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 :


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

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