Changes

:* The '''ground''' of ''L'' is a [[sequence]] of ''k'' [[nonempty]] [[set]]s, ''X''₁, …, ''X''_''k'', called the ''domains'' of the relation ''L''.

:* The '''ground''' of ''L'' is a [[sequence]] of ''k'' [[nonempty]] [[set]]s, ''X''₁, …, ''X''_''k'', called the ''domains'' of the relation ''L''.

:* The '''figure''' of ''L'' is a [[subset]] of the [[cartesian product]] taken over the domains of ''L'', that is, ''F''(''L'') ⊆ ''G''(''L'') = ''X''₁ \times; … \times; ''X''_''k''.

:* The '''figure''' of ''L'' is a [[subset]] of the [[cartesian product]] taken over the domains of ''L'', that is, ''F''(''L'') ⊆ ''G''(''L'') = ''X''₁ × … × ''X''_''k''.

Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''), ''G''(''L'')), but it is customary in loose speech to use the single name ''L'' in a systematically equivocal fashion, taking it to denote either the couple ''L'' = (''F''(''L''), ''G''(''L'')) or the figure ''F''(''L'').  There is usually no confusion about this so long as the ground of the relation can be gathered from context.

Strictly speaking, then, the relation ''L'' consists of a couple of things, ''L'' = (''F''(''L''), ''G''(''L'')), but it is customary in loose speech to use the single name ''L'' in a systematically equivocal fashion, taking it to denote either the couple ''L'' = (''F''(''L''), ''G''(''L'')) or the figure ''F''(''L'').  There is usually no confusion about this so long as the ground of the relation can be gathered from context.

The formal definition of a '''finite arity relation''', specifically, a '''k-ary relation''' can now be stated.

The formal definition of a '''finite arity relation''', specifically, a '''k-ary relation''' can now be stated.

* '''Definition.'''  A '''k-ary relation''' ''L'' over the nonempty sets ''X''₁, …, ''X''_''k'' is a (1+''k'')-tuple ''L'' = (''F''(''L''), ''X''₁, …, ''X''_''k'') where ''F''(''L'') is a subset of the cartesian product ''X''₁ \times; … \times; ''X''_''k''.  If all of the ''X''_''j'' for ''j'' = 1 to ''k'' are the same set ''X'', then ''L'' is more simply called a '''''k''-ary relation over ''X'''''.  The set ''F''(''L'') is called the ''figure'' of ''L'' and, providing that the sequence of sets ''X''₁, …, ''X''_''k'' is fixed throughout a given discussion or determinate in context, one may regard the relation ''L'' as being determined by its figure ''F''(''L'').

* '''Definition.'''  A '''k-ary relation''' ''L'' over the nonempty sets ''X''₁, …, ''X''_''k'' is a (1+''k'')-tuple ''L'' = (''F''(''L''), ''X''₁, …, ''X''_''k'') where ''F''(''L'') is a subset of the cartesian product ''X''₁ × … × ''X''_''k''.  If all of the ''X''_''j'' for ''j'' = 1 to ''k'' are the same set ''X'', then ''L'' is more simply called a '''''k''-ary relation over ''X'''''.  The set ''F''(''L'') is called the ''figure'' of ''L'' and, providing that the sequence of sets ''X''₁, …, ''X''_''k'' is fixed throughout a given discussion or determinate in context, one may regard the relation ''L'' as being determined by its figure ''F''(''L'').

The formal definition simply repeats more concisely what was said above, merely unwrapping the conceptual packaging of the relation's ground to define the relation in 1 + ''k'' parts, as ''L'' = (''F''(''X''), ''X''₁, …, ''X''_''k''), rather than just the two, as ''L'' = (''F''(''L''), ''G''(''L'')).

The formal definition simply repeats more concisely what was said above, merely unwrapping the conceptual packaging of the relation's ground to define the relation in 1 + ''k'' parts, as ''L'' = (''F''(''X''), ''X''₁, …, ''X''_''k''), rather than just the two, as ''L'' = (''F''(''L''), ''G''(''L'')).

