Changes

|}

|}

Then we rewrote these permutations &mdash; being functions <math>f : X \to X</math> they can also be recognized as being 2-adic relations <math>f \subseteq X \times X</math> &mdash; in ''relative form'', in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance:

Then we rewrote these permutations -- being functions f : X --> X

they can also be recognized as being 2-adic relations f c X x  X --

in "relative form", in effect, in the manner to which Peirce would

have made us accostumed had he been given a relative half-a-chance:

    e = A:A + B:B + C:C

\operatorname{e}

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

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

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

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

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

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

These days one is much more likely to encounter the natural representation of <math>S_3\!</math> in the form of a ''linear representation'', that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these:

 

 

 

 

 

These days one is much more likely to encounter the natural representation

of S_3 in the form of a "linear representation", that is, as a family of

linear transformations that map the elements of a suitable vector space

into each other, all of which would in turn usually be represented by

a set of matrices like these:

Table 2.  Matrix Representations of Permutations in Sym(3)

Table 2.  Matrix Representations of Permutations in Sym(3)

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

The key to the mysteries of these matrices is revealed by noting that their

The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlaid on a place-mat marked like so:

coefficient entries are arrayed and overlayed on a place mat marked like so:

    | A:A A:B A:C |

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

    | B:A B:B B:C |

    | C:A C:B C:C |

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

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

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

|}

Of course, the place-settings of convenience at different symposia may vary.

Of course, the place-settings of convenience at different symposia may vary.

==Note 24==

==Note 24==