Definition:Homomorphism (Abstract Algebra)
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Definition
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a mapping from one algebraic structure $\left({S, \circ}\right)$ to another $\left({T, *}\right)$.
If $\circ$ has the morphism property under $\phi$, that is:
- $\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$
then $\phi$ is a homomorphism.
This can be generalised to algebraic structures with more than one operation:
Let:
- $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$
- $\left({T, *_1, *_2, \ldots, *_n}\right)$
Let $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({T, *_1, *_2, \ldots, *_n}\right)$ be a mapping from $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$ to $\left({T, *_1, *_2, \ldots, *_n}\right)$.
If, $\forall k \in \left[{1 \,.\,.\, n}\right]$, $\circ_k$ has the morphism property under $\phi$, that is:
- $\forall x, y \in S: \phi \left({x \circ_k y}\right) = \phi \left({x}\right) *_k \phi \left({y}\right)$
then $\phi$ is a homomorphism.
Semigroup Homomorphism
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups.
Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$.
That is, $\forall a, b \in S$:
- $\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$
Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a semigroup homomorphism.
Group Homomorphism
Let $\left({G, \circ}\right)$ and $\left({H, *}\right)$ be groups.
Let $\phi: G \to H$ be a mapping such that $\circ$ has the morphism property under $\phi$.
That is, $\forall a, b \in R$:
- $\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$
Then $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ is a group homomorphism.
Ring Homomorphism
Let $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ be rings.
Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$.
That is, $\forall a, b \in R$:
| \((1):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \phi \left({a + b}\right)\) | \(=\) | \(\displaystyle \phi \left({a}\right) \oplus \phi \left({b}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | ||
| \((2):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \phi \left({a \circ b}\right)\) | \(=\) | \(\displaystyle \phi \left({a}\right) * \phi \left({b}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Then $\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$ is a ring homomorphism.
Field Homomorphism
Let $\left({F, +, \times}\right)$ and $\left({K, \oplus, \otimes}\right)$ be fields.
Let $\phi: F \to K$ be a mapping such that both $+$ and $\times$ have the morphism property under $\phi$.
That is, $\forall a, b \in F$:
| \((1):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \phi \left({a + b}\right)\) | \(=\) | \(\displaystyle \phi \left({a}\right) \oplus \phi \left({b}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | ||
| \((2):\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \phi \left({a \times b}\right)\) | \(=\) | \(\displaystyle \phi \left({a}\right) \otimes \phi \left({b}\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Then $\phi: \left({F, +, \times}\right) \to \left({K, \oplus, \otimes}\right)$ is a field homomorphism.
F-Homomorphism
Let $R, S$ be rings with unity.
Let $F$ be a subfield of both $R$ and $S$.
Then a ring homomorphism $\varphi: R \to S$ is called an $F$-homomorphism if:
- $\forall a \in F: \varphi \left({a}\right) = a$.
That is, $\varphi \restriction_F = I_F$ where:
- $\varphi \restriction_F$ is the restriction of $\varphi$ to $F$
- $I_F$ is the identity mapping on $F$.
R-Algebraic Structure Homomorphism
Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures.
Then $\phi$ is an $R$-algebraic structure homomorphism iff:
- $(1): \quad \forall k: k \in \left[{1 .. n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$
- $(2): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$
Note that this definition also applies to modules and vector spaces.
G-Module Homomorphism
Let $\left({G, \cdot}\right)$ be a group.
Let $\left({V, \phi}\right)$ and $\left({W, \mu}\right)$ be $G$-modules.
Then a linear mapping $f: V \to W$ is called a $G$-module homomorphism iff:
- $\forall g \in G,\ \forall v \in V: f \left({\phi \left({g, v}\right)}\right) = \mu \left({g, f\left({v}\right)}\right)$
Image
As a homomorphism is a mapping, and therefore a relation, we define the image of a homomorphism in the same way:
- $\operatorname{Im} \left({\phi}\right) = \left\{{t \in T: \exists s \in S: t = \phi \left({s}\right)}\right\}$
Homomorphism as Cartesian Product
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a mapping from one algebraic structure $\left({S, \circ}\right)$ to another $\left({T, *}\right)$.
We define the cartesian product $\phi \times \phi: S \times S \to T \times T$ as:
- $\forall \left({x, y}\right) \in S \times S: \left({\phi \times \phi}\right) \left({x, y}\right) = \left({\phi \left({x}\right), \phi \left({y}\right)}\right)$
Hence we can state that $\phi$ is a homomorphism iff:
- $\ast \left({\left({\phi \times \phi}\right) \left({x, y}\right)}\right) = \phi \left({\circ \left({x, y}\right)}\right)$
using the notation $\circ \left({x, y}\right)$ to denote the operation $x \circ y$.
The point of doing this is so we can illustrate what is going on in a commutative diagram:
Thus we see that $\phi$ is a homomorphism iff both of the composite mappings from $S \times S$ to $T$ have the same effect on all elements of $S \times S$.
Also see
- Epimorphism: a surjective homomorphism
- Monomorphism: an injective homomorphism
- Isomorphism: a bijective homomorphism
- Endomorphism: a homomorphism from an algebraic structure to itself
- Automorphism: an isomorphism from an algebraic structure to itself.
Linguistic Note
The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.
Thus homomorphism means similar structure.
Sources
- Iain T. Adamson: Introduction to Field Theory (1964): $\S 1.3$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 12$
