Talk:Homomorphism

Identity is not always preserved

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)

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)

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)

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)

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)

Poorly Explained

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)

Isomorphism if and only if both a monomorphism and an epimorphism

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, Dascalescu, Sorin; Nastasescu, Constantin; Raianu, Serban (2000). Hopf Algebra: An Introduction. CRC Press. p. 363. ISBN 0824704819.. Deltahedron (talk) 18:14, 11 July 2014 (UTC)

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)

In fact, the wrong definitions have been introduced by this edit [1]. 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)

Sorry, but I don't understand your point. First, a definition can't be wrong, but unusual at worst. Second, the case of category theory is handled in an own subsection where the inclusion Z-->Q is even used as example. The text there appears to say exactly the same as Deltahedron. So, where is the problem? - Jochen Burghardt (talk) 22:28, 11 July 2014 (UTC) Btw: It was my edit, and I don't think I deserve to be called a troll. - Jochen Burghardt (talk) 22:32, 11 July 2014 (UTC)

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)

Please have a look at the version immediately before my edit. It says:

"... However, the definitions in category theory are somewhat technical. In the important special case of module homomorphisms, and for some other classes of homomorphisms, there are much simpler descriptions, as follows: ... An epimorphism (sometimes called a cover) is a surjective homomorphism. ... For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; the first three descriptions do not. ..."

So the non-equivalent meanings of (e.g.) epimorphism were present already before my edit, as was the definition of epimorphism to be a surjective homomorphism (which happens to be the definition I am familar with).

If you look at Epimorphism, it gives the right-cancellation definition in the context of category theory, and notes a deviating definition (viz. being surjective) by "Many authors in abstract algebra and universal algebra". This is consistent with the result of my edit.

You are right that we should supply a citation for the non-category wording. I can offer the following one: Garrett Birkhoff, "Lattice Theory", Providence, Am. Math. Soc., Vol.25 of Colloquium Publications, 1967, Sect.VI.3, p.134 (although the overall issue of the book are lattices, Sect.VI is on "Universal Algebra", and VI.3 is on "Morphisms"). Birkhoff says:

A morphism of A onto B is called an epimorphism; a one-one morphism is called a monomorphism. An isomorphism of A with B is defined to be a one-one morphism from A onto B; an automorphism is an isomorphism of an algebra with itself, and an endomorphism is a morphism of an algebra into itself.

- Jochen Burghardt (talk) 07:09, 12 July 2014 (UTC)

Inaccuracy (remainder)

This has already been pointed out in 2008 at Talk:Homomorphism/Archive 1#Incorrect definitions of special types of homomorphisms but, as a remainder, I just want to point out the definitions of an isomorphism and an epimorphisms given in the article are ... wrong. The simplest way to see it to look them at some algebra textbooks. But more conceptually the definitions should not refer to the underlying set; this allows in particular a smooth transition from sets to some more structured objects; for instance, one can talk about a homomorphism between sheaves of modules. -- Taku (talk) 20:28, 30 November 2016 (UTC)

I agree. Moreover, the "equivalently" assertion for epimorphisms is wrong, even for modules. I'll try to correct the section, if you do not do it before me. D.Lazard (talk) 09:20, 1 December 2016 (UTC)

Being not a native English speaker, the only relevant textbook I have at hand is Birkhoff.1967, cited above, where D.Lazard and I had almost the same diccsussion in 2014. As a result, I then tried to fix section Homomorphism#Types according to Birkhoff's definitions. If they are unusual today, we need other citations supporting that - nobody gave some since 2014.

In any case, I suggest to keep the remarks (and also the proofs) on relations between the category theory and the (i.e. Birkhoff's) abstract algebra notions; this is possible even if "iso", "epi", "mono" is used only for the former, and only "bijective", "onto", "one-one" for the latter. Moreover, the article's current version uses terms like "generally", "often" etc. in places where definitions are expected, and we should avoid that.

