[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DAL.  Note 19

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

To construct the regular representations of S_3,
we pick up from the data of its operation table,
DAL 17, Table 17-b, at either one of these sites:

http://stderr.org/pipermail/inquiry/2004-May/001419.html
http://forum.wolframscience.com/showthread.php?postid=1321#post1321

Just by way of staying clear about what we are doing,
let's return to the recipe that we worked out before:

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
    the empty slot of the rheme <_, y>, with
    the effect of producing the saturated
    rheme <x, y> that evaluates to xy.

2.  Sub(x, <y, _>) puts any specified x into
    the empty slot of the rheme <y, _>, with
    the effect of producing the saturated
    rheme <y, x> that evaluates to yx.

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

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

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

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

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

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

In (2), 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 yx, for y in G, all of which
is summarily notated as x = {<y : yx> : y in G}.
The pairs <y : yx> can be found by picking an x
on the right margin of the group operation table
and considering its effects on each y in turn as
these run along the left margin.  This generates
the regular post-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:i  +  i:j  +  j:h

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

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

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

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

If the ante-rep looks different from the post-rep,
it is just as it should be, as S_3 is non-abelian
(non-commutative), and so the two representations
differ in the details of their practical effects,
though, of course, being representations of the
same abstract group, they must be isomorphic.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o