Homomorphism

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Not to be confused with homeomorphism.

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). The word homomorphism comes from the Greek language: homoios meaning "same" and morphe meaning "shape".

1 Informal discussion
2 Formal definition
3 Types of homomorphisms
4 Kernel of a homomorphism
5 See also
6 References

[edit] Informal discussion

Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. These functions are known as homomorphisms.

For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b). For example, f(x) = 3x is one such homomorphism, since f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b). Note that this homomorphism maps the natural numbers back into themselves.

Homomorphisms do not have to map between sets which have the same operations. For example, operation-preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication. A function which preserves operation should have this property: f(a + b) = f(a) * f(b), since addition is the operation in the first set and multiplication is the operation in the second. Given the laws of exponents, f(x) = e^x satisfies this condition : 2 + 3 = 5 translates into e² * e³ = e⁵.

A particularly important property of homomorphisms is that if an identity element is present, it is always preserved, that is, mapped to the identity. Note in the first example f(0) = 0, and 0 is the additive identity. In the second example, f(0) = 1, since 0 is the additive identity, and 1 is the multiplicative identity.

If we are considering multiple operations on a set, then all operations must be preserved for a function to be considered as a homomorphism. Even though the set may be the same, the same function might be a homomorphism, say, in group theory (sets with a single operation) but not in ring theory (sets with two related operations), because it fails to preserve the additional operation that ring theory considers.

[edit] Formal definition

A homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure; i.e. properties like identity elements, inverse elements, and binary operations.

N.B. Some authors use the word homomorphism in a larger context than that of algebra. Some take it to mean any kind of structure preserving map (such as continuous maps in topology), or even a more abstract kind of map—what we term a morphism—used in category theory. This article only treats the algebraic context. For more general usage see the morphism article.

For example; if one considers two sets $X$ and $Y$ with a single binary operation defined on them (an algebraic structure known as a magma), a homomorphism is a map $\phi: X \rightarrow Y$ such that

$\phi(u \cdot v) = \phi(u) \circ \phi(v)$

where $\cdot$ is the operation on $X$ and $\circ$ is the operation on $Y$ .

Each type of algebraic structure has its own type of homomorphism. For specific definitions see:

group homomorphism
ring homomorphism
module homomorphism
linear operator (a homomorphism on vector spaces)
algebra homomorphism

The notion of a homomorphism can be given a formal definition in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this setting, a homomorphism $\phi: A \rightarrow B$ is a map between two algebraic structures of the same type such that

$\phi(f_A(x_1, \ldots, x_n)) = f_B(\phi(x_1), \ldots, \phi(x_n))\,$

for each n-ary operation $f$ and for all $x i$ in $A$ .

[edit] Types of homomorphisms

An isomorphism is a bijective homomorphism. Two objects are said to be isomorphic if there is an isomorphism between them. Isomorphic objects are completely indistinguishable as far as the structure in question is concerned.

An epimorphism is a surjective homomorphism.

A monomorphism is an injective homomorphism.

A homomorphism from an object to itself is called an endomorphism.

An endomorphism which is also an isomorphism is called an automorphism.

The above terms are used in an analogous fashion in category theory, however, the definitions in category theory are more subtle; see the article on morphism for more details.

Note that in the larger context of structure preserving maps, it is generally insufficient to define an isomorphism as a bijective morphism. One must also require that the inverse is a morphism of the same type. In the algebraic setting (at least within the context of universal algebra) this extra condition is automatically satisfied.

Relationships between different kinds of homomorphisms.
H = set of Homomorphisms, M = set of Monomorphisms,
P = set of ePimorphisms, S = set of iSomorphisms,
N = set of eNdomorphisms, A = set of Automorphisms.
Notice that: M ∩ P = S, S ∩ N = A, P ∩ N = A,
M ∩ N \ A contains only infinite homomorphisms, and
P ∩ N \ A is empty.

[edit] Kernel of a homomorphism

Main article: Kernel (algebra)

Any homomorphism f : X → Y defines an equivalence relation ~ on X by a ~ b iff f(a) = f(b). The relation ~ is called the kernel of f. It is a congruence relation on X. The quotient set X/~ can then be given an object-structure in a natural way, i.e. [x] * [y] = [x * y]. In that case the image of X in Y under the homomorphism f is necessarily isomorphic to X/~; this fact is one of the isomorphism theorems. Note in some cases (e.g. groups or rings), a single equivalence class K suffices to specify the structure of the quotient; so we can write it X/K. (X/K is usually read as "X mod K".) Also in these cases, it is K, rather than ~, that is called the kernel of f (cf. normal subgroup, ideal).