Definition:Isomorphism (Abstract Algebra)
This page is about Isomorphism in the context of Abstract Algebra. For other uses, see Isomorphism.
Definition
An isomorphism is a homomorphism which is a bijection.
That is, it is a mapping which is both a monomorphism and an epimorphism.
An algebraic structure $\struct {S, \circ}$ is isomorphic to another algebraic structure $\struct {T, *}$ if and only if there exists an isomorphism from $\struct {S, \circ}$ to $\struct {T, *}$, and we can write $S \cong T$ (although notation may vary).
Semigroup Isomorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups.
Let $\phi: S \to T$ be a (semigroup) homomorphism.
Then $\phi$ is a semigroup isomorphism if and only if $\phi$ is a bijection.
Monoid Isomorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be monoids.
Let $\phi: S \to T$ be a (monoid) homomorphism.
Then $\phi$ is a monoid isomorphism if and only if $\phi$ is a bijection.
Group Isomorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: G \to H$ be a (group) homomorphism.
Then $\phi$ is a group isomorphism if and only if $\phi$ is a bijection.
Ring Isomorphism
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.
Let $\phi: R \to S$ be a (ring) homomorphism.
Then $\phi$ is a ring isomorphism if and only if $\phi$ is a bijection.
$F$-Isomorphism
Let $R, S$ be rings with unity.
Let $F$ be a subfield of both $R$ and $S$.
Let $\phi: R \to S$ be an $F$-homomorphism such that $\phi$ is bijective.
Then $\phi$ is an $F$-isomorphism.
The relationship between $R$ and $S$ is denoted $R \cong_F S$.
Field Isomorphism
Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields.
Let $\phi: F \to K$ be a (field) homomorphism.
Then $\phi$ is a field isomorphism if and only if $\phi$ is a bijection.
R-Algebraic Structure Isomorphism
Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures.
Let $\phi: S \to T$ be an $R$-algebraic structure homomorphism.
Then $\phi$ is an $R$-algebraic structure isomorphism if and only if $\phi$ is a bijection.
Ordered Structure Isomorphism
An ordered structure isomorphism from an ordered structure $\struct {S, \circ, \preceq}$ to another $\struct {T, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both:
- $(1): \quad$ An isomorphism, that is a bijective homomorphism, from the structure $\struct {S, \circ}$ to the structure $\struct {T, *}$
- $(2): \quad$ An order isomorphism from the ordered set $\struct {S, \preceq}$ to the ordered set $\struct {T, \preccurlyeq}$.
Isomorphic Copy
Let $\phi: S \to T$ be an isomorphism.
Let $x \in S$.
Then $\map \phi x \in T$ is known as the isomorphic copy of $x$ (under $\phi$).
Also see
Linguistic Note
The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.
Thus isomorphism means equal structure.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras