ j\!</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}\!</math> is equal to <math>c\!</math> for all <math>x \in X_j,\!</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c\!</math> for all <math>x \in X_j.\!</math>

In similar fashion, it is possible to define the numerical incidence properties <math>(< c)\text{-regular at}~ j,\!</math> <math>(> c)\text{-regular at}~ j,\!</math> and so on.  For ease of reference, a few of these definitions are recorded below.

In a similar fashion, it is possible to define the numerical incidence properties <math>(< c)\text{-regular at}~ j,\!</math> <math>(> c)\text{-regular at}~ j,\!</math> and so on.  For ease of reference, a few of these definitions are recorded below.

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The definition of local flags can be broadened to give a definition of ''regional flags''.  Suppose <math>L \subseteq X_1 \times \ldots \times X_k\!</math> and choose a subset <math>M \subseteq X_j.\!</math> Let <math>L_{M \,\text{at}\, j}\!</math> be a subset of <math>L\!</math> called the ''flag of <math>L\!</math> with <math>M\!</math> at <math>j,\!</math>'' or the ''<math>M \,\text{at}\, j\!</math> flag of <math>L,\!</math>'' defined as follows.

The definition of "local flags" can be broadened to give a definition of "regional flags".  Suppose R c X1x...xXn and choose a subset M c Xi.  Let "R&M@i" denote a subset of R called the "flag of R with M at i", or the "M@i flag of R", defined as:

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

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

Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties.  Let R c SxT be an arbitrary dyadic relation.  The following properties of R can then be defined:

Returning to dyadic relations, it is useful to describe some familiar classes of objects in terms of their local and numerical incidence properties.  Let R c SxT be an arbitrary dyadic relation.  The following properties of R can then be defined: