Changes

By way of collecting a short-term pay-off for all the work &mdash; not to mention all the peirce-spiration &mdash; 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.

By way of collecting a short-term pay-off for all the work &mdash; not to mention all the peirce-spiration &mdash; 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.

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

{| align="center" cellpadding="10" width="90%"

| align="center" |

| align="center" |

<pre>

<pre>

Writing this table in relative form generates the following natural representation of <math>S_3.\!</math>

Writing this table in relative form generates the following natural representation of <math>S_3.\!</math>

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

{| align="center" cellpadding="10" width="90%"

|

| align="center" |

<math>\begin{matrix}

<math>\begin{matrix}

\operatorname{e}

\operatorname{e}

& = & \operatorname{A}:\operatorname{A}

& = & \operatorname{A}\!:\!\operatorname{A}

& + & \operatorname{B}:\operatorname{B}

& + & \operatorname{B}\!:\!\operatorname{B}

& + & \operatorname{C}:\operatorname{C}

& + & \operatorname{C}\!:\!\operatorname{C}

\\[4pt]

\\[4pt]

\operatorname{f}

\operatorname{f}

& = & \operatorname{A}:\operatorname{C}

& = & \operatorname{A}\!:\!\operatorname{C}

& + & \operatorname{B}:\operatorname{A}

& + & \operatorname{B}\!:\!\operatorname{A}

& + & \operatorname{C}:\operatorname{B}

& + & \operatorname{C}\!:\!\operatorname{B}

\\[4pt]

\\[4pt]

\operatorname{g}

\operatorname{g}

& = & \operatorname{A}:\operatorname{B}

& = & \operatorname{A}\!:\!\operatorname{B}

& + & \operatorname{B}:\operatorname{C}

& + & \operatorname{B}\!:\!\operatorname{C}

& + & \operatorname{C}:\operatorname{A}

& + & \operatorname{C}\!:\!\operatorname{A}

\\[4pt]

\\[4pt]

\operatorname{h}

\operatorname{h}

& = & \operatorname{A}:\operatorname{A}

& = & \operatorname{A}\!:\!\operatorname{A}

& + & \operatorname{B}:\operatorname{C}

& + & \operatorname{B}\!:\!\operatorname{C}

& + & \operatorname{C}:\operatorname{B}

& + & \operatorname{C}\!:\!\operatorname{B}

\\[4pt]

\\[4pt]

\operatorname{i}

\operatorname{i}

& = & \operatorname{A}:\operatorname{C}

& = & \operatorname{A}\!:\!\operatorname{C}

& + & \operatorname{B}:\operatorname{B}

& + & \operatorname{B}\!:\!\operatorname{B}

& + & \operatorname{C}:\operatorname{A}

& + & \operatorname{C}\!:\!\operatorname{A}

\\[4pt]

\\[4pt]

\operatorname{j}

\operatorname{j}

& = & \operatorname{A}:\operatorname{B}

& = & \operatorname{A}\!:\!\operatorname{B}

& + & \operatorname{B}:\operatorname{A}

& + & \operatorname{B}\!:\!\operatorname{A}

& + & \operatorname{C}:\operatorname{C}

& + & \operatorname{C}\!:\!\operatorname{C}

\end{matrix}</math>

\end{matrix}</math>

|}

|}