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==Note 15==
==Note 15==
In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(i\!:\!j),</math> as <math>i, j\!</math> range over the universe of discourse, would be referred to as the ''umbral elements'' of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the "ingredients". When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:
In Peirce's time, and even in some circles of mathematics today,
the information indicated by the elementary relatives (i:j), as
i, j range over the universe of discourse, would be referred to
as the "umbral elements" of the algebraic operation represented
by the matrix, though I seem to recall that Peirce preferred to
call these terms the "ingredients". When this ordered basis is
understood well enough, one will tend to drop any mention of it
from the matrix itself, leaving us nothing but these bare bones:
{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>
M \quad = \quad
\begin{bmatrix}
1 & 1 & 0
\\
0 & 1 & 1
\\
1 & 0 & 1
\end{bmatrix}
</math>
|}
The various representations of <math>M\!</math> are nothing more than alternative ways of specifying its basic ingredients, namely, the following aggregate of elementary relatives:
{| align="center" cellpadding="6" width="90%"
| align="center" |
<math>\begin{array}{*{13}{c}}
M
& = & a\!:\!a
& + & b\!:\!b
& + & c\!:\!c
& + & a\!:\!b
& + & b\!:\!c
& + & c\!:\!a
\end{array}</math>
|}
Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
Up to this point, we are 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>m_{ij} = 1,\!</math> to say that <math>i\!</math> is a marker for <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 customarily read them in
logical contexts: i is the relate, j is the correlate, and in
our current example we reading i:j, or more exactly, m_ij = 1,
to say that i is a marker for j. This is the mode of reading
that we call "multiplying on the left".
<pre>
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
application, however, Peirce converts to the alternative mode of
