In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication.
An antiautomorphism is an invertible antihomomorphism, i.e. an antiisomorphism, from a set to itself.
Informally, an antihomomorphism is a map that switches the order of multiplication.
(reversing the operation before or after applying the map is equivalent).
and acting as the identity on maps is a functor (indeed, an involution).
The map that sends x to x−1 is an example of a group antiautomorphism.
Another important example is the transpose operation in linear algebra, which takes row vectors to column vectors.
With matrices, an example of an antiautomorphism is given by the transpose map.
Since inversion and transposing both give antiautomorphisms, their composition is an automorphism.
[1] For algebras over a field K, φ must be a K-linear map of the underlying vector space.
If the underlying field has an involution, one can instead ask φ to be conjugate-linear, as in conjugate transpose, below.
For example, in any group the map that sends x to its inverse x−1 is an involutive antiautomorphism.
A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples.
If the source X or the target Y is commutative, then an antihomomorphism is the same thing as a homomorphism.