Changes

This operation table is abstractly the same as, or isomorphic to, the versions with the <math>\operatorname{E}_{ij}</math> operators and the <math>\operatorname{T}_{ij}</math> transformations that we discussed earlier.  That is to say, the story is the same &mdash; only the names have been changed.  An abstract group can have a multitude of significantly and superficially different representations.  Even after we have long forgotten the details of the particular representation that we may have come in with, there are species of concrete representations, called the ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

This operation table is abstractly the same as, or isomorphic to, the versions with the <math>\operatorname{E}_{ij}</math> operators and the <math>\operatorname{T}_{ij}</math> transformations that we discussed earlier.  That is to say, the story is the same &mdash; only the names have been changed.  An abstract group can have a multitude of significantly and superficially different representations.  Even after we have long forgotten the details of the particular representation that we may have come in with, there are species of concrete representations, called the ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

For example, select a group element from the top margin of the Table, and "consider its effects" on each of the group elements as they are listed along the left margin.  We may record these effects as Peirce usually did, as a ''logical aggregate'' of elementary dyadic relatives, that is to say, a logical disjunction or sum whose terms represent the ordered pairs of <math>\operatorname{input} : \operatorname{output}</math> transactions that are produced by each group element in turn.  This yields what is usually known as one of the ''regular representations'' of the group, specifically, the ''first'', the ''post-'', or the ''right'' regular representation.  It has long been conventional to organize the terms in the form of a matrix:

For example, select a group element from the top margin of the Table,

and "consider its effects" on each of the group elements as they are

listed along the left margin.  We may record these effects as Peirce

usually did, as a logical "aggregate" of elementary dyadic relatives,

that is to say, a disjunction or a logical sum whose terms represent

the ordered pairs of <input : output> transactions that are produced

by each group element in turn.  This yields what is usually known as

one of the "regular representations" of the group, specifically, the

"first", the "post-", or the "right" regular representation.  It has

long been conventional to organize the terms in the form of a matrix:

Reading "+" as a logical disjunction:

Reading "<math>+\!</math>" as a logical disjunction:

G = e + f + g + h,

| <math>\mathrm{G} ~=~ \mathrm{e} + \mathrm{f} + \mathrm{g} + \mathrm{h}</math>

And so, by expanding effects, we get:

And so, by expanding effects, we get:

G = e:e + f:f + g:g + h:h

 

  + e:f + f:e + g:h + h:g

 

\mathrm{G}

  + e:g + f:h + g:e + h:f

& = & \mathrm{e}:\mathrm{e}

 

& + & \mathrm{f}:\mathrm{f}

  + e:h + f:g + g:f + h:e

& + & \mathrm{g}:\mathrm{g}

& + & \mathrm{h}:\mathrm{h}

& + & \mathrm{e}:\mathrm{f}

& + & \mathrm{f}:\mathrm{e}

& + & \mathrm{g}:\mathrm{h}

& + & \mathrm{h}:\mathrm{g}

& + & \mathrm{e}:\mathrm{g}

& + & \mathrm{f}:\mathrm{h}

& + & \mathrm{g}:\mathrm{e}

& + & \mathrm{h}:\mathrm{f}

& + & \mathrm{e}:\mathrm{h}

& + & \mathrm{f}:\mathrm{g}

& + & \mathrm{g}:\mathrm{f}

& + & \mathrm{h}:\mathrm{e}

More on the pragmatic maxim as a representation principle later.

More on the pragmatic maxim as a representation principle later.

====Note 10====

====Note 10====