[Axiom-developer] group theory ordering

[root]

Chuck,

Thanks for the notes. I realize that there are several different ways
to order the lattice of group based on different metrics. I have an
example from topology (which I hope to keyboard later today). It comes
from an excellent book called "Counterexamples in Topology" by Steen
and Seebach (Springer-Verlag, 1978; Dover 1995) on page 16. They
order the groups based on axioms. They also have order diagrams 
based on properties such as connectedness and compactness which are
entirely different. I'm trying to derive similar information for our
area and it appears nobody has done so yet. As Dylan Thurston pointed
out I still have several points of confusion.

One point of confusion appears to be that some terms are defined as
as axioms and some are defined as properties. Sometimes I see writers
use abelian as an axiom and sometimes I see it used as a property.
I'm beginning to see that Axiom has this confusion also.

I'm looking to attack the problem of ordering these things for 3 reasons:

1) I need to understand (and classify) these groups in some systematic ways
   so I can get a better handle on the pile of results and algorithms.
2) I need to understand (and classify) these groups so I can figure out
   a category structure that Axiom can use to construct these groups and
   order the algorithms.
3) It seems like a good domain to build this beast I'm calling a "crystal".
   Picture a huge ball of string in space with many knots (the
   ball being a graph of the many relationships and the knots being
   clusters of axioms and properties related to a concept. Now "wrap" the
   knotted string with a "crystal" with many facets. Each facet extracts
   a different structural relationship of the concepts in the ball. This
   whole thing (viewer and network; aka facets and string) I call crystal.
   Some facets show the math, some the lattice, some the underlying code.

Virtually every area Axiom has touched has the same problem (though less
intense) as Infinite Group Theory. Everything is ordered by everything
else. How you want to order it depends on how you want to think about it.
I want to look at Infinite Groups ordered in many different ways.

So crystal is an attempt to let you string things together in all of the
ways you want to think about them (the ball) and view them in all of the
ways you want to see them (the facets). Hard problems within crystal are
things like automatically classifying concepts (by using a metric like
subsumption) so (a) they fit in the "appropriate" place and (b) you can
find them again. You want it to be automatic because you want the machine
to do the work. Axiom already automatically extracts a great deal of
information when it compiles things and even more when you contruct
type towers of domains. At the moment this information is exported as
"databases". The idea of a "database" is too limiting to order the kind
of math we need to play with. Semantic networks are closer but, like the
ball itself, they aren't the only way to build the structure.

Of course, none of this is magic which means that I have to understand,
extract, categorize, code, compile, classify, and view the concepts.
I've done a complete reduction of Axiom's categories and domains and
I'm looking at the compiler output to see what I can automatically
extract.

I'll look at the information on your website. 

Thanks,
Tim