[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 16

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We've been exploring the applications of a certain technique
for clarifying abstruse concepts, a rough-cut version of the
pragmatic maxim that I've been accustomed to refer to as the
"operationalization" of ideas.  The basic idea is to replace
the question of "What it is", which modest people comprehend
is far beyond their powers to answer any time soon, with the
question of "What it does", which most people know at least
a modicum about.

In the case of regular representations of groups we found
a non-plussing surplus of answers to sort our way through.
So let us track back one more time to see if we can learn
any lessons that might carry over to more realistic cases.

Here is is the operation table of V_4 once again:

o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` * ` % ` e ` | ` f ` | ` g ` | ` h ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o=======o=======o=======o=======o=======o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` e ` % ` e ` | ` f ` | ` g ` | ` h ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` f ` % ` f ` | ` e ` | ` h ` | ` g ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` g ` % ` g ` | ` h ` | ` e ` | ` f ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
| ` h ` % ` h ` | ` g ` | ` f ` | ` e ` |
| ` ` ` % ` ` ` | ` ` ` | ` ` ` | ` ` ` |
o-------o-------o-------o-------o-------o

A group operation table is really just a device for recording
a certain 3-adic relation, specifically, the set of 3-tuples
of the form <x, y, z> that satisfy the equation x * y = z,
where the sign "*" that indicates the group operation is
frequently omitted in contexts where it is understood.

In the case of V_4 = (G, *), where G is the "underlying set"
{e, f, g, h}, we have the 3-adic relation L(V_4) c G x G x G
whose triples are listed below:

   e:e:e
   e:f:f
   e:g:g
   e:h:h

   f:e:f
   f:f:e
   f:g:h
   f:h:g

   g:e:g
   g:f:h
   g:g:e
   g:h:f

   h:e:h
   h:f:g
   h:g:f
   h:h:e

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 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

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

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

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

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

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

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

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

Jon Awbrey

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