Definition:Antihomomorphism

Definition

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

Then $\phi$ is an antihomomorphism iff:

$\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({y}\right) * \phi \left({x}\right)$

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

Group Antihomomorphism

If both $\left({S, \circ}\right)$ and $\left({T, *}\right)$ are groups, then an antihomomorphism $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is called a group antihomomorphism.

Ring Antihomomorphism

Let $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ be rings.

Then $\phi: R \to S$ is a ring antihomomorphism iff:

$\forall a, b \in R: \phi \left({a + b}\right) = \phi \left({a}\right) \oplus \phi \left({b}\right)$

$\forall a, b \in R: \phi \left({a \circ b}\right) = \phi \left({b}\right) * \phi \left({a}\right)$

Field Homomorphism

If both $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ are fields, then a ring antihomomorphism $\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$ is called a field antihomomorphism.