On Wed, 6 Aug 2014 19:00:22 -0700 (PDT)
rjf <fate...@gmail.com> wrote:
> On Tuesday, August 5, 2014 7:59:00 PM UTC-7, Robert Bradshaw wrote:
> >
> > On Tue, Aug 5, 2014 at 6:36 PM, rjf <fate...@gmail.com>
> > wrote:
...
> > > Which one, -3 or 10?
> > > They can't both be canonical.
> >
> > Yes, they can "both" be canonical: they are equal (as elements of
> > Z/13Z). There is the choice of internal representation, and what
> > string to use when displaying them to the user, but that doesn't
> > affect its canonicity as a mathematical object.
> >
> I don't understand this.
> I use the word canonical to mean unique distinguished exemplar.
> So there can't be two. If there are two distinguishable items, then one or
> the other or neither might be canonical. Not both.
It seems to me that both of you mean "unique distinguished exemplar"
when speaking of canonical. The problem seems to be that for "unique"
to make sense you should agree on an ambient set. For RJF the ambient
set seems to be ZZ, the set of integers, while for Robert the ambient
set seems to be ZZ/13ZZ, the set of integers modulo 13.
Let's do a more elaborate example: Let ZZ be the ring of integers and
let 3ZZ be its ideal {3k : k in ZZ}. Take ZZ/3ZZ to be the quotient
ring (that is, the ring of integers modulo 3, which happens to be a
field), and take b to be the element represented by the integer 2.
Then we have:
I) The integer 2 is not a canonical representation of b.
II) The element 2 of {0,1,2} is a canonical representation of b.
III) The element b of ZZ/3ZZ is a canonical representation of b.
IV) The element 2 of ZZ/3ZZ is a canonical representation of b.
V) The element 5 of ZZ/3ZZ is a canonical representation of b.
VI) The elements 2 and 5 of ZZ/3ZZ are both canonical representations
of b.
These are the proofs:
I) The integer 5, which is not 2, also represents b.
II) The elements 0 and 1 of {0,1,2} represent elements of ZZ/3ZZ that
are not b.
III) This is a trivial: If c is an element of ZZ/3ZZ that is b, then it
is b.
IV) The element 2 of ZZ/3ZZ - that is to say, the element represented
by the integer 2 - is b. The result follows by assertion III.
V) Analogous to IV.
VI) Immediate from IV and V.
The importance of agreeing on the ambient set becomes apparent when
comparing assertions I and IV.
The words "of ZZ/3ZZ" are important indeed in assertion VI: If we had
read "of ZZ", then this assertion would be pointing directly at its own
counterexample.
Regards,
Erik Massop
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