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disps, the rubik's cube, and group theory

I finally found a rubik's cube last night!  (at ThinkerToys).  As a
result I started reading (skimming) my book on Cubik math while
waiting for my fileIns.  I've been wanting to understand group theory
for awhile because what little I knew of it rang with my insights into
dispattivity a while back.

I just ran across the 4 laws necessary to define a group: closure law,
associativity law, identity law, and inverse law.  These fit
properties that I settled on when working on tree-spaces.

closure:  	combining a dsp with another dsp returns a dsp.
associative:  	the property required to balance enfilades and ents
identity:  	necessary to take a coordinate (dsp in the same group)
		down and up levels in the enfilade without changing
		it.  X*I = I*X = X
inverse:	required to transform into and out of a local
		coordinate space.  X*i(X) = i(X)*X = i.  

I don't know yet whether groups require a UNIQUE inverse for each
element.  I'm pretty sure they don't (for fairly straightforward
mathematical reasons).  If they don't, then partially ordered
coordinate spaces are also addressed by elements of a group.

Something in the phrasing of the description of a group caught my eye.
This may be totally bogus, but it led to the following...

Groups are the theoretical structure underlying the ability to define
coordinate spaces (primarily because of closure, I suppose).  Hmmm.
I've tried a couple of times to articulate my intuition and keep
deleting it.  Essentially, a purely set-based model such as I believe
the relational model to be cannot adequately describe coordinate
spaces well.  When I better understand the relationship between
general set theory and group theory, I'll articulate this better:
does the abstraction of group theory subsume the abstractions of set
theory, and what does that mean for us?

slight subject change: I just realized and can now articulate my
continuing discomfort with ID space Orgls:  the IDs are not elements
of a group.  I don't know of an operation, *, that has all the above
properties defined across the ID space.  I'm sure it's not
coincidental that enfilades don't buy us anything for ID spaces (until
we add group properties for implementation reasons).  

In a few days I'll probably have wait time with which to continue my
reading.  Are enclosures really sub-groups in disguise?  Where is the
parallel mathematical construct for wids hidden?  (I'm pretty sure
they aren't groups:  they don't need an identity elements or