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 which has an inverse as an antihomomorphism; this coincides with it being a bijection from an object to itself.
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[edit] Definition
Informally, an antihomomorphism is map that switches the order of multiplication.
Formally, an antihomomorphism between X and Y is a homomorphism 
, where Yop equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as 
and the multiplication on Yop as * , we have 
. The object Yop is called the opposite object to Y. (Respectively, opposite group, opposite algebra, etc.)
This definition is equivalent to a homomorphism 
(reversing the operation before or after applying the map is equivalent). Formally, sending X to Xop and acting as the identity on maps is a functor (indeed, an involution).
[edit] Examples
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. Another important example is the transpose operation in linear algebra which takes row vectors to column vectors. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed.
In ring theory, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So φ : X → Y is a ring antihomomorphism if and only if:
- φ(1) = 1
- 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. If the underlying field has an involution, one can instead ask φ to be conjugate-linear, as in conjugate transpose, below.
[edit] Involutions
It is frequently the case that antiautomorphisms are involutions, i.e. the square of the antiautomorphism is the identity map; these are also called involutive antiautomorphisms.
- The map that sends x to its inverse x−1 is an involutive antiautomorphism in any group.
A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples.
[edit] Properties
If the target Y is commutative, then an antihomomorphism is the same thing as a homomorphism and an antiautomorphism is the same thing as an automorphism.
The composition of two antihomomorphisms is always a homomorphism, since reversing the order twice preserves order. The composition of an antihomomorphism with an automorphism gives another antiautomorphism.
