Changes

It is part of the definition of a group that the 3-adic relation ''L'' ⊆ ''G''³ is actually a function ''L'' : ''G'' × ''G'' → ''G''.  It is from this functional perspective that we can see an easy way to derive the two regular representations.

It is part of the definition of a group that the 3-adic relation ''L'' ⊆ ''G''³ is actually a function ''L'' : ''G'' × ''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'' × ''G'' → ''G'', we can define a couple of substitution operators:

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

# Sub(''x'',&nbsp;«_,&nbsp;''y''») puts any specified ''x'' into the empty slot of the rheme «_,&nbsp;''y''», with the effect of producing the saturated rheme «''x'',&nbsp;''y''» that evaluates to ''xy''.

    the empty slot of the rheme <_, y>, with

# Sub(''x'',&nbsp;«''y'',&nbsp;_») puts any specified ''x'' into the empty slot of the rheme «''y'',&nbsp;_», with the effect of producing the saturated rheme «''y'',&nbsp;''x''» that evaluates to ''yx''.

    the effect of producing the saturated

    rheme <x, y> that evaluates to xy.

In (1), we consider the effects of each ''x'' in its practical bearing on contexts of the form «_,&nbsp;''y''», as ''y'' ranges over ''G'', and the effects are such that ''x'' takes «_,&nbsp;''y''» into ''xy'', for ''y'' in ''G'', all of which is summarily notated as ''x''&nbsp;=&nbsp;{(''y''&nbsp;:&nbsp;''xy'')&nbsp;:&nbsp;''y''&nbsp;in&nbsp;''G''}. The pairs (''y''&nbsp;:&nbsp;''xy'') 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:

 

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 xy, for y in G, all of which

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

The pairs (y : xy) 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

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

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

   j  =  e:j  +  f:h  +  g:i  +  h:f  +  i:g  +  j: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, _>,