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 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 x·y, for y in G, all of which

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

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

from the left margin of the group operation table

and considering its effects on each y in turn as

these run along the right margin.  This produces

the regular ante-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:j + i:h + j:i

 

\operatorname{e}

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

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

 

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

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

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

 

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

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

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

 

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

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

\operatorname{f}

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

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

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

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

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

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

\operatorname{g}

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

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

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

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

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

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

\operatorname{h}

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

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

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

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

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

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

\operatorname{i}

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

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

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

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

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

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

\operatorname{j}

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

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

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

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

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

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

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

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

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

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