Changes

# <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.</math>

# <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.</math>

In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>x \cdot y,</math> for <math>y\!</math> in <math>G,\!</math> all of which is summarily notated as <math>x = \{ (y : x \cdot y) : y \in G \}.</math>  The pairs <math>(y : x \cdot y)</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin.  This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:

In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>x \cdot y,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : x \cdot y) ~|~ y \in G \}.</math>  The pairs <math>(y : x \cdot y)</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin.  This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:

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In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>y \cdot x,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : y \cdot x) ~|~ y \in G \}.</math>  The pairs <math>(y : y \cdot x)</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin.  This generates the ''regular post-representation'' of <math>S_3,\!</math> like so:

In (2), we consider the effects of each x in its

practical bearing on contexts of the form <y, _>,

as y ranges over G, and the effects are such that

x takes <y, _> into y·x, for y in G, all of which

is summarily notated as x = {(y : y·x) : y in G}.

The pairs (y : y·x) can be found by picking an x

on the right margin of the group operation table

and considering its effects on each y in turn as

these run along the left margin.  This generates

the regular post-representation of S_3, like so:

e   =   e:e + f:f + g:g + h:h + i:i + j:j

 

f   =   e:f + f:g + g:e + h:i + i:j + j:h

 

\operatorname{e}

g   =   e:g + f:e + g:f + h:j + i:h + j:i

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

 

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

h   =   e:h + f:j + g:i + h:e + i:g + j:f

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

 

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

i   =   e:i + f:h + g:j + h:f + i:e + j:g

& + & \operatorname{i}:\operatorname{i}

 

& + & \operatorname{j}:\operatorname{j}

j   =   e:j + f:i + g:h + h:g + i:f + j:e

\operatorname{f}

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

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

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

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

& + & \operatorname{i}:\operatorname{j}

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

\operatorname{g}

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

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

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

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

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

& + & \operatorname{j}:\operatorname{i}

\operatorname{h}

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

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

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

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

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

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

\operatorname{i}

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

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

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

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

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

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

\operatorname{j}

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

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

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

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

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

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

If the ante-rep looks different from the post-rep,

If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3\!</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.

it is just as it should be, as S_3 is non-abelian

(non-commutative), and so the two representations

differ in the details of their practical effects,

though, of course, being representations of the

same abstract group, they must be isomorphic.

==Note 22==

==Note 22==