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RE: [Axiom-developer] Set Any and SXHASH

From: Bill Page
Subject: RE: [Axiom-developer] Set Any and SXHASH
Date: Fri, 6 Apr 2007 17:18:14 -0400

On April 6, 2007 8:04 AM Waldek Hebisch wrote:
> ...
> Bill Page wrote [about using LEXGREATERP as an ordering]:
> > 
> > Yes. Why do you say that this would not be consistent with
> > domain equality?
> >
> Domain elements may be mathematically equal but have different
> representations.  AFAICS this happens for example for general
> fractions.

Ah yes, in this case there is no "canonical" representative.

Marin gives another examnple like that in:

Martin refers to the description of "canonical" in HyperDoc:

"canonical is true if and only if distinct elements have
 distinct data structures. For example, a domain of
 mathematical objects which has the canonical attribute means
 means that two objects are mathematically equal if and only
 if their data structures are equal."

I guess one should interpret "distinct" as a synonym for x ~= y.
So this means that for some domain D with representation Rep

  x,y:D; x = y implies rep(x) = rep(y) and
         rep(x) = rep(y) implies x = y

where by '=' I am specifically referring to the functions
exported by D and Rep, respectively. 

> > What problems would occur if elements of all finite sets
> > were ordered according to this ordering?
> > 
> We may miss duplicates and declare equal sets as unequal.

Yes, I see that you are right if the domain is not canonical.
HyperDoc only shows 5 domains with canonical, FourierSeries,
Fraction, Integer, RomanNumeral, and SingleInteger; although
I expect there are actually many more domains that could be
given this attribute.

I assume that when you wrote "formal fractions" above, you
were not refering to Axiom's 'Fraction' domain?

Besides, the "canonical" attribute, it seems to me that there
is something else interesting here that should to be stated
more formally.

I suppose that it might be possible to design a domain in
which two representives of a given domain are equal but that
the corresponding members of the domain are not equal. But
this is somehow not "natural". So a "natural" representation
would be one for which

  x,y:D; rep(x) = rep(y) implies x = y

I was thinking about something like this when I wrote:

I am still looking for any computer sciense publication that
addresses this issue of representation in a formal mathematical

Bill Page.

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