Changes

===Commentary Note 11.5===

===Commentary Note 11.5===

It always helps me to draw lots of pictures of stuff, so let's extract the somewhat overly compressed bits of the "Relations In General" thread that we'll need right away for the applications to Peirce's 1870 LOR, and draw what icons we can within the frame of Ascii.

It always helps me to draw lots of pictures of stuff,

so let's extract the somewhat overly compressed bits

of the "Relations In General" thread that we'll need

right away for the applications to Peirce's 1870 LOR,

and draw what icons we can within the frame of Ascii.

For the immediate present, we may start with 2-adic relations

For the immediate present, we may start with 2-adic relations and describe the customary species of relations and functions in terms of their local and numerical incidence properties.

and describe the customary species of relations and functions

in terms of their local and numerical incidence properties.

Let P c X x Y be an arbitrary 2-adic relation.

Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of P can be defined:

The following properties of P can be defined:

P is "total" at X     iff   P is (>=1)-regular at X.

| ''P'' is "total" at ''X''

| ''P'' is (≥1)-regular at ''X''.

| iff

| ''P'' is (≥1)-regular at ''Y''.

| ''P'' is (≤1)-regular at ''X''.

| ''P'' is (≤1)-regular at ''Y''.

To illustrate these properties, let us fashion a "generic enough" example of a 2-adic relation, ''E'' ⊆ ''X'' × ''Y'', where ''X'' = ''Y'' = {0, 1, …, 8, 9}, and where the bigraph picture of ''E'' looks like this:

 

 

 

To illustrate these properties, let us fashion

a "generic enough" example of a 2-adic relation,

E c X x Y, where X = Y = {0, 1, ..., 8, 9}, and

where the bigraph picture of E looks like this:

0  1  2  3  4  5  6  7  8  9

0  1  2  3  4  5  6  7  8  9

o  o  o  o  o  o  o  o  o  o  X

o  o  o  o  o  o  o  o  o  o  X

o  o  o  o  o  o  o  o  o  o  Y

o  o  o  o  o  o  o  o  o  o  Y

0  1  2  3  4  5  6  7  8  9

0  1  2  3  4  5  6  7  8  9

If we scan along the X dimension we see that the "Y incidence degrees"

If we scan along the ''X'' dimension we see that the "''Y'' incidence degrees" of the ''X'' nodes 0 through 9 are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0, in order.

of the X nodes 0 through 9 are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0, in order.

If we scan along the Y dimension we see that the "X incidence degrees"

If we scan along the ''Y'' dimension we see that the "''X'' incidence degrees" of the Y nodes 0 through 9 are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0, in order.

of the Y nodes 0 through 9 are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0, in order.

Thus, E is not total at either X or Y,

Thus, ''E'' is not total at either ''X'' or ''Y'', since there are nodes in both ''X'' and ''Y'' having incidence degrees that equal 0.

since there are nodes in both X and Y

having incidence degrees that equal 0.

Also, E is not tubular at either X or Y,

Also, ''E'' is not tubular at either ''X'' or ''Y'', since there exist nodes in both ''X'' and ''Y'' having incidence degrees greater than 1.

since there exist nodes in both X and Y

having incidence degrees greater than 1.

Clearly, then, E cannot qualify as a pre-function

Clearly, then, ''E'' cannot qualify as a pre-function or a function on either of its relational domains.

or a function on either of its relational domains.

===Commentary Note 11.6===

===Commentary Note 11.6===