Given two sets with some structure imposed on it - for instance, a relation, or an operation, or an algebra - a homomorphism is a mapping from one to the other that preserves that structure.

E.g., the natural numbers can be mapped to the even natural numbers by the mapping x -> 2x; this is a homomorphism with respect to the usual ordering of numbers.

x -> 2^x is a homomorphism to map addition onto multiplication.

Isomorphisms are the most common.

Prologue

A homomorphism is a function that demonstrates a similarity between two abstract structures (like groups, rings, fields, or modules) and shows that the two are equivalent in most of the important ways. It is used in abstract algebra for classification purposes. There are many properties which are invariant under homomorphism, and for these properties the following argument is useful:

The "Argument from Homomorphism"

1. X is a structure which has a magic property. We don't know whether or not Y (another structure) also has this magic property.
2. We prove X is homomorphic to Y
3. The magic property is invariant under homomorphism.
4. Therefore, Y also has the magic property.

This is typically much easier than proving that Y has a property 'from scratch.'

Note: I so made the name "Argument from Homomorphism" up. The argument is valid, however.

Backstory

Homomorphism comes from the fake-greek portmanteau 'ομομορφωσις (homomorphôsis), which means 'similar-shape.' It invariably sends immature boys into fits of giggles.

The simplest sort of homomorphism is that which occurs between groups. Because they were so useful for simplifying the work done in group theory, the concept was extended to cover all sorts of things. In category theory, many types of homomorphisms are represented by arrows (but not all arrows are homomorphisms! As with many things categorical, "Best not to think about it!").

Group Homomorphisms

Groups are amazingly popular things in abstract algebra. Basically, you take a set of things (which could be numbers, permutations, intervals, rotations in the plane, or — well, anything) along with a binary operation that plays nice with the set of things you're working with. By convention, no matter what the operation really is (it could be addition, multiplication, function composition, conjugation, convolution, or any number of things), it is represented by multiplication. The notation for a group is an ordered pair, the first part being the set and the second part being the operation.

So a homomorphism for groups is a function which forms a correspondence between the elements of two groups. Let's represent the two groups with G: (G, •) and H: (H, ×). That way we can keep the operation in G and the operation in H straight. In maths terms, the correspondence is written like this:

φ(a • b) = φ(a) × φ(b)

Read: 'multiplication' in G acts like 'multiplication' in H.

φ is a function that takes things in G and turns them into things in H. If you were to draw a picture of a homomorphism, it would look like this:

   G                  H
+------+          +------+
| a    |          |      |
|  >-c-------------->d   |
| b    |          |      |
+------+          +------+

      is the same as

+------+          +------+
| a---------------->e    |
|      |          |  >-d |
| b---------------->f    |
+------+          +------+

φ must also have one other property to be a group homomorphism. It must be surjective (which, like, see) so that the conversion from G to H is complete. Technically, though, homomorphisms only need the first property.

Uses for our new toy

Homomorphisms are really the tip of the iceberg for things in abstract algebra. You can use them to make quotient groups, (though you don't need to); finding the fibers of a group, along with the kernel; to reach the isomorphism theorems (isomorphism is a stronger type of homomorphism); and other sorts of fun things like that.

But all of that stuff is rather abstract. The best reason to use homomorphisms is laziness. As I mentioned earlier, it is typically much easier to generate homomorphisms and use the "argument from homomorphism" and show the property holds for all the groups that respond to your homomorphism.

                  /
                 /
                /
               /     All the structures
 X -> P(X) -> φ      homomorphic to X
               \     
                \
                 \
                  \

Metanarrative

• If you're interested in pseudo-practical application, check out isomorphism and the isomorphism theorems, the heart and soul of abstract algebra.
• For breeding groups together to make more groups, look at quotient groups, along with kernel and normal subgroup. These lead up to the first isomorphism theorem, anyways.
• surjective, injective, and bijective — the Holy Trinity of function classifications.
• epimorphism and monomorphism, which are isomorphism's parents.
• ring homomorphism, for bigger and better things.
• lattice, if you're in this for the pretty pictures.
• And finally, abstract algebra, for the view outside.

Ho`mo*mor"phism (?), n. [See Homomorphous.]

1. Biol.

Same as Homomorphy.

2. Bot.

The possession, in one species of plants, of only one kind of flowers; -- opposed to heteromorphism, dimorphism, and trimorphism.

3. Zool.

The possession of but one kind of larvae or young, as in most insects.

<-- 4. (Math) A special type of mapping of one mathematical set into or onto another set . . . -->

 

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