Differences between homomorphisms and functions

Question

Incredibly, I haven't been able to find a good entry online comparing these two mathematical structures. Their relative familiarity doesn't make their boundaries less fuzzy. Here is a coarse attempt:

From Wikipedia,

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

A homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).

Based solely on these definitions it would seem that:

A function is injective (?) - this sounds incorrect. Yet, from general use, it would seem that both functions and homomorphisms can be injective or surjective, bijective, or neither.
Functions seem more limited in their algebraic structure, while the maps in homomorphisms can be defined more broadly.
Homomorphisms seem to encompass relations between a broader range of algebraic structures.
The property of homomorphisms $\varphi(a\circ b)=\varphi(a)\circ\varphi(b)$ is not necessarily a characteristic of functions.

@Harry Alli No, this is not what I meant. In the Wikipedia definition, it reads, "with the property that each input is related to exactly one output." This would seem to indicate that if $f(a)=f(b) \implies a=b,$ but that doesn't seem to be the case: there are injective and non-injective functions... — Antoni Parellada, Feb 02 '18 at 06:08
That does not indicate that. You are interpreting it as each output is related to exactly one input, the converse of the Wikipedian statement, which is not true — Harry Alli, Feb 02 '18 at 06:14
All homomorphisms are functions, but not all functions are homomorphisms. In particular, homomorphisms, by definition, must be functions between algebraic structures, and moreover they must satisfy the multiplicative criterion, i.e. $\phi(ab) = \phi(a)\phi(b)$. Functions in general are defined on sets, and algebraic structures are special kinds of sets. Lastly, functions in general don't have to be "multiplicative". This is to say, homomorphisms are functions with those two extra stipulations. — Kaj Hansen, Feb 02 '18 at 06:45

score 5 · Accepted Answer · answered Feb 02 '18 at 06:12

Ok this is a tough topic (for me) so if I quote/mention/state something wrongly please comment and (if I can) I will edit my answer.

Functions are a relation (trying to be a bit informal) sending everything in domain uniquely to something in range. Except the bolded words, there are no other hidden criteria.

Homomorphism is something trickier. Firstly, we will not say "it is a homomorphism" without any context. It will not even make sense to say so. For example, we may say "$\varphi$ is a group homomorphism" (and later on in courses we take, we will omit the word to save us from troubles writing the word).

Homomorphism is something defined to "preserve" the structure of that thing (object) that we are describing. For example, in group, we requires $f(a+b)=f(a)+f(b)$ since we want the addition structure to be "sent" by the homomorphism. Note that in fact we also need $f(0)=0$ as an axiom of group homomorphism, but we got luck that $f(a+b)=f(a)+f(b)$ automatically implies $f(0)=0$ so we are good.

For vector space, homomorphism needs to respect addition $f(a+b)=f(a)+f(b)$ and multiplication by field elements $f(\lambda a)=\lambda f(a)$. Also, we need $f(0)=0$ and $f(1\cdot a)=1\cdot f(a)$ but they are both automatic.

For fields, homomorphism needs to respect addition $f(a+b)=f(a)+f(b)$, multiplication $f(ab)=f(a)f(b)$, $f(0)=0$ (automatic) and $f(1)=1$.

I cannot list examples forever (I mean there are still many objects in mathematics) but we can summarise that homomorphism is something that preserve certain structure for the category of objects.

I deliberately use the italicised words since it is tightly related to a subject called "Category theory". It is certainly not advisory for beginners (like me) to read deeply in that theory since examples are not enough to support ideas presented inside. However, I guess it can answer your question (to certain extent) to satisfy our curiosity.

Your answer is excellent. It could only be improved if you added some comments re: "limitations" of a function compared to Hom, as in one of the comments to the OP. — Antoni Parellada, Feb 02 '18 at 13:31
@mathkoan yeah tbh I am new to the language of category and I find it is kind of fascinating to conclude things about structure into axioms. Indeed we can construct Hom-s which is not necessarily the homomorphism, eg we can define Hom on rings with group homomorphism (that "forgetting" about ring structure). But by mentioning homomorphism my mind pops up Hom and the structure-preserving properties etc so I wrote the answer. Thank you for your comment. — ElfHog, Feb 02 '18 at 22:55

score 1 · Answer 2 · answered Feb 02 '18 at 06:07

You've interpreted "each input is related to exactly one output" as "for each output, there is exactly one input which is related to that output", which is wrong. It is intended to be interpreted as "for each input, there is exactly one output which is related to that input".
It's the other way round. A homomorphism is a function which satisfies some additional constraints (such as "$f$ takes the identity to the identity", if you're in a group). But see the next point.
Well, sort of. Most people take algebraic structures to be "sets with some additional structure". In that case, every homomorphism is a function of the underlying sets, such that the function satisfies some extra conditions.
Correct. Indeed, a general function $\varphi$ on a set $X$ need not have any available notion of $\cdot$ for the notation $a \cdot b$ to make sense. It's only in an algebraic context that the $\cdot$ becomes defined.

Differences between homomorphisms and functions

2 Answers2