Talk:Homomorphism

message: Consider the map f:(Z,.)--->(Z,.) (Z: set of integers , . : multiplication) between monoids given by f(x)=0. This is a homomorphism since f(x.y) = f(x).f(y) but identity is not preserved since f(1) = 0. The statement that identity is preservd seems wrong to me.
--Shahab 12:14, 1 September 2006 (UTC)
replies: text: Doesn't a homomorphism need to preserve identity? The definition of monoid homomorphisms in the article on monoids requires this; the same goes for the definition of functors (and it says in the article about functors that one can think of them as homomorphisms between categories). 95.88.237.20 (talk) 17:36, 12 February 2013 (UTC)
replies: text: In any variety (or a more general class) of algebras, homomorphisms are required to preserve all operations in the signature of the variety. The signature of monoids includes the constant for identity, hence monoid homomorphisms have to preserve identity elements. The signature of, say, semigroups does not include identity, hence semigroup homomorphisms do not have to preserve identity (even if the two semigroups happen to be monoids). Shahab’s map is a semigroup homomorphism, but not a monoid homomorphism.—Emil J. 17:55, 12 February 2013 (UTC)
replies: text: Indeed, this is quite a nice illustration of the necessity for care over the codomain as well as the domain of a map.  The map x → 0 is a monoid homomorphism from Z,× to the monoid {0},× but not to the monoid Z,×.  Of course, {0},× is a monoid, and a subsemigroup of Z,×, but not a submonoid of Z,× since the identity of {0},× is not the identity of Z,×.  Deltahedron (talk) 20:33, 12 February 2013 (UTC)
replies: text: The wording under the Definition section is confusing in this respect. It suggests on first reading, for example, that a group homomorphism only applies between two groups, and not between say two rings (though on further thinking, of course every ring is a group under addition). EmilJ and Deltahedron make it clear here; we could do with this clarity in the article. I'll try to reword the definition to make it clearer that the structure that is preserved by a homomorphism is nominated (and only implied as a default), and not specified by the objects. In particular, it should be clear that a "monoid homomorphism" is a homomorphism that preserves all structure that is required by the definition of monoids. My attempt may break subtler aspects, please feel free to rescue the correction. — Quondum (talk) 08:31, 24 September 2013 (UTC)
thread_title: Identity is not always preserved

message: This page should follow the example of the Group page and have formal definitions as well as examples because definitions may be helpful, but understanding won't come with a plethora of unorganized information. Therazzz (talk) 22:20, 23 May 2014 (UTC)
thread_title: Poorly Explained

message: The statement that "in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism" needs qualification. The definition in terms of injective and surjective maps makes this true, but the categorical definition in terms of left and right cancellation does not.  For example, the injection i : Z → Q is a ring monomorphism and epimorphism in the latter sense, but not an isomorphism.  See, for example, .  Deltahedron (talk) 18:14, 11 July 2014 (UTC)
replies: text: I agree. I have tagged the disputed definitions. It seems, from the comments in the text that previously these definitions were explicitly restricted to modules and that the present version is due to a troll that believed that abstract algebra is reduced to module theory. I have not yet got the time to check the history. D.Lazard (talk) 19:49, 11 July 2014 (UTC)

text: In fact, the wrong definitions have been introduced by this edit . Should we restore the previous version of the section or rewrite the section to have a correct version? D.Lazard (talk) 22:10, 11 July 2014 (UTC)

text: Your edit assert implicitly that "monomorphism", "epimorphism", ... have two non-equivalent meanings in mathematics. This is this implicit assertion which is wrong. Saying, as you did that in abstract algebra an epimorphism is always surjective, implies that this is true for rings or that the study of rings do not belong to abstract algebra; both are wrong. I remove the word of troll. Your edit is only original research that is highly confusing for the reader and has been inserted with a summary saying that the previous correct version was confusing. D.Lazard (talk) 23:19, 11 July 2014 (UTC)
thread_title:  Isomorphism if and only if both a monomorphism and an epimorphism 

message: I have completely rewritten, and expanded the section "Types", now renamed "Special homomorphisms". In my opinion, this solves the questions discussed in the previous section. Nevertheless some further work would be useful, in particular for improving references and settling the points that I have listed at the end of the preceding section. D.Lazard (talk) 14:26, 6 January 2017 (UTC)
thread_title:  New section "Special homomorphisms" 

message: Some visitors might be looking for a quick understanding of these concepts in the common case of sets, groups, etc. For beginners especially, would including this figure be helpful? Book here. Also might address the removed figure mentioned above. Adam Marsh (talk) 23:53, 26 January 2018 (UTC)
replies: text: This figure is not convenient for this article, because it is wrong in this context. In fact, a large part of this article is devoted to showing that monomorphism and epimorphism are not synonymous of injective and surjective (respectively). Moreover, this figure introduces a confusion between "function" and "homomorphism": looking at the figure, a beginner could think that the hyperbolic sine is an automorphism of the reals, which is blatantly wrong. D.Lazard (talk) 10:38, 27 January 2018 (UTC)
replies: text: Ok. I suppose an implicit question I had was whether the category theoretic definitions might be better placed either under a separate subheading or under morphism as opposed to here. I would guess that at least some readers landing here might be confused by the introduction of the more general definitions, and could be looking for the algebraic definitions (injective / surjective). Not sure I understand the comment about functions, the assumption was that the figure would be placed in context, which is clearly that of homomorphisms, not functions. Adam Marsh (talk) 04:00, 1 February 2018 (UTC)
thread_title:  Figure 

message: In January 2017, I edited the article for clarifying the relationships monomorphism–injection and epimorphism–surjection. Reading my edits again, I remark that I have left open the question of (non-necessarily commutative) groups: is every epimorphism (categorical meaning) of groups a surjection? More precisely, do there exist a right-cancelable group homomorphism that is not surjective? Does anybody know an answer to this question? D.Lazard (talk) 10:31, 7 May 2018 (UTC)
replies: text: I have found a reference, and I'll edit the article accordingly. D.Lazard (talk) 13:14, 7 May 2018 (UTC)
thread_title:  Epimorphisms of groups 

message: I think it would be good if we could add a section on the history of the homomorphism concept. In particular: Who invented the concept? Which motivation did he have? For which specific purposes is the concept used today? Zaunlen (talk) 23:48, 7 November 2019 (UTC)
thread_title:  Section on the history and application of the homomorphism concept?