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 \phi\colon X \to Y^{\mbox{op}}, where Yop equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as \cdot and the multiplication on Yop as * , we have x*y := y\cdot x. The object Yop is called the opposite object to Y. (Respectively, opposite group, opposite algebra, etc.)

This definition is equivalent to a homomorphism \phi\colon X^{\mbox{op}} \to Y (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.

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.

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