If there isn't sufficient evidence that Birkhoff's terminology isn't used any longer today, I suggest just to revert D.Lazard's edits. The ring homomorphism example was already present in the 15-Jul-2016 version, and the module homomorphism example can easily be added to the same paragraph; a more explicit warning about deviating notions in abtract algebra and category theory then should be added there, too. - Jochen Burghardt (talk) 13:43, 6 December 2016 (UTC)

To editor Jochen Burghardt: In your above citation, it is not said what are A and B. The sentence is undoubtful correct if they are sets, modules or groups. If Birkhoff allows more general categories for them, this definition is clearly in contradiction with the present standard of mathematics. One must keep in mind that when he wrote his book, the language of categories was not yet popularized in all mathematics. Therefore, this book is not reliable for the modern terminology. Moreover, for some authors the term "homomorphism" is used only for algebraic structures, while, for others, this is a synonymous of "morphism". As we cannot know the background of the readers, we must have a formulation which is correct, whichever is this background. This is the reason for giving definitions that are correct in any case, and then explain that "in general" isomorphism = bijective, monomorphism = injective and epimorphism = surjective, with a short explanation of what "in general" means in each case. This is what I have tried to do with my edit. In any case, I am strongly against reverting to previous version, as 1/ it may be confusing for people who do not care that the given definition do not apply for rings and non-algebraic structures. 2/ it contains the wrong assertions that every epimorphism is a split epimorphism and that every monomorphism is a split monomorphism (this is true for vector spaces, but not for groups an modules). D.Lazard (talk) 15:32, 6 December 2016 (UTC)

@D.Lazard: So I understand you havn't an Englush textbook available either? Possibly, TakuyaMurata (Taku) can help with a few citations of algebra textbooks? - Jochen Burghardt (talk) 18:59, 6 December 2016 (UTC)

This is not a problem of language nor a problem of availability of textbooks. The point is that this article is about a general definition of homomorphism, and that most definitions given in textbooks are about specific kinds of homomorphisms, commonly, homomorphisms of groups, vector spaces and modules. As all definitions are equivalent in these cases, these textbooks are not useful for sourcing a general definition. As far as I know, there exist three theories providing a general definition of isomorphism, epimorphism and monomorphism. The first one is Bourbaki's definition of algebraic structures. It has never been accepted by the mathematicians community, nor really used by other authors, and thus cannot be used as a source. The second one is the theory of universal algebra. Although being the object of some active research, universal algebra has rarely been considered in the main stream of mathematics, and its terminology is not widely accepted; a witness of this is that the term "variety" has a completely different meaning in universal algebra and in the main stream of mathematics (algebraic variety). Thus universal algebra, and Birkhoff book may not be used as a source for the terminology of this article. The last general definition of homorphisms that I know is that of category theory. It is presently used in many fields of mathematics, from number theory (Wiles's proof of Fermat's Last Theorem) to topology (via homological algebra). It follows that the only terminology that make sense for this article is that of category theory. Any other terminology would be confusing for many readers. However, I agree that if some sources use a terminology which is not compatible, this must be added to the article in a sentence like "Some author use the following different definition ...". D.Lazard (talk) 10:55, 7 December 2016 (UTC)

@D.Lazard: Ok, now I understand your point of view. What about the following suggestion:

We use adjectives to name the (homo)morphism properties, as far as possible, i.e. "monic", "epic" (this is what S. Mac Lane uses in the cited book, p.19), "injective", "surjective", as there is no confusion about them. We discuss the differences and relations between them (such as "surjective implies epic, but nor vice versa: e.g. ring homomorphism Z-->Q"). We comment on different terminologies and warn about confusions, using the nouns ("monomorphism", "epimorphism") for that. We could e.g. write "In category theory and some abstract algebra books, "monomorphism" and "epimorphism" refers to a monic and epic morphism, respectively,^{(some citations)} while other abstract algebra books use these notions to refer to an injective and surjective homomorphism, respectively.^{Birkhoff.1967, Stanley.Sankappanavar.2012:29,43}" We could extend the former image to include all kinds of morphisms discussed, by adding "inj" and "surj" and extending the areas for "mon" and "epi" (then meaning "monic" and "epic") appropriately. And of course we should warn that "injective" and "surjective" don't make sense in category theory (if I remember right - ?).

Our overall goal should be that a by reading the section under consideration, the confusion (that caused our repeated discussions here) can be overcome. - Jochen Burghardt (talk) 13:45, 7 December 2016 (UTC)

I agree with your overall goal, but not with the suggested use of epic and monic. In fact, making a difference between "being an epimorphism" and "being epic" is WP:OR, and cannot be introduced in Wikipedia. IMO, the problem with the present state of section "Types" is that the bulleted list implies a very short description, and that this description, although correct (in my opinion) it too short for being clearly understood by a non-expert reader. Therefore, I suggest to merge the subsection in the section and to split the section in subsections, one for each item of the bulleted list. This would allow to clarify everything, to provide as many examples as needed, and so on. D.Lazard (talk) 15:16, 7 December 2016 (UTC)

