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)
 * I was trying to organize things, not to pretend I know more than I do. If you don't think it's valuable to have a morphism category, then fine. But I feel rather strongly that an isomorphism category, including not only algebraic but also topological, etc., ones is quite reasonable. Certainly PM's move of making "isomorphism" a subcategory of abstract algebra and removing the abstract algebra categorization from pages defining isomorphisms goes the wrong way. --Dfeuer (talk) 17:24, 18 January 2013 (UTC)
 * Agreed on the part of isos - if I'm not mistaken, I didn't cancel your edits on those. In hindsight the current categorisation of "isomorphism" was unfortunate - OTOH, it is an understandable move for people not knowing about CT. Your efforts in organizing the site are appreciated; I just think that there are ample other areas you're (reasonably) knowledgeable about yearning for a similar treatment. Attacking those will be easier for everyone and prevents needless shouting and annoyance. --Lord_Farin (talk) 17:48, 18 January 2013 (UTC)


 * I would argue against the "certainly" above because it's not certainty at all, it's merely your opinion, which I think you'll find not everybody else shares.


 * There is limited use in creating a subcategory based on nothing else but coincidence of language. True, the concept of "isomorphism" in general means the same thing wherever it is encountered: it means "same form". But whereas there is an "isomorphism" in abstract algebra, there is an "isomorphism" in graph theory and whatever else you can dream up, '''but that in no way makes it sensible to lump everything which relates to "isomorphism" in the same category, that is undeniably stupid.


 * If you do feel the burning need to create a category called "isomorphism" then make it a subcategory of "Definitions", rather than a subcategory of everything it is blatantly not a subcategory of. (If it's a subcategory of abstract algebraical homomorphisms, it is not a subcategory of Graph Theory, and so on.


 * Categories evolve. It is a questionable approach to generate a category just because there is a definition for it. Instead, you look at an existing category and see that there are many results inside it which all relate to the same concept (like e.g. Union in the category of Set Theory, or Conjunction in the category of Propositional Logic. When this happens, it makes sense to break the category down and thence to reduce the number of results in any given category.


 * So please lay off your organising of the categories until you've created some useful material to put in them. --prime mover (talk) 18:31, 18 January 2013 (UTC)

That is more or less what I tried to suggest, or at least has significant overlap. Also, please stop quibbling about while you both make yourselves liable to accusation of hypocrisy by criticizing one another's language, and in the process use words of similar intonation. Just stop it; I'm done with it already. Point made, case buried; let's proceed to a constructive and welcoming environment. It's not hard; at worst it costs some more effort. IMHO it's worth it because PW will stay a more pleasant place for it. --Lord_Farin (talk) 22:01, 18 January 2013 (UTC)


 * I will attempt not to react to hostile language by more hostility, and also try not to react to what I perceive as ill-thought-out and unnecessary fiddling with perfectly adequate stuff with further hostile language. In return it would be nice to not have to continually have to keep checking that stuff. If something is the way it is, there is often a reason for it (whether just because it evolved like it) and unless something is actually wrong it is usually completely unnecessary to rewrite it. --prime mover (talk) 22:15, 18 January 2013 (UTC)

Closedness
It doesn't make sense to say
 * $\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$

if $S$ is not closed. --barto (talk) (contribs) 19:13, 13 January 2018 (EST)


 * Definition:Algebraic Structure has been replaced by Definition:Magma throughout this page. --prime mover (talk) 15:29, 25 October 2018 (EDT)