I would like to thank Eric Massop for his excellent and enlightening posts.
On Saturday, August 9, 2014 11:58:20 PM UTC+2, Erik Massop wrote:
>
> On Wed, 6 Aug 2014 19:00:22 -0700 (PDT)
> rjf <fat...@gmail.com <javascript:>> 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 <fat...@gmail.com <javascript:>>
> > > 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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