No principal objection, but I see a problem of presentation: in each subsection we'd repeat something like "some authors call it '...', and some call it '...'", and the information was lost that the same authors that call an injective homomorphism "monomorphism" also use "epimorphism" for a surjective one. It is easier to present that information if we first split into subsections like e.g. "category theory approaches" and e.g. "set theory approaches". The suffix "approaches" is intended to reflect that the notions inspired by category theory are nowadays used beyond that field, in particular in abstract algebra (this was Taku's and your point above, if I understand you right); maybe you have better title suggestions to express this intention. - Jochen Burghardt (talk) 20:12, 7 December 2016 (UTC)

The presentation of this materials (epimorphisms, etc) is still not optimal. I have come to wonder: do we really need to give definitions of epimorphisms and such for some hypothetical algebraic object? Since, as I want to stress, an algebraic object need not have an underlying set, the set-theoretic definition cannot be used for that purpose (this was a point I wanted to make). It seems a solution among contemporary textbook authors (Dummit-Foote, Eisenbud) is simply to avoid such problematic terms like an epimorphism and use more down-to-earth unambiguous terms like surjective homomorphisms or injective morphisms. They then discuss category theory and mentions, for instance, a subjective module homomorphism is precisely an epimorphism in the category of modules. Put in another way, do people use terms like epimorphisms aside from the category-theoretic meaning? Yes, some do but that's limited to historical sources. (There is no neee to use historical terms in Wikipedia.) -- Taku (talk) 03:50, 11 December 2016 (UTC)

@TakuyaMurata: I think it is a good idea to use unambiguous names in definitions, and that "surjective", "injective" (from your citations), and "left" and "right cancellable" (from Mac Lane's "Categories for the Working Mathematician") is a good choice. Based on these definitions, we could then say that "epimorphism" and "monomorphism" in earlier days meant "surjective" and "injective" homomorphism ("set-theoretically inspired meaning"), but has got a second ("category-theoretically inspired") meaning with the raise of category theory, viz. "left" and "right cancellable" (homo)morphism, respectively. If I understand D.Lazard and you right, it is your point that the latter meaning is more elegant and therefore gets more and more widespread in abstract algebra, even beyond pure category theory; this should be made clear in the article (provided there is evidence for it).

Concerning your second question, please note that up to now the article only has citations that support the set-theoretic meanings (viz. Birkhoff.1967, which is admittedly historical, and Burris.Sankappanavar.2012, which is undoubtedly not), apart from the pure category-theory book Saunders.1971 (which moreover isn't much more contemporary than Birkhoff.1967). So, if you can provide citations that support your point, please do! In particular, could you please give full references of the "Dummit-Foote, Eisenbud" textbook(s)? Since the set-theory meaning is still (widely, I believe, based on the current distribution of citations) in use, the article should mention it; from your point of view, readers should be warned of the unelegent meaning still found in the literature. - Jochen Burghardt (talk) 09:56, 11 December 2016 (UTC)

To editor Jochen Burghardt: I agree with your suggestion of using "surjective", "injective", "left" and "right cancelable". However, we have also to discuss all the names, their variants, and their relationships. As this discussion cannot be done in a single line, this is the main reason for splitting the section into one subsection by concept.

IMO, most of this discussion results of an ambiguity in the first line of the article: the definition of "algebraic structure", given in the linked article, is not recalled here. Thus it is unclear that, here, an algebraic structure has a underlying set, and only one, while, for many mathematicians, "algebraic structure" is either not clearly defined or is much larger concept. Thus, we have first to state clearly that in this article, an algebraic structure has one and only one underlying set.

About "isomorphisms": It is a (rather easy) theorem that, for these algebraic structures, bijective is equivalent with having an inverse. This must be presented as a theorem, and possibly proved.

About "monomorphisms": It is also a theorem that, for algebraic structures for which all operations are everywhere defined (varieties of universal algebra), "monomorphism" is equivalent with "injective". I do not know of an explicit source, but the proof may proceed as follows. Given a variety of algebraic structure, a free object over x is an algebraic structure of the variety that has a distinguished element x and has the property that for any element a of any algebraic structure of the variety, there exists an homomorphism from the free object to the algebraic structure that maps x to a. The existence of a free object over x implies, almost immediately that every monomorphism is injective, and thus that "monomorphism" is equivalent with "injective". The free object over x, which is unique up to an isomorphism is, for sets, any singleton, for semigroups, the semigroup of positive integers, for monoids, the monoid of nonnegative integers, for groups, the additive group of the integers, for rings, the polynomial ring in x with integer coefficients, and for rngs, the subset of the preceding consisting of the polynomials without constant terms. For vector spaces and modules, the free object over x is the vector space or free module which has {x} as a base. In general, the free object over x may be constructed by considering the formal application of the operations of the structure to x. This cannot be done if some operations are not everywhere defined, such as the multiplicative inverse in the case of field.

