Date: Aug 24, 2006 8:36 AM
Author: Ben
Subject: Re: [CoT]: Homomorphism and Isomorphism


Arturo Magidin wrote:
> Cc:
>
> In article <1156316940.028777.289440@m73g2000cwd.googlegroups.com>,
> Ben <partho.choudhury@gmail.com> wrote:
>
> >According to Lang:
> >
> >If G and G' are monoids (with a law of composition/closed binary
> >operator *), then G is monoid-homomorphic if there exists a mapping
> >f:G->G' such that f(x*y) = f(x)*f(y) for all x,y in G (the domain of
> >f). If G is a group, then it is a group-homomorphism.
>
> I have Lang's "Algebra" in front of me (3rd Edition). I suspect you
> are talking about what is in pp. 10. Lang does NOT use the term
> "monoid homomorphic", which you seem to be trying to use in the sense
> of a binary relation among monoids. Rather, Lang says:
>
>   "Let G, G' be monoids. A ->monoid-homomorphism<- [boldface in the
>   original] (or simply ->homomorphism<-) of G into G' is a mapping
>   f:G->G' such that f(xy)=f(x)f(y) for all x,y in G, and mapping the
>   unit element of G into that of G'. If G,G' are groups, a
>   ->group-homomorphism<- of G into G' is simply a monoid-homomorphism.
>
>   "We sometimes says: 'Let f:G->G' be a group-homomorphism' to mean:
>   'Let G,G' be groups, and let f be a homomorphism from G into G'.'"
>
> As you can see, the term "monoid-homomorphic" does not seem to
> appear.

Mea Culpa: I guess "monoid-homomorphic" monoids or "group-homomorphic"
groups G aint what I was meaning.....rather it is the mapping f which
is being characterized as homomorphic ("A ->monoid-homomorphism<-
[boldface in the original] (or simply ->homomorphism<-) of G into G' is
a mapping f:G->G' such that .....").

>
> >However, LeVeque extends the definition to include the fact that
> >f:G->G' is also a surjection as a necessary/minimum crietrion to define
> >a homomorphism.
>
> Well, I have LeVeque in front of me as well. LeVeque does not speak
> about monoids, so I assume you mean groups or rings.

Yes, thats right....he is targeting rings.....

> The reference I
> find is (coincidentally also) in pp. 10 is about rings. It reads
> (changing the greek lower-case phi into an f, and representing
> boldface in the original with "->" and "<-" as above):
>
>    "Suppose it is possible find a map* f such that to every element a
>    in S there corresponds a unique image f(a) in S', and for which
>
>          every element of S' is the image of some element of S;
>          f preserves addition: if a+b = c then f(a)+f(b)=f(c);
>          f preserves multiplication: if ab=c, then f(a)f(b)=f(c).
>
>    "Then f is called a ->homomorphism<- of S ONTO S' [emphasis ADDED],
>    and if the correspondence is 1-1 (meanting that also for each a' in
>    S' there is a unique a in S such that f(a)=a'), then f is called an
>    ->isomorphism<- and S and S' are said to be isomorphic."
>
> The footnote reads in part:
>
>    "* A map from a set A ->into<- [italics in the original] a set B is
>    simply a function defined on A, taking values in B. The map is said
>    to be ->onto<- [italics in the original] if every b in B is the
>    image (function-value) of some a in A."
>
> You will note that LeVeque is defining an ONTO homomorphism, not
> merely a homomorphism, whereas Lang is defining a general homomorphism
> INTO the target. This is the difference.

Ok....I guess I missed the nuance on the specific reference to onto
homomorphism (Even then, I think it does sound weird to directly jump
to surjective homomorphisms even before speaking about homomorphisms in
general!!!). But then, I guess this hidden nuance does clear the
cobwebs!!!


>
> >LeVeque gors on to define a group-isomorphism independently as the case
> >when f is an injection. Nowhere is it implied that for an isomorphism
> >to occur, it also has to be homomorphic in the first case.
>
> No, you are misreading it. LeVeque's phrasing clearly indicates that
> the function f is expected to have satisfied the conditions labeled
> (H) in addition to the new condition being added of being a bijection.
>

gotcha!!!!

> >However, Lang insists that isomorphism is possible/defined only if it
> >is already a homomorphism (Homomorphic groups are also isomorphic
> >if.....).
>
> LeVeque requires this as well. You are misreading it.
>
> >Upto this point the defnitions by Lang and LeVeque have only one point
> >of conflict or paradox:
> >
> >Is homomorphism a necessary condition for isomorphism to occur, as
> >suggested by Lang.
>
> Lang does not "suggest" it. He says it plainly: "Let G,G' be
> monoids. A homomorhpism f:G->G' is called an ->isomorphism<- if there
> exists a homomorphism g:G'->G such that fog and gof are the identity
> mappings." LeVeque ALSO requires the isomorphism to be a homomorphism,
> as should be clear from his phrasing: the clause on isomorphisms is
> not even a separate sentence, but is simply a dependent clause on the
> sentence defining (onto) homorphisms.
>

Yes, I get it.....I missed the "also" inside the paranthesis in
LeVeque's note on isomorphism the first time around!!!

> -- 
>