Changes

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This 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 took up earlier.  That is to say, the story is the same, only the names have been changed.  An abstract group can have a variety of significantly and superficially different representations.  But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

This table is abstractly the same as, or isomorphic to, the versions

with the E_ij operators and the T_ij transformations that we took up

earlier.  That is to say, the story is the same, only the names have

been changed.  An abstract group can have a variety of significantly

and superficially different representations.  But even after we have

long forgotten the details of any particular representation there is

a type of concrete representations, called "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,

To see how a regular representation is constructed from the abstract operation table, 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, as a logical disjunction or boolean 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 forms one of the two possible ''regular representations'' of the group, in this case the one that is called the ''post-regular representation'' or the ''right regular representation''.  It has long been conventional to organize the terms of this logical aggregate in the form of a matrix:

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,

\operatorname{G}

& = & \operatorname{e}

& + & \operatorname{f}

& + & \operatorname{g}

& + & \operatorname{h}

And so, by expanding effects, we get:

And so, by expanding effects, we get:

  G =

 

| align="center" |

  e:e + f:f + g:g + h:h +

 

  e:f + f:e + g:h + h:g +

& = & \operatorname{e}:\operatorname{e}

 

& + & \operatorname{f}:\operatorname{f}

  e:g + f:h + g:e + h:f +

& + & \operatorname{g}:\operatorname{g}

 

& + & \operatorname{h}:\operatorname{h}

  e:h + f:g + g:f + h:e

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

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

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

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

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

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

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

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

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

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

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

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

More on the pragmatic maxim as a representation principle later.

More on the pragmatic maxim as a representation principle later.

==Note 13==

==Note 13==