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Revision as of 16:52, 4 June 2009
, 16:52, 4 June 2009→Note 20: markup
==Note 20==
==Note 20==
By way of collecting a short-term pay-off for all the work — not to mention all the peirce-spiration — that we sweated out over the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations of the symmetric group on three letters, <math>S_3 = \operatorname{Sym}(3).</math> After doing the usual bit of compare and contrast among these divers representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.
<pre>
<pre>
Table 1. Permutations or Substitutions in Sym {A, B, C}
Table 1. Permutations or Substitutions in Sym {A, B, C}
o---------o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o---------o
| | | | | | |
| | | | | | |
o---------o---------o---------o---------o---------o---------o
o---------o---------o---------o---------o---------o---------o
</pre>
Writing this table in relative form generates
Writing this table in relative form generates the following natural representation of <math>S_3.\!</math>
the following "natural representation" of S_3.
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{matrix}
\operatorname{e}
& = & \operatorname{A}:\operatorname{A}
& + & \operatorname{B}:\operatorname{B}
& + & \operatorname{C}:\operatorname{C}
\\[4pt]
\operatorname{f}
& = & \operatorname{A}:\operatorname{C}
& + & \operatorname{B}:\operatorname{A}
& + & \operatorname{C}:\operatorname{B}
\\[4pt]
\operatorname{g}
& = & \operatorname{A}:\operatorname{B}
& + & \operatorname{B}:\operatorname{C}
& + & \operatorname{C}:\operatorname{A}
\\[4pt]
\operatorname{h}
& = & \operatorname{A}:\operatorname{A}
& + & \operatorname{B}:\operatorname{C}
& + & \operatorname{C}:\operatorname{B}
\\[4pt]
\operatorname{i}
& = & \operatorname{A}:\operatorname{C}
& + & \operatorname{B}:\operatorname{B}
& + & \operatorname{C}:\operatorname{A}
\\[4pt]
\operatorname{j}
& = & \operatorname{A}:\operatorname{B}
& + & \operatorname{B}:\operatorname{A}
& + & \operatorname{C}:\operatorname{C}
\end{matrix}</math>
|}
I have without stopping to think about it written out this natural representation of <math>S_3\!</math> in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as <math>X:Y\!</math> constitutes the turning of <math>X\!</math> into <math>Y.\!</math> It is possible that the next time we check in with CSP that we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.
I have without stopping to think about it written out this natural
representation of S_3 in the style that comes most naturally to me,
to wit, the "right" way, whereby an ordered pair configured as X:Y
constitutes the turning of X into Y. It is possible that the next
time we check in with CSP that we will have to adjust our sense of
direction, but that will be an easy enough bridge to cross when we
come to it.
==Note 21==
==Note 21==
