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*To*: <us>*Subject*: disps, the rubik's cube, and group theory*From*: Eric Dean Tribble <tribble>*Date*: Sun, 17 Sep 89 05:11:30 PDT

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 inverses). g'mornin...zzzz.... dean

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