quotient map
Aleks Kleyn
quotient set. it may be quotient group or quotient ring. I expected to call
map from set to quotient set quotient map. However when I did research on
google I see that all references on quotient map mention topology. my
question is how we call such map when we do not use topology?
--
Aleks Kleyn
http://www.geocities.com/aleks_kleyn
William Elliot
> when we have equivalence on the set we call set of classes of equivalence
> quotient set. it may be quotient group or quotient ring. I expected to call
> map from set to quotient set quotient map. However when I did research on
> google I see that all references on quotient map mention topology. my
> question is how we call such map when we do not use topology?
In algebra it's called a homomorphism.
homomorphism : isomorphism :: quotient map : homeomorphsim
Aleks Kleyn
--
Aleks Kleyn
http://www.geocities.com/aleks_kleyn
"William Elliot" <ma...@rdrop.remove.com> wrote in message
news:2008122220...@agora.rdrop.com...
David C. Ullrich
<ma...@rdrop.remove.com> wrote:
>On Mon, 22 Dec 2008, Aleks Kleyn wrote:
>
>> when we have equivalence on the set we call set of classes of equivalence
>> quotient set. it may be quotient group or quotient ring. I expected to call
>> map from set to quotient set quotient map. However when I did research on
>> google I see that all references on quotient map mention topology. my
>> question is how we call such map when we do not use topology?
>
>In algebra it's called a homomorphism.
>homomorphism : isomorphism :: quotient map : homeomorphsim
Not really - homomorphisms in algebra need not be quotient
maps.
>
>
>> --
>>
>> Aleks Kleyn
>> http://www.geocities.com/aleks_kleyn
>>
>>
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
William Elliot
> <ma...@rdrop.remove.com> wrote:
>> On Mon, 22 Dec 2008, Aleks Kleyn wrote:
>>
>>> when we have equivalence on the set we call set of classes of equivalence
>>> quotient set. it may be quotient group or quotient ring. I expected to call
>>> map from set to quotient set quotient map. However when I did research on
>>> google I see that all references on quotient map mention topology. my
>>> question is how we call such map when we do not use topology?
>>
>> In algebra it's called a homomorphism.
>
> Not really - homomorphisms in algebra need not be quotient maps.
surjective homomorphism : isomorphism :: quotient map : homeomorphism
However if f is a not surjective homomorphism, there's still the
partition of the domain and the quotient is isomorphic to a subgroup
of the codomain.
Were a quotient map not surjective, the neglected remainer of the codomain
would be a discrete subspace of the codomain and the quotient space
would still embed into the codomain space.
subspace : subgroup :: embeds : isomorphic to a subgroup
David C. Ullrich
<ma...@rdrop.remove.com> wrote:
>On Tue, 23 Dec 2008, David C. Ullrich wrote:
>> <ma...@rdrop.remove.com> wrote:
>>> On Mon, 22 Dec 2008, Aleks Kleyn wrote:
>>>
>>>> when we have equivalence on the set we call set of classes of equivalence
>>>> quotient set. it may be quotient group or quotient ring. I expected to call
>>>> map from set to quotient set quotient map. However when I did research on
>>>> google I see that all references on quotient map mention topology. my
>>>> question is how we call such map when we do not use topology?
>>>
>>> In algebra it's called a homomorphism.
>>> homomorphism : isomorphism :: quotient map : homeomorphism
>>
>> Not really - homomorphisms in algebra need not be quotient maps.
>
>surjective homomorphism : isomorphism :: quotient map : homeomorphism
>
>However if f is a not surjective homomorphism, there's still the
>partition of the domain and the quotient is isomorphic to a subgroup
>of the codomain.
Whatever. That doesn't imply that quotient maps are called
homomophisms in algebra, as you said. That's simply not
true.
>Were a quotient map not surjective, the neglected remainer of the codomain
>would be a discrete subspace of the codomain and the quotient space
>would still embed into the codomain space.
