Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism

Definition

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.

That is, $\phi$ is a semigroup isomorphism if and only if $\phi$ is both a monomorphism and an epimorphism.

If $S$ is isomorphic to $T$, then the notation $S \cong T$ can be used (although notation varies).

Examples

Structure with Two Operations

Let $\struct {S_1, \circ_1, *_1}$ and $\struct {S_2, \circ_2, *_2}$ be algebraic structures such that:

$\struct {S_1, \circ_1}$ is isomorphic to $\struct {S_2, \circ_2}$

$\struct {S_1, *_1}$ is isomorphic to $\struct {S_2, *_2}$

Then it is not necessarily the case that $\struct {S_1, \circ_1, *_1}$ is isomorphic to $\struct {S_2, \circ_2, *_2}$.

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

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$: Definition $2.1$