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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

relation L c G^3 is actually a function L : G x G -> G.

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 L : G x G -> G,

we can define a couple of substitution operators:

1.  Sub(x, <_, y>) puts any specified x into

# <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

# <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

    rheme <x, y> that evaluates to x·y.

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 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 across the top margin.  This aspect of

pragmatic definition we recognize as the regular

ante-representation:

     e  =  e:e  +  f:f  +  g:g  +  h:h

     e  =  e:e  +  f:f  +  g:g  +  h:h

     h  =  e:h  +  f:g  +  g:f  +  h:e

     h  =  e:h  +  f:g  +  g:f  +  h:e

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

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, _>,

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

from the top margin of the group operation table

and considering its effects on each y in turn as

these run down the left margin.  This aspect of

pragmatic definition we recognize as the regular

post-representation:

     e  =  e:e  +  f:f  +  g:g  +  h:h

     e  =  e:e  +  f:f  +  g:g  +  h:h

     h  =  e:h  +  f:g  +  g:f  +  h:e

     h  =  e:h  +  f:g  +  g:f  +  h:e

If the ante-rep looks the same as the post-rep,

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,

that is because V_4 is abelian (commutative),

and so the two representations have the very

same effects on each point of their bearing.

==Note 19==

==Note 19==