Definition:Antihomomorphism

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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.


Group Antihomomorphism

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


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

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


Ring Antihomomorphism

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


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

\(\, \displaystyle \forall a, b \in R: \, \) \(\displaystyle \map \phi {a + b}\) \(=\) \(\displaystyle \map \phi a \oplus \map \phi b\) $\quad$ $\quad$
\(\, \displaystyle \forall a, b \in R: \, \) \(\displaystyle \map \phi {a \circ b}\) \(=\) \(\displaystyle \map \phi b * \map \phi a\) $\quad$ $\quad$


Field Antihomomorphism

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.