Definition:Antihomomorphism

Definition

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from one algebraic structure $\struct {S, \circ}$ to another $\struct {T, *}$.

Then $\phi$ is an antihomomorphism if and only if:

$\forall x, y \in S: \map \phi {x \circ y} = \map \phi y * \map \phi x$

For structures with more than one operation, $\phi$ may be antihomomorphic for a subset of those operations.

Let $\struct {S, \circ}$ and $\struct {T, *}$ be groups.

Then $\phi: S \to T$ is a group antihomomorphism if and only if:

$\forall x, y \in S:\map \phi {x \circ y} = \map \phi y * \map \phi x$

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.

Then $\phi: R \to S$ is a ring antihomomorphism if and only if:

$\ds \forall a, b \in R: \, $

$\ds \map \phi {a + b}$

$=$

$\ds \map \phi a \oplus \map \phi b$

$\ds \forall a, b \in R: \, $

$\ds \map \phi {a \circ b}$

$=$

$\ds \map \phi b * \map \phi a$

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be fields.

Then a ring antihomomorphism $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is called a field antihomomorphism.