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<pre>
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It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math>  It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
It is part of the definition of a group that the 3-adic
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relation L c G^3 is actually a function L : G x G -> G.
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It is from this functional perspective that we can see
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an easy way to derive the two regular representations.
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Since we have a function of the type L : G x G -> G,
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we can define a couple of substitution operators:
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1.  Sub(x, <_, y>) puts any specified x into
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# <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>x \cdot y.</math>
    the empty slot of the rheme <_, y>, with
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# <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>
    the effect of producing the saturated
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    rheme <x, y> that evaluates to x·y.
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2.  Sub(x, <y, _>) puts any specified x into
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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:
    the empty slot of the rheme <y, >, with
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    the effect of producing the saturated
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    rheme <y, x> that evaluates to y·x.
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In (1), we consider the effects of each x in its
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practical bearing on contexts of the form <_, y>,
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as y ranges over G, and the effects are such that
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x takes <_, y> into x·y, for y in G, all of which
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is summarily notated as x = {(y : x·y) : y in G}.
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The pairs (y : x·y) can be found by picking an x
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from the left margin of the group operation table
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and considering its effects on each y in turn as
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these run across the top margin.  This aspect of
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pragmatic definition we recognize as the regular
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ante-representation:
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<pre>
 
     e  =  e:e  +  f:f  +  g:g  +  h:h
 
     e  =  e:e  +  f:f  +  g:g  +  h:h
   Line 3,443: Line 3,421:     
     h  =  e:h  +  f:g  +  g:f  +  h:e
 
     h  =  e:h  +  f:g  +  g:f  +  h:e
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</pre>
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In (2), we consider the effects of each x in its
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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> 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:
practical bearing on contexts of the form <y, _>,
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as y ranges over G, and the effects are such that
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x takes <y, _> into y·x, for y in G, all of which
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is summarily notated as x = {(y : y·x) : y in G}.
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The pairs (y : y·x) can be found by picking an x
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from the top margin of the group operation table
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and considering its effects on each y in turn as
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these run down the left margin.  This aspect of
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pragmatic definition we recognize as the regular
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post-representation:
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<pre>
 
     e  =  e:e  +  f:f  +  g:g  +  h:h
 
     e  =  e:e  +  f:f  +  g:g  +  h:h
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     h  =  e:h  +  f:g  +  g:f  +  h:e
 
     h  =  e:h  +  f:g  +  g:f  +  h:e
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</pre>
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If the ante-rep looks the same as the post-rep,
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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.
now that I'm writing them in the same dialect,
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that is because V_4 is abelian (commutative),
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and so the two representations have the very
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same effects on each point of their bearing.
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</pre>
      
==Note 19==
 
==Note 19==
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