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: http://wiki.axiom-developer.org/[EMAIL PROTECTED] .axiom-developer.org 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: http://wiki.axiom-developer.org/RepAndPer I am still looking for any computer sciense publication that addresses this issue of representation in a formal mathematical way. Regards, Bill Page. _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
