[Axiom-developer] Re: [#358 Variable is apparently always assumed to be positive?]Functions and Segments

[William Sit]

Bill Page wrote:
> 
> William,
> 
> Thanks for adding your comments to the wiki. I have
> re-arranged this a little by separating the comments about
> segment from those about functions so that now your
> comments about segments appears here:
> 
> http://wiki.axiom-developer.org/SandBoxFloatSegment

I wonder why you did not move it to a separate IssueTracker?
It is bug in the implementation of 'expand' (incompatibility
with 'BY').

> 
> On June 7, 2007 9:21 PM William Sit wrote:
> 
> > ...
> > I am overwhelmed by the amount of Axiom emails and most
> > I  could not follow. I don't even have the guts to try
> > compiling the newest versions. To me, it is very
> > confusing, but it does seem that Axiom is in excellent
> > hands.
> >
> 
> Jump in any time, your contributions are like a "breath
> of fresh air" ;-)

Would you jump into the ocean without knowing how to swim
(even with lifeguards around), or would you rather work on
some beauties at the beach? 

[...]

> I added a slightly changed version of Segment to this
> page:
> 
> http://wiki.axiom-developer.org/SandBoxFloatSegment
> 
> All I did was require that Segment be defined over
> a Ring and then re-iterpreted BY as specifying the
> separation of elements of the segment.
> 
> This seems more "categorical" to me given what I
> expect to be the normal use-case although it is in
> some ways more restrictive than the original as you
> pointed out in your email. 

I made some comments on your construction, see
http://wiki.axiom-developer.org/SandBoxFloatSegment

> Can you imagine an actual
> application where it is interesting to define a
> segment over something that is not a Ring?


I gave a simple example of MWF as a segment of M..S by 2. 

In general, one can define a segment in any chain (linearly
ordered set) or even a partially ordered set such as the set
of some algebraic substructures between two substructures,
ordered by inclusion. Such substructures can be subfields,
ideals in a ring, subsets of a set, etc.  For example, in
solving an equation by radicals, one may want to select a
shortest chain in the poset of algebraic subextensions of a
field where a solution by radical exists to minimize the
number of radicals used. Or, one may want to compute 

     [dim i for i in p..q] 

for the set of prime ideals between two primes p and q in a
ring. One can then select a maximal chain in this poset, and
later construct the "lying over" chain to this chain in an
overring. Of course, one has to be able to perform "expand"
in these situations. For specific posets, such as subsets of
a finite set, this may be possible. The 'BY' construction
(expansion by adding) may not be that meaningful in this
context (the meaning of "add" is usually missing), although
it certainly make sense to do expansion by counting. That is
why there should be two 'BY' constructs. Unfortunately,
putting them in the same package would cause confusion when
the parameter of SEGMENT is Integer.


William