Antihomomorphism
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In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism that is a bijection from an object to itself.
In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism,
- φ(xy) = φ(y)φ(x)
for all x,y in X.
The map that sends x to x-1 is an example of a group antiautomorphism.
In ring theory, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So if φ : X → Y is a ring homomorphism,
- φ(x+y) = φ(x)+φ(y)
- φ(xy) = φ(y)φ(x)
for all x,y in X. For algebras over a field K, φ must be a K-linear map of the underlying vector space.
The operation of matrix transpose is an example of a ring antiautomorphism.
Note that if the multiplication in the range of φ is commutative, then an antihomomorphism is the same thing as a homomorphism and an antiautomorphism is the same thing as an automorphism.
One can also define an antihomomorphism as an homomorphism from X to the opposite object Yop (which is identical to X but with multiplication reversed).
The composition of two antihomomorphisms is always an homomorphism, since reversing the order twice preserves order. The composition of an antihomomorphism with an automorphism gives another antiautomorphism.
It is frequently the case that antiautomorphisms are involutions, i.e. the square of the antiautomorphism is the identity map.
Examples
- The map that sends x to its inverse x−1 is an antiautomorphism in any group.
- The transpose map (or conjugate transpose map) is an antiautomorphism on the algebra of square matrices.
- The Hermitian adjoint is an antiautomorphism on the algebra of linear operators on a Hilbert space.
- More generally, the *-involution in any star-algebra is an antiautomorphism.
- The conjugation involution in any Cayley-Dickson algebra such as the quaternions and octonions.