I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do you agree with the following definitions of the homomorphism subtypes? If so, is there a trick to memorise them?

Let $V_1, V_2$ be vector spaces over a common field. We consider a function $f : V_1 \rightarrow V_2$. Now, $f$ is a homomorphism iff $f$ is linear (linear-algebra-linear, not calculus-linear).

epimorphism = homomorphism + surjective

monomorphism = homomorphism + injective

isomorphism = epimorphism + monomorphism

endomorphism = homomorphism + (domain = codomain)

automorphism = endomorphism + isomorphism

The article Algebra homomorphism enumerates (in its first sentence) homogeneity and additivity but also a third property. The third property seems to be missing in my definition (definition based on the book).

By the way, should I use the term map instead of function?

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    $\begingroup$ Regarding the conflict with Wikipedia, the third property is only relevant to algebras (a special type of vector space which has a way of multiplying two vectors to form a new vector). A homomorphism of algebras needs to respect the behaviour of this multiplication operation, but a homomorphism of vector spaces doesn't need to care, as a generic vector space doesn't possess a way to multiply vectors. $\endgroup$ – Erick Wong Aug 26 '14 at 15:38
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    $\begingroup$ It's more usual to call a homomorphism between vector spaces a 'linear transformation' or just a 'linear map'. If there is more structure being considered, such as an algebra structure, then 'homomorphism' is more usual. $\endgroup$ – Dan Rust Aug 26 '14 at 15:41
  • $\begingroup$ ... monomorphism mnemonic : for every (out/in)put, there is only one (mono-) corresponding (in/out)put $\endgroup$ – DracoMalfoy Aug 26 '14 at 15:57
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    $\begingroup$ @DanielRust But a map between groups is still typically called a 'homomorphism', isn't it? I'm not sure that the distinction is correlated to the amount of structure. $\endgroup$ – Erick Wong Aug 27 '14 at 4:40
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    $\begingroup$ @ErickWong Yes, if there is no vector space structure at all then all bets are off. My comment only applies in that case. For some reason history has singled out vector spaces in particular to have their morphisms given a different name to most other algebraic structures, but in the literature it is definitely true that linear transformation/map are the most commonly used name. $\endgroup$ – Dan Rust Aug 27 '14 at 9:14

As for "how do you remember them?":

Every single one of those words is self-explanatory if you have a basic knowledge of Greek and Latin roots in English. This site is pretty handy, as would any dictionary.* (These might all be Greek: I didn't check.)

  • "homo-" meaning "same" (think about what homogenization means)
  • "epi" meaning "upon" ("epidemic"/"epicenter")
  • "mono" meaning "one" ("monologue")
  • "iso" meaning "identical/equal" ("isobar")
  • "endo" meaning "inside/inner" ("endoskeleton"))
  • "auto" meaning "self" ("automobile")

and finally "morphism" meaning "form/shape".

Every category of objects has its own special version of these things. The one thing that changes in between categories is what the morphism preserves.

A morphism is only good for that algebraic category if it preserves the basic features of the objects. So for example in the category of groups, a homomorphism only has to preserve the product. For rings, it has to preserve both operations. For algebras, it has to preserve both operations and it has to be linear with respect to the field. For vector spaces it has to preserve addition and scaling.

$^\ast$ Actually, I took a college course on Greek and Latin roots in English, and I would recommend it to any student: it's really useful.

Addendum: As for "function" versus "map": they are usually interchangeable. Some specific books or disciplines might find different uses for them, but in most contexts they mean the same thing.

  • $\begingroup$ Many thanks! I am still thinking about "iso-/identical" vs. "homo-/same" ... maybe for "iso" we could say "equals" (or "equivalent") instead of "identical" ... $\endgroup$ – DracoMalfoy Aug 26 '14 at 18:51
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    $\begingroup$ @DracoMalfoy Sure, I added that in for you. $\endgroup$ – rschwieb Aug 26 '14 at 19:24

You found a different definition of homomorphism because you were looking at the wiki page on Algebras. If you want an explicit definition of "homomorphisms between vector spaces", you should be looking at the definition of module homorphisms.

The main difference between modules and algebras is that in algebras (unlike in modules, and in particular, vector spaces) you are able to multiply two elements. Notice, however, that if we take the definition of homomorphisms and disregard the multiplication of two elements (since you can't multiply vectors), then this just gives you linearity.

The definitions you have written are certainly correct in the context of linear algebra. However, all of these terms have slightly more general definitions that used in algebraic topics such as category theory.


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