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disps, the rubik's cube, and group theory

A note to skeptics before diving in:  General enfilade theory is the
basis of Xanadu's technology.  By implementing one data-structure that
can efficiently index and edit coordinate spaces defined by dsps, we
can rapidly accommodate new media and representations.  As the
underlying principles are worked out in further detail, the
implementation is simplified and generalized.  The particular
coordinate space we should be able to address is that of Directed,
Acyclic Graphs (DAGs).  If we can handle DAGs, we will use them to
address historical trace documents.

If positions in the DAG are denoted with turning directions from some
root (the home in enclosure terminology), then two distinct points
might have two completely different paths connecting them.  The
turning directions describing each of these paths would be a dsp, and
if they happened to be from the root, then they could be considered
coordinates in the DAGspace.  Note that the use of turning directions
for dsps is derived f

   Wait a minute.  A group is a set.  The definition of equality between
   group elements is "they are the same set element".  If you use a
   definition of equality which defines equivalence classes of some set S,
   then your focus is on the set E of equivalence classes of S, not
   directly on the set S.  E might also be a group.

Hmmm.  Yes.  See my motivation for all this below.

   In any case, how can the definition of equality be based on the
   definition of equality?:

By induction.  Since my definition wasn't an induction, I've come
up with a better one.

	A = B iff A*inverse(B) = I and 
	and we define the identity element I to be unique.

I'll continue this later...