Changes

===Commentary Note 9.5===

===Commentary Note 9.5===

Peirce's comma operation, in its application to an absolute term, is tantamount to the representation of that term's denotation as an idempotent transformation, which is commonly represented as a diagonal matrix.  This is why I call it the "diagonal extension".

Peirce's comma operation, in its application to an absolute term, is tantamount to the representation of that term's denotation as an idempotent transformation, which is commonly represented as a diagonal matrix.  This is why I call it the ''diagonal extension''.

An idempotent element ''x'' is given by the abstract condition that ''xx'' = ''x'', but we commonly encounter such elements in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients.

An idempotent element <math>x\!</math> is given by the abstract condition that <math>xx = x,\!</math> but we commonly encounter such elements in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients.

Let's see how all of this looks from the graphical and matrical perspectives.

Let's see how all of this looks from the graphical and matrical perspectives.

Absolute terms:

Absolute terms:

:{| cellpadding="4"

{| align="center" cellspacing="6" width="90%"

| 1

|

| = || "anybody"

| = || B +, C +, D +, E +, I +, J +, O

\mathbf{1}

& =     & \text{anything}

| m

& =     & \mathrm{B}

| = || "man"

& +\!\!, & \mathrm{C}

| = || C +, I +, J +, O

& +\!\!, & \mathrm{D}

& +\!\!, & \mathrm{E}

| n

& +\!\!, & \mathrm{I}

| = || "noble"

& +\!\!, & \mathrm{J}

| = || C +, D +, O

& +\!\!, & \mathrm{O}

| w

\mathrm{m}

| = || "woman"

& =     & \text{man}

| = || B +, D +, E

& =     & \mathrm{C}

& +\!\!, & \mathrm{I}

& +\!\!, & \mathrm{J}

& +\!\!, & \mathrm{O}

\mathrm{n}

& =     & \text{noble}

& =     & \mathrm{C}

& +\!\!, & \mathrm{D}

& +\!\!, & \mathrm{O}

\mathrm{w}

& =     & \text{woman}

& =     & \mathrm{B}

& +\!\!, & \mathrm{D}

& +\!\!, & \mathrm{E}

|}

|}