Changes

Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{~~~~}\, {}^{\prime\prime}</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.

Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{~~~~}\, {}^{\prime\prime}</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.

Up to this point, we're still reading the elementary relatives of the form <math>I:J\!</math> in the way that Peirce reads them in logical contexts: <math>I\!</math> is the relate, <math>J\!</math> is the correlate, and in our current example we read <math>I:J,\!</math> or more exactly, <math>\mathit{n}_{ij} = 1,\!</math> to say that <math>I\!</math> is a noder of <math>J.\!</math>  This is the mode of reading that we call ''multiplying on the left''.

Up to this point, we are still reading the elementary relatives of

the form I:J in the way that Peirce reads them in logical contexts:

I is the relate, J is the correlate, and in our current example we

read I:J, or more exactly, n_ij = 1, to say that I is a noder of J.

This is the mode of reading that we call "multiplying on the left".

In the algebraic, permutational, or transformational contexts of

In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>I\!</math> the relate and <math>J\!</math> the correlate, the elementary relative <math>I:J\!</math> now means that <math>I\!</math> gets changed into <math>J.\!</math>  In this scheme of reading, the transformation <math>\mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}</math> is a permutation of the aggregate <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C},</math> or what we would now call the set <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> in particular, it is the permutation that is otherwise notated as follows:

application, however, Peirce converts to the alternative mode of

reading, although still calling I the relate and J the correlate,

the elementary relative I:J now means that I gets changed into J.

In this scheme of reading, the transformation A:B + B:C + C:A is

a permutation of the aggregate $1$ = A + B + C, or what we would

now call the set {A, B, C}, in particular, it is the permutation

that is otherwise notated as:

( A B C )

( B C A )

\mathrm{A} & \mathrm{B} & \mathrm{C}

\mathrm{B} & \mathrm{C} & \mathrm{A}

This is consistent with the convention that Peirce uses in

This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324&ndash;327).

the paper "On a Class of Multiple Algebras" (CP 3.324-327).

==Note 16==

==Note 16==