A '''local incidence property''' (LIP) of a relation ''L'' is a property that depends in turn on the properties of special subsets of ''L'' that are known as its ''local flags''.  The local flags of a relation are defined in the following way:

A '''local incidence property''' (LIP) of a relation ''L'' is a property that depends in turn on the properties of special subsets of ''L'' that are 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'' ⊆ ''X''₁ \times; … \times; ''X''_''k''.

Let ''L'' be a ''k''-place relation ''L'' ⊆ ''X''₁ × … × ''X''_''k''.

Select a relational domain ''X''_''j'' and one of its elements ''x''.  Then ''L''_''x''.''j'' is a subset of ''L'' that is referred to as the ''flag'' of ''L'' with ''x'' at ''j'', or the ''x''.''j''-flag of ''L'', an object which has the following definition:

Select a relational domain ''X''_''j'' and one of its elements ''x''.  Then ''L''_''x''.''j'' is a subset of ''L'' that is referred to as the ''flag'' of ''L'' with ''x'' at ''j'', or the ''x''.''j''-flag of ''L'', an object which has the following definition:

Any property ''C'' of the local flag ''L''_''x''.''j'' ⊆ ''L'' is said to be a ''local incidence property'' of ''L'' with respect to the locus ''x'' at ''j''.

Any property ''C'' of the local flag ''L''_''x''.''j'' ⊆ ''L'' is said to be a ''local incidence property'' of ''L'' with respect to the locus ''x'' at ''j''.

A ''k''-adic relation ''L'' ⊆ ''X''₁ × … × ''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''.   

A ''k''-adic relation ''L'' ⊆ ''X''₁ × … × ''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''.

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''.

The definition of a local flag can be broadened from a point ''x'' in ''X''_''j'' to a subset ''M'' of ''X''_''j'', arriving at the definition of a ''regional flag'' in the following way:

The definition of a local flag can be broadened from a point ''x'' in ''X''_''j'' to a subset ''M'' of ''X''_''j'', arriving at the definition of a ''regional flag'' in the following way:

Suppose that ''L'' ⊆ ''X''₁ × … × ''X''_''k'', and choose a subset ''M'' ⊆ ''X''_''j''.  Then ''L''_''M''.''j'' is a subset of ''L'' that is said to be the ''flag'' of ''L'' with ''M'' at ''j'', or the ''M''.''j''-flag of ''L'', an object which has the following  definition:

Suppose that ''L'' ⊆ ''X''₁ × … × ''X''_''k'', and choose a subset ''M'' ⊆ ''X''_''j''.  Then ''L''_''M''.''j'' is a subset of ''L'' that is said to be the ''flag'' of ''L'' with ''M'' at ''j'', or the ''M''.''j''-flag of ''L'', an object which has the following  definition:

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Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and their numerical incidence properties.  Let ''L'' ⊆ ''S'' × ''T'' be an arbitrary 2-adic relation.  The following properties of ''L'' can be defined:

Returning to 2-adic relations, it is useful to describe some familiar classes of objects in terms of their local and their numerical incidence properties.  Let ''L'' ⊆ ''S'' × ''T'' be an arbitrary 2-adic relation.  The following properties of ''L'' can be defined:

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If ''L'' is a prefunction ''p'' : ''S'' <math>\rightharpoonup</math> ''T'' that happens to be total at ''S'', then ''L'' is called a ''function'' from ''S'' to ''T'', indicated by writing ''L'' = ''f'' : ''S'' &rarr; ''T''.  To say that a relation ''L'' &sube; ''S'' × ''T'' is ''totally tubular'' at ''S'' is to say that it is 1-regular at ''S''.  Thus, we may formalize the following definition:

If ''L'' is a prefunction ''p'' : ''S'' <math>\rightharpoonup</math> ''T'' that happens to be total at ''S'', then ''L'' is called a ''function'' from ''S'' to ''T'', indicated by writing ''L'' = ''f'' : ''S'' &rarr; ''T''.  To say that a relation ''L'' &sube; ''S'' &times; ''T'' is ''totally tubular'' at ''S'' is to say that it is 1-regular at ''S''.  Thus, we may formalize the following definition:

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