Definition:Isomorphism (Abstract Algebra)

From ProofWiki
Jump to navigation Jump to search

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 $\left({S, \circ}\right)$ is isomorphic to another algebraic structure $\left({T, *}\right)$ iff there exists an isomorphism from $\left({S, \circ}\right)$ to $\left({T, *}\right)$, and we can write $S \cong T$ (although notation may vary).


Semigroup Isomorphism

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups.

Let $\phi: S \to T$ be a (semigroup) homomorphism.


Then $\phi$ is a semigroup isomorphism iff $\phi$ is a bijection.


Monoid Isomorphism

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ 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 $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ 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 $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures.

Let $\phi: S \to T$ be an $R$-algebraic structure homomorphism.


Then $\phi$ is an $R$-algebraic structure isomorphism iff $\phi$ is a bijection.


Field Isomorphism

Let $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ 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.


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