It would be useful to find an explicit reference for this result (it is certainly not new). If we do not find such a reference, I suggest to include the proof in the article, because this would avoid the vague "in may cases" D.Lazard (talk) 14:10, 11 December 2016 (UTC)

(Sorry, by "Dummit-Foote, I meant their "abstract algebra" and by "Eisenbud", "communicative algebra view toward algebraic geometry". I have deliberately avoided old, foreign or other less popular textbooks. The fact they avoid the term "epimorphism" is a good indication that we should do the same in Wikipedia.)

I think the problem of the confusing state of the sectio stems from our attempt to give general definitions of isomorphism, epimorphism, etc. for general algebraic structures. Since the latter is not well-defined, that's not possible and we shouldn't try that. An "epimorphism" has specific meaning in the category theory but some authors, especially in historical sources, use it also mean a surjective homomorphism. We should stick to such a commentary on the terminology and not try to give a mathematical exposition.

As for an isomorphism, before defining it, if we like, we can explain it along the line: if a homomorphism is a bijection between the underlying sets (note algebraic structure is not a set!!), then the inverse function is necessarily a homomorphism. And, as D.Lazard suggested, with a proof, since the proof is illuminating. --- Taku (talk) 21:54, 12 December 2016 (UTC)

@TakuyaMurata: Could you please give a full citation of your books? Apparently, they are the only references up to now that could support your and D.Lazard's claim that category theory notions are commonly used in abstract algebra, so we badly need them. I've been searching in our library, but found only these books, all sticking to set-based notions:

• Wolfgang Wechsler (1992). Wilfried Brauer and Grzegorz Rozenberg and Arto Salomaa (ed.). Universal Algebra for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Vol. 25. Berlin: Springer. ISBN 3-540-54280-9. — defines injection, surjection (p.3); unsorted term algebra, unsorted homomorphism, endomorphism (p.17); many-sorted algebra, many-sorted homomorphism (p.246) — the latter may be a good source for algebras with more than one carrier set
• David Gries and Fred B. Schneider (1994). A Logical Approach to Discrete Math. Texts and Monographs in Computer Science. Berlin: Springer. ISBN 3-540-94115-0. — defines algebra (p.387, single carrier-set only), homomorphsim (p.395, for algebras with operations of an arity up to 2 only), surjective=onto function and injective=one-to-one function in general (p.282)
• Ethan D. Bloch (2000). Proofs and Fundamentals —- A First Course in Abstract Mathematics. Boston: Birkhäuser. ISBN 978-0-8176-4111-5. — doesn't handle abstract algebras (although he mentions them informally on p.56,59); defines particular algebras, like group (p.258) and group homomorphism (p.267); defines injective=one-to-one=monic function and surjective=onto=epic function (p.161) in general

In order to ease discussion, I suggest to upload scans of the most relevant pages locally for a short time to this page - I guess this could be allowed by the fair use rule. Or does somebody see a problem with that?

I agree to include some elementary proofs into the article; in the version of 26 Nov 2016, there were already three proofs of that kind. If they are re-inserted, they should get a more explicit header such that it becomes obvious what they are about without having to read the corresponding section in the article. (I guess, due to the latter problem, D.Lazard removed them on 1 Dec.) - Jochen Burghardt (talk) 13:14, 3 January 2017 (UTC)

It is not because I am against proofs in WP that I have removed these three proofs. It because they were confusing, as the statements that were supposed to be proved were lacking.

By the way, I have started to write a new version for the section "Types" (see User:D.Lazard/sandbox). Presently, only the end of the (collapsed) proof for monomorphism, and the subsection about epimorphisms are is sill lacking. Here are some comments about this rewriting.

• I have tried to avoid the above debate in stating that the two definitions of monomorphism are essentially equivalent. As I do not know a good reference for this equivalence, I have included a proof, collapsed, because it is too technical and not really interesting for most readers.
• For epimorphisms, I do not know a condition implying the equivalence, nor the details of how to write the subsection.
• I have encountered a problem of terminology: firstly it is not clear, what exactly means "algebraic structure", a specific instance, as the additive group of the integers or the whole class, as "structure of group". In general, and in Algebraic structure, an algebraic structure is a triple consisting of a set, the operations and the axioms satisfied by these operations. But it seems that there is no commonly accepted term for the class of algebraic structures sharing the same operations and the same axioms. Bourbaki uses "species of algebraic structure", but it seems that this terminology is generally ignored. In universal algebra, one use the term of variety, but the term is too restricted, as fields do not form a variety. Moreover most people interested in algebra do not know the terminology of universal algebra. Thus, for not being too technical, some caution is needed for using the terminology of universal algebra in this article. Because of this, I am not fully satisfied by some of my sentences, and some help would be welcome.

I intend to replace section "Types" by my new version. Before that, it would be better to have a consensus that this replacement will improve the article. D.Lazard (talk) 15:32, 3 January 2017 (UTC)

New section "Special homomorphisms"

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)

Figure

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)

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)

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)

References

Epimorphisms of groups

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)

I have found a reference, and I'll edit the article accordingly. D.Lazard (talk) 13:14, 7 May 2018 (UTC)

All algebraic structures are homomorphic

Should we include the following in the article?

Let $(R,+_{R},\times _{R})$ and $(S,+_{S},\times _{S})$ be rings and let $\phi :R\to S$ be the trivial homomorphism, i.e. the homomorphism mapping every element of $R$ to the additive identity $0_{S}$ of $S$. This is indeed a homomorphism; we have for all $r,r'\in R$, $\phi (r+_{R}r')=\phi (r)+_{S}\phi (r')$ and $\phi (r\times _{R}r')=\phi (r)\times _{S}\phi (r')$ since both sides of both equations evaluate to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 0_S} . Indeed this works not only with rings or groups, but with any algebraic structure; you can add as many operations as you want, binary or otherwise, and the argument still works. Therefore all groups are homomorphic, as are all rings, and indeed all algebraic structures. This is why we have a notation $\cong$ for isomorphy but no notation for homomorphy. Joel Brennan (talk) 20:01, 5 July 2018 (UTC)

Being homomorphic is not a relevant concept, as, for most structures, there are many homomorphisms between any two objects. Your example is not a ring homomorphism, as $\phi (1_{R})\neq 1_{S}.$ There is no ring homomorphism between two fields of different characteristic. Your example works only for structures having a zero object. Nevertheless a few words may be added to the article for saying that many algebraic structures have zero (homo)morphisms.

Thank you for correcting my example; I was not aware that preservation of the multiplicative identity was required in the definition of ring homomorphism, and as it turns out neither was the lecturer for my rings course at univeristy, whom I have now informed. However does this not then mean that we also should alter the definition of a general homomorphism in this article; the general definition given here does not line up with the definition of ring homomorphism given in its corresponding article. Unless of course the definition given on that wikipedia page is really of a non-zero homomorphism, which should be clarified over there. Joel Brennan (talk) 20:36, 7 July 2018 (UTC)

The definition that is given here must not be changed. However, it is useful to clarify that is applies also to 0-ary operations. I have added a paragraph in the article for this clarification. D.Lazard (talk) 07:26, 8 July 2018 (UTC)

Thank you that is a perfect explanation. Joel Brennan (talk) 11:07, 8 July 2018 (UTC)

Section on the history and application of the homomorphism concept?

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)

Operation version?

@Jochen Burghardt: What is operation version? The sentence is:

Formally, a map $f:A\to B$ preserves an operation $\mu$ of arity k, defined on both $A$ and $B$ if

$f(\mu _{A}(a_{1},\ldots ,a_{k}))=\mu _{B}(f(a_{1}),\ldots ,f(a_{k})),$

This sentence says that "preserves ««an»» operation". «an» means "a single" or "untyped" or "unversioned". If we should use versions, the sentence should be modified and make it

preserves operations $\mu _{A}and\mu _{B}$ (versions of $\mu$) of arity k,

There is something wrong here: we should either change the sentence, or change the formula, or somehow clarify the meaning of subscripts in one or two sentences. Hooman Mallahzadeh (talk) 16:47, 31 May 2022 (UTC)

The fact that A and B have the same type, means that there is a one to one correspondence between their operations, and that the corresponding operations have the same arity and satisfy the same axioms. In most cases, the corresponding operations have the same name, but this is not required. An example is the exponential function, which is an homomorphism from the additive group of the complex numbers to the multiplicative group of the nonzero complex numbers. So, in the definition, one must distinguish between an operation on A, and the corresponding operation on B. The easiest way for that is to index the name of the operation with the name of the structure.

I agree that presently, the formulation is confusing, but introducing the concept of "operation version" does not solve anything. Instead, the beginning of the section shoud be rewritten in the line of what precedes. D.Lazard (talk) 17:16, 31 May 2022 (UTC)