Definition talk:Homomorphism (Abstract Algebra)

Is it modern convention to say something like $\varphi(a)=a~\forall ~a\in F$? If so I think I'm going to go and kill myself.

I learned to write it $\forall a\in F: \varphi(a)=a$ so as to emphasise the scope of the $\forall$. Otherwise (in more complicated constructions) the precise meaning of a nested sequence of $\forall, \exists$ etc. can become ambiguous. If the fundamentals of logic are no longer considered part of mathematical rigor then it's a disappointing development. --Matt Westwood 07:33, 20 December 2008 (UTC)

Ummm, "statement for all elements" and "for all elements, statement" seem the same to me, though were the statement to include "there exists" you wouldn't catch me dead commuting the two. Such is the difference between convergence and uniform convergence, continuity and uniform continuity, etc. What I've done here is fine given the context of the statement. --Grambottle 16:15, 20 December 2008 (UTC)

Yeah but it's not so obvious what it means. --Matt Westwood 19:47, 20 December 2008 (UTC)

I agree that "for all elements, statement" is probably better. If I use the other, I try always put in some space (\;\;\; seems tp be a reasonable amount of space in mathtext) or a comma. --Cynic-(talk) 01:49, 21 December 2008 (UTC)

Good touch-up on the F-homomorphism part. I just switched the notation from $I_F$ to $1_F$ as $1_F$ is much more common. This is one of those instances where I hate the convention and use it reluctantly on this site (also, will be using a set $I_F(E,\Omega)$ later on). I would be much much happier if convention was $\iota_F$ instead of $1_F$ but such is not the case. --Grambottle 15:47, 21 December 2008 (UTC)

I hate $1_F$ as much as you do, I don't care how much more common it is (common as in peasanty), $1_F$ is also often used as the unity of the field $F$ (which seriously confused me in this definition till I figured it out). (Another reason for defining, or referring to a definition, everything and I mean everything that we use). The definition for Identity Mapping has already been defined as $I_F$. Why should we be slaves to conventions that suck? Who runs the world anyway, us or those farty old buffers who have the temerity to call themselves "professors"? --Matt Westwood 16:28, 21 December 2008 (UTC)

Morphisms
I believe homomorphisms are morphisms. Certainly Wikipedia says so, and Quora. I know these are not definitive sources, but I have also heard such elsewhere. --Dfeuer (talk) 06:15, 18 January 2013 (UTC)


 * Learn some mathematics from a proper source. We are not in the business of providing another version of Shitipedia, which is undeniably the completely wrong source to get any information (you might as well learn quantum mechanics from the National Enquirer) and using Quora as a source for any sort of question on anything scholastic is so stupidly risible that it invalidates your pretension to rise above the level of subhuman. --prime mover (talk) 06:43, 18 January 2013 (UTC)


 * Again, PM, get it together and soften your tone. As for the fact that homomorphisms are morphisms: it's true. It's also true that $u \le v$ is a morphism in a poset category. A mapping is a morphism in the Definition:Category of Sets. Morphisms are everywhere. It's not helpful to add virtually all pages defining a mapping to the category Category:Definitions/Morphisms. There admittedly is currently a lack of application of CT to other areas covered on PW, but a more structured approach approved of by most or all of the maintenance team would be the way to go. I hope you see the distinction.
 * In general, maintenance in categories you wouldn't say to "master" to some extent (at the very least, to have consulted/worked through one or more source works) is not a good idea. Your work is appreciated, but currently it's best for all of us that you leave CT alone and work on areas we're convinced you are proficient in (or where the first few of your edits make us conclude you are). Thanks. --Lord_Farin (talk) 08:50, 18 January 2013 (UTC)