Changes

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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 across the top margin.  This aspect of pragmatic definition we recognize as the regular ante-representation:→

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 across the top margin.  This aspect of pragmatic definition we recognize as the regular ante-representation:

<pre>

    e = e:e + f:f + g:g + h:h

 

<math>\begin{matrix}

    f = e:f + f:e + g:h + h:g

\mathrm{e}

 

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

    g = e:g + f:h + g:e + h:f

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

 

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

    h = e:h + f:g + g:f + h:e

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

</pre>

\mathrm{f}

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

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

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

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

\mathrm{g}

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

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

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

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

\mathrm{h}

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

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

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

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

\end{matrix}</math>

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> from the top margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run down the left margin.  This aspect of pragmatic definition we recognize as the regular post-representation:

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> from the top margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run down the left margin.  This aspect of pragmatic definition we recognize as the regular post-representation:

<pre>

    e = e:e + f:f + g:g + h:h

 

<math>\begin{matrix}

    f = e:f + f:e + g:h + h:g

\mathrm{e}

 

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

    g = e:g + f:h + g:e + h:f

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

 

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

    h = e:h + f:g + g:f + h:e

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

</pre>

\mathrm{f}

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

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

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

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

\mathrm{g}

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

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

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

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

\mathrm{h}

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

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

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

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

\end{matrix}</math>

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.