>
>subspace : subgroup :: embeds : isomorphic to a subgroup
David C. Ullrich
Axel Vogt
> when we have equivalence on the set we call set of classes of
> equivalence quotient set. it may be quotient group or quotient ring. I
> expected to call map from set to quotient set quotient map. However when
> I did research on google I see that all references on quotient map
> mention topology. my question is how we call such map when we do not use
> topology?
You can call S -> S/~ a quotient map and it is always surjective
(and if I remember correctly old lectures we learned it that way
for groups in Algebra, like Z/nZ).
However the names are not consistent: in Algebra that is sometimes
often called a factor group or ring or what is under discussion,
while by quotient ring or quotient field something like Rationals
is meant: passing from integers to fractions (localization is a
different name as well).
If you want some ugly presentation (= complicate to read) you can
look into Lang's Algebra book (though the book is great).
Another wording is "S modulo equivalence" or residual map ...
So what ... that you do not find a convenient source may simply
say others find so basic they do not write about it, may be you
complete your search term by s.th. like 'lecture, undergraduate'
and find different answers.
Just take it as an example that the nomen clatura is not quite
unique, be encouraged to ask - and pass to essentials :-)
Arturo Magidin
Axel Vogt <&nor...@axelvogt.de> wrote:
>However the names are not consistent: in Algebra that is sometimes
>often called a factor group or ring or what is under discussion,
>while by quotient ring or quotient field something like Rationals
>is meant: passing from integers to fractions (localization is a
>different name as well).
I usually see them refered to as "field of quotients" or "ring of
quotients", instead.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
Axel Vogt
> In article <6rd2jr...@mid.individual.net>,
> Axel Vogt <&nor...@axelvogt.de> wrote:
>
>> However the names are not consistent: in Algebra that is sometimes
>> often called a factor group or ring or what is under discussion,
>> while by quotient ring or quotient field something like Rationals
>> is meant: passing from integers to fractions (localization is a
>> different name as well).
>
> I usually see them refered to as "field of quotients" or "ring of
> quotients", instead.
>
May be my language is to Germish ... (we call it "Quotientenring" etc).
Arturo Magidin
I've ->also<- seen them refered to as "quotient field"; it's just that
in my experience "field of quotients" is much more common.
Aleks Kleyn
such statement like
canonical map onto factor group
Thank everybody.
--
Aleks Kleyn
http://www.geocities.com/aleks_kleyn
"Aleks Kleyn" <Aleks...@MailAps.org> wrote in message
news:49506059$0$20282$607e...@cv.net...
Bill Dubuque
>In article <6rd333...@mid.individual.net>,
>Axel Vogt <&nor...@axelvogt.de> wrote:
>>Arturo Magidin wrote:
>>>Axel Vogt <&nor...@axelvogt.de> wrote:
>>>>
>>>> However the names are not consistent: in Algebra that is sometimes
>>>> often called a factor group or ring or what is under discussion,
>>>> while by quotient ring or quotient field something like Rationals
>>>> is meant: passing from integers to fractions (localization is a
>>>> different name as well).
>>>
>>> I usually see them refered to as "field of quotients"
>>> or "ring of quotients", instead.
>>
>> May be my language is to Germish ... (we call it "Quotientenring" etc).
>
> I've ->also<- seen them refered to as "quotient field"; it's just
> that in my experience "field of quotients" is much more common.
My experience is the opposite - confirmed via a Google Books
search: 1076 books contain "quotient field" [1]
but only 675 books contain "field of quotients" [2]
Moreover, "field of quotients" appears to be dying out
since the ratio increases to 4:1 (400:111) over the
past 5 years [3],[4], and to 23:1 (138:6) in 2008.
--Bill Dubuque
[1] http://google.com/books?q=quotient-field
[2] http://google.com/books?q=field-of-quotients
[3] http://google.com/books?q=quotient-field+date:2003-2008
[2] http://google.com/books?q=field-of-quotients+date:2003-2008
Mariano Suárez-Alvarez
> Arturo Magidin <magi...@math.berkeley.edu> wrote:
> >In article <6rd333Fncb...@mid.individual.net>,
>
> >>>> However the names are not consistent: in Algebra that is sometimes
> >>>> often called a factor group or ring or what is under discussion,
> >>>> while by quotient ring or quotient field something like Rationals
> >>>> is meant: passing from integers to fractions (localization is a
> >>>> different name as well).
>
> >>> I usually see them refered to as "field of quotients"
> >>> or "ring of quotients", instead.
>
> >> May be my language is to Germish ... (we call it "Quotientenring" etc).
>
> > I've ->also<- seen them refered to as "quotient field"; it's just
> > that in my experience "field of quotients" is much more common.
>
> My experience is the opposite - confirmed via a Google Books
> search: 1076 books contain "quotient field" [1]
> but only 675 books contain "field of quotients" [2]
>
> Moreover, "field of quotients" appears to be dying out
> since the ratio increases to 4:1 (400:111) over the
> past 5 years [3],[4], and to 23:1 (138:6) in 2008.
>
> --Bill Dubuque
>
> [1]http://google.com/books?q=quotient-field
> [2]http://google.com/books?q=field-of-quotients
> [3]http://google.com/books?q=quotient-field+date:2003-2008
> [2]http://google.com/books?q=field-of-quotients+date:2003-2008
I tend to hear and read "field of fractions".
-- m
Bill Dubuque
>Arturo Magidin <mag...@math.berkeley.edu> wrote:
>>In article <6rd333...@mid.individual.net>,
>>Axel Vogt <&nor...@axelvogt.de> wrote:
>>>Arturo Magidin wrote:
>>>>Axel Vogt <&nor...@axelvogt.de> wrote:
>>>>>
>>>>> However the names are not consistent: in Algebra that is sometimes
>>>>> often called a factor group or ring or what is under discussion,
>>>>> while by quotient ring or quotient field something like Rationals
>>>>> is meant: passing from integers to fractions (localization is a
>>>>> different name as well).
>>>>
>>>> I usually see them refered to as "field of quotients"
>>>> or "ring of quotients", instead.
>>>
>>> May be my language is to Germish ... (we call it "Quotientenring" etc).
>>
>> I've ->also<- seen them refered to as "quotient field"; it's just
>> that in my experience "field of quotients" is much more common.
>
> My experience is the opposite - confirmed via a Google Books
> search: 1076 books contain "quotient field" [1]
> but only 675 books contain "field of quotients" [2]
>
> Moreover, "field of quotients" appears to be dying out
> since the ratio increases to 4:1 (400:111) over the
> past 5 years [3],[4], and to 23:1 (138:6) in 2008.
This strong preference for "quotient field" vs. "field of quotients",
esp. recently, is also confirmed by other math literature searches:
Zentralblatt Math
yields 8:1 (948:112) forever
12:1 (194:16) for 2003-2008 bi:"quotient field" & py:2003-2008
15:1 (15:1) for 2008
AMS Math Reviews
yields 6:1 (2151:364) forever
11:1 (258:24) for 2003-2008
8:1 (8:1) for 2008
Google Scholar
yields 6:1 (7810:1260) forever
9:1 (1880:221) for 2003-2008
13:1 (257:20) for 2008
Google Books (as above)
yields 2:1 (2076:675) forever
4:1 (400:111) for 2003-2008
23:1 (138:6) for 2008
arxiv: 13:1 (3060:244) forever, via Google search.
Thus I'm surprised that Arturo and Mariano's experience is the opposite.
Perhaps there's a local preference opposite to the above global preference
in certain subfields of math. Or perhaps their experience is influenced
by texts in their first (non-English?) language.
G. A. Edgar
7790 quotient field
1260 field of quotients
4930 field of fractions
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Ken Pledger
"Aleks Kleyn" <Aleks...@MailAps.org> wrote:
> when we have equivalence on the set we call set of classes of equivalence
> quotient set. it may be quotient group or quotient ring. I expected to call
> map from set to quotient set quotient map. However when I did research on
> google I see that all references on quotient map mention topology. my
> question is how we call such map when we do not use topology?
There seems to be no standard name. However, such a mapping from
a group to a quotient group is called a "natural homomorphism"; so by
analogy I use "natural mapping" for the more general version which you
describe.
Ken